34th International Symposium on Computational Geometry (SoCG 2018), SoCG 2018, June 11-14, 2018, Budapest, Hungary
SoCG 2018
June 11-14, 2018
Budapest, Hungary
Symposium on Computational Geometry
SoCG
http://www.computational-geometry.org/
https://dblp.org/db/conf/compgeom
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Bettina
Speckmann
Bettina Speckmann
Csaba D.
Tóth
Csaba D. Tóth
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
99
2018
978-3-95977-066-8
https://www.dagstuhl.de/dagpub/978-3-95977-066-8
Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks
Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks
Front Matter
Table of Contents
Foreword
Conference Organization
Additional Reviewers
Acknowledgement of Support
Invited Talks
0:i-0:xi
Front Matter
Bettina
Speckmann
Bettina Speckmann
Csaba D.
Tóth
Csaba D. Tóth
10.4230/LIPIcs.SoCG.2018.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm
We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from alpha-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an epsilon-sample on the bounding surface, with a weak sigma-sparsity condition, we work with the balls of radius delta times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of epsilon, sigma and delta, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples.
sampling conditions
surface reconstruction
polyhedral meshing
Voronoi
1:1-1:16
Regular Paper
Ahmed
Abdelkader
Ahmed Abdelkader
Chandrajit L.
Bajaj
Chandrajit L. Bajaj
Mohamed S.
Ebeida
Mohamed S. Ebeida
Ahmed H.
Mahmoud
Ahmed H. Mahmoud
Scott A.
Mitchell
Scott A. Mitchell
John D.
Owens
John D. Owens
Ahmad A.
Rushdi
Ahmad A. Rushdi
10.4230/LIPIcs.SoCG.2018.1
A. Abdelkader, C. Bajaj, M. Ebeida, A. Mahmoud, S. Mitchell, J. Owens, and A. Rushdi. Sampling conditions for conforming Voronoi meshing by the VoroCrust algorithm. CoRR, arXiv:1803.06078, 2018. URL: http://arxiv.org/abs/1803.06078.
http://arxiv.org/abs/1803.06078
A. Abdelkader, C. Bajaj, M. Ebeida, A. Mahmoud, S. Mitchell, J. Owens, and A. Rushdi. VoroCrust Illustrated: Theory and Challenges (Multimedia Contribution). In 34th International Symposium on Computational Geometry (SoCG 2018), pages 77:1-77:4, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.77.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.77
A. Abdelkader, C. Bajaj, M. Ebeida, A. Mahmoud, S. Mitchell, J. Owens, and A. Rushdi. VoroCrust: Voronoi meshing without clipping. Manuscript, In preparation.
A. Abdelkader, C. Bajaj, M. Ebeida, and S. Mitchell. A Seed Placement Strategy for Conforming Voronoi Meshing. In Canadian Conference on Computational Geometry, 2017.
A. Abdelkader, A. Mahmoud, A. Rushdi, S. Mitchell, J. Owens, and M. Ebeida. A constrained resampling strategy for mesh improvement. Computer Graphics Forum, 36(5):189-201, 2017.
O. Aichholzer, F. Aurenhammer, B. Kornberger, S. Plantinga, G. Rote, A. Sturm, and G. Vegter. Recovering structure from r-sampled objects. Computer Graphics Forum, 28(5):1349-1360, 2009.
N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering. Discrete &Computational Geometry, 22(4):481-504, dec 1999.
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N. Amenta, S. Choi, and R.-K. Kolluri. The power crust, unions of balls, and the medial axis transform. Computational Geometry, 19(2):127-153, 2001.
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H. Edelsbrunner. The union of balls and its dual shape. Discrete & Computational Geometry, 13(3):415-440, Jun 1995.
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G. Manzini, A. Russo, and N. Sukumar. New perspectives on polygonal and polyhedral finite element methods. Mathematical Models and Methods in Applied Sciences, 24(08):1665-1699, 2014.
R. Merland, G. Caumon, B. Lévy, and P. Collon-Drouaillet. Voronoi grids conforming to 3D structural features. Computational Geosciences, 18(3):373-383, 2014.
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C. Rycroft. Voro++: A three-dimensional Voronoi cell library in C++. Chaos, 19(4):-, 2009. Software available online at URL: http://math.lbl.gov/voro++/.
http://math.lbl.gov/voro++/
M. Sents and C. Gable. Coupling LaGrit Unstructured Mesh Generation and Model Setup with TOUGH2 Flow and Transport. Comput. Geosci., 108(C):42-49, 2017.
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Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs
We consider a well studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and an integer d >= 1, in the maximum diameter-bounded subgraph problem (MaxDBS for short), the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For d=1, this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor n^{1-epsilon}, for any epsilon > 0. Moreover, it is known that, for any d >= 2, it is NP-hard to approximate MaxDBS within a factor n^{1/2 - epsilon}, for any epsilon > 0.
In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems and several geometric properties of unit disk graphs.
Approximation algorithms
maximum diameter-bounded subgraph
unit disk graphs
fractional Helly theorem
VC-dimension
2:1-2:12
Regular Paper
A. Karim
Abu-Affash
A. Karim Abu-Affash
Paz
Carmi
Paz Carmi
Anil
Maheshwari
Anil Maheshwari
Pat
Morin
Pat Morin
Michiel
Smid
Michiel Smid
Shakhar
Smorodinsky
Shakhar Smorodinsky
10.4230/LIPIcs.SoCG.2018.2
E. Ackerman. On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom., 41:365-375, 2009.
P. K. Agarwal, B. Aronov, J. Pach, and M. Sharir. Quasi-planar graphs have linear number of edges. Combinatorica, 17:1-9, 1997.
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Vietoris-Rips and Cech Complexes of Metric Gluings
We study Vietoris-Rips and Cech complexes of metric wedge sums and metric gluings. We show that the Vietoris-Rips (resp. Cech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris-Rips (resp. Cech) complexes. We also provide generalizations for certain metric gluings, i.e. when two metric spaces are glued together along a common isometric subset. As our main example, we deduce the homotopy type of the Vietoris-Rips complex of two metric graphs glued together along a sufficiently short path. As a result, we can describe the persistent homology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs.
Vietoris-Rips and Cech complexes
metric space gluings and wedge sums
metric graphs
persistent homology
3:1-3:15
Regular Paper
Michal
Adamaszek
Michal Adamaszek
Henry
Adams
Henry Adams
Ellen
Gasparovic
Ellen Gasparovic
Maria
Gommel
Maria Gommel
Emilie
Purvine
Emilie Purvine
Radmila
Sazdanovic
Radmila Sazdanovic
Bei
Wang
Bei Wang
Yusu
Wang
Yusu Wang
Lori
Ziegelmeier
Lori Ziegelmeier
10.4230/LIPIcs.SoCG.2018.3
Mridul Aanjaneya, Frederic Chazal, Daniel Chen, Marc Glisse, Leonidas Guibas, and Dmitry Morozon. Metric graph reconstruction from noisy data. International Journal of Computational Geometry and Applications, 22(04):305-325, 2012.
Michał Adamaszek. Clique complexes and graph powers. Israel Journal of Mathematics, 196(1):295-319, 2013.
Michał Adamaszek and Henry Adams. The Vietoris-Rips complexes of a circle. Pacific Journal of Mathematics, 290:1-40, 2017.
Michał Adamaszek, Henry Adams, Florian Frick, Chris Peterson, and Corrine Previte-Johnson. Nerve complexes of circular arcs. Discrete & Computational Geometry, 56(2):251-273, 2016.
Michał Adamaszek, Henry Adams, Ellen Gasparovic, Marix Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier. Vietoris-Rips and Čech complexes of metric gluings. arXiv:1712.06224, 2018.
Eric Babson and Dmitry N. Kozlov. Complexes of graph homomorphisms. Israel Journal of Mathematics, 152(1):285-312, 2006.
Jonathan Ariel Barmak and Elias Gabriel Minian. Simple homotopy types and finite spaces. Advances in Mathematics, 218:87-104, 2008.
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Michael Lesnick, Raul Rabadan, and Daniel Rosenbloom. Quantifying genetic innovation: Mathematical foundations for the topological study of reticulate evolution. In preparation, 2018.
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Steve Oudot and Elchana Solomon. Barcode embeddings, persistence distortion, and inverse problems for metric graphs. arXiv: 1712.03630, 2017.
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Improved Dynamic Geodesic Nearest Neighbor Searching in a Simple Polygon
We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set S of point sites in a static simple polygon P. Our data structure allows us to insert a new site in S, delete a site from S, and ask for the site in S closest to an arbitrary query point q in P. All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in P. Our data structure achieves polylogarithmic update and query times, and uses O(n log^3n log m + m) space, where n is the number of sites in S and m is the number of vertices in P. The crucial ingredient in our data structure is an implicit representation of a vertical shallow cutting of the geodesic distance functions. We show that such an implicit representation exists, and that we can compute it efficiently.
data structure
simple polygon
geodesic distance
nearest neighbor searching
shallow cutting
4:1-4:14
Regular Paper
Pankaj K.
Agarwal
Pankaj K. Agarwal
Lars
Arge
Lars Arge
Frank
Staals
Frank Staals
10.4230/LIPIcs.SoCG.2018.4
Pankaj K. Agarwal and Jiří Matoušek. Dynamic Half-Space Range Reporting and its Applications. Algorithmica, 13(4):325-345, 1995.
Pankaj Agarwal K., Lars Arge, and Frank Staals. Improved dynamic geodesic nearest neighbor searching in a simple polygon. CoRR, abs/1803.05765, 2018. URL: http://arxiv.org/abs/1803.05765.
http://arxiv.org/abs/1803.05765
Lars Arge and Frank Staals. Dynamic geodesic nearest neighbor searching in a simple polygon. CoRR, abs/1707.02961, 2017. URL: http://arxiv.org/abs/1707.02961.
http://arxiv.org/abs/1707.02961
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Sariel Har-Peled. Geometric Approximation Algorithms, volume 173. American mathematical society Boston, 2011.
Sariel Har-Peled and Micha Sharir. Relative (p,ε)-approximations in Geometry. Discrete & Computational Geometry, 45(3):462-496, Apr 2011.
John Hershberger. A new data structure for shortest path queries in a simple polygon. Information Processing Letters, 38(5):231-235, 1991.
John Hershberger and Subhash Suri. An Optimal Algorithm for Euclidean Shortest Paths in the Plane. SIAM Journal on Computing, 28(6):2215-2256, 1999.
Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications. In Proc. 28th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2017.
Rolf Klein and Andrzej Lingas. Hamiltonian abstract Voronoi diagrams in linear time, pages 11-19. Springer Berlin Heidelberg, Berlin, Heidelberg, 1994.
Der-Tsai Lee. On k-nearest neighbor voronoi diagrams in the plane. IEEE Transactions on Computers, C-31(6):478-487, June 1982.
Chih-Hung Liu and D. T. Lee. Higher-order geodesic voronoi diagrams in a polygonal domain with holes. In Proc. 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1633-1645, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.117.
http://dx.doi.org/10.1137/1.9781611973105.117
Jiří Matoušek. Reporting points in halfspaces. Computational Geometry Theory and Applications, 2(3):169-186, 1992.
Eunjin Oh and Hee-Kap Ahn. Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon. In Proc. 33rd International Symposium on Computational Geometry, volume 77 of Leibniz International Proceedings in Informatics, pages 52:1-52:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.
Evanthia Papadopoulou and Der-Tsai Lee. A New Approach for the Geodesic Voronoi Diagram of Points in a Simple Polygon and Other Restricted Polygonal Domains. Algorithmica, 20(4):319-352, 1998.
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O~(n^{1/3})-Space Algorithm for the Grid Graph Reachability Problem
The directed graph reachability problem takes as input an n-vertex directed graph G=(V,E), and two distinguished vertices s and t. The problem is to determine whether there exists a path from s to t in G. This is a canonical complete problem for class NL. Asano et al. proposed an O~(sqrt{n}) space and polynomial time algorithm for the directed grid and planar graph reachability problem. The main result of this paper is to show that the directed graph reachability problem restricted to grid graphs can be solved in polynomial time using only O~(n^{1/3}) space.
graph reachability
grid graph
graph algorithm
sublinear space algorithm
5:1-5:13
Regular Paper
Ryo
Ashida
Ryo Ashida
Kotaro
Nakagawa
Kotaro Nakagawa
10.4230/LIPIcs.SoCG.2018.5
Eric Allender, David A Mix Barrington, Tanmoy Chakraborty, Samir Datta, and Sambuddha Roy. Planar and grid graph reachability problems. Theory of Computing Systems, 45(4):675-723, 2009.
Tetsuo Asano and Benjamin Doerr. Memory-constrained algorithms for shortest path problem. In CCCG, 2011.
Tetsuo Asano, David Kirkpatrick, Kotaro Nakagawa, and Osamu Watanabe. Õ(√n)-space and polynomial-time algorithm for planar directed graph reachability. In International Symposium on Mathematical Foundations of Computer Science, pages 45-56. Springer, 2014.
Greg Barnes, Jonathan F Buss, Walter L Ruzzo, and Baruch Schieber. A sublinear space, polynomial time algorithm for directed st connectivity. SIAM Journal on Computing, 27(5):1273-1282, 1998.
Chris Bourke, Raghunath Tewari, and NV Vinodchandran. Directed planar reachability is in unambiguous log-space. ACM Transactions on Computation Theory (TOCT), 1(1):4, 2009.
Gerard Jennhwa Chang. Algorithmic aspects of domination in graphs. Handbook of Combinatorial Optimization, pages 221-282, 2013.
Tatsuya Imai, Kotaro Nakagawa, Aduri Pavan, NV Vinodchandran, and Osamu Watanabe. An O(n^1/2+ε)-space and polynomial-time algorithm for directed planar reachability. In Computational Complexity (CCC), 2013 IEEE Conference on, pages 277-286. IEEE, 2013.
Omer Reingold. Undirected connectivity in log-space. Journal of the ACM (JACM), 55(4):17, 2008.
Derrick Stolee and NV Vinodchandran. Space-efficient algorithms for reachability in surface-embedded graphs. In Computational Complexity (CCC), 2012 IEEE 27th Annual Conference on, pages 326-333. IEEE, 2012.
Leslie G Valiant. Universality considerations in vlsi circuits. IEEE Transactions on Computers, 100(2):135-140, 1981.
Avi Wigderson. The complexity of graph connectivity. Mathematical Foundations of Computer Science 1992, pages 112-132, 1992.
Creative Commons Attribution 3.0 Unported license
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The Reverse Kakeya Problem
We prove a generalization of Pál's 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360° inside Q. We also prove a lower bound of Omega(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.
Kakeya problem
convex
isodynamic point
turning
6:1-6:13
Regular Paper
Sang Won
Bae
Sang Won Bae
Sergio
Cabello
Sergio Cabello
Otfried
Cheong
Otfried Cheong
Yoonsung
Choi
Yoonsung Choi
Fabian
Stehn
Fabian Stehn
Sang Duk
Yoon
Sang Duk Yoon
10.4230/LIPIcs.SoCG.2018.6
P. K. Agarwal, N. Amenta, and M. Sharir. Largest placement of one convex polygon inside another. Discrete & Computational Geometry, 19:95-104, 1998. URL: http://dx.doi.org/10.1007/PL00009337.
http://dx.doi.org/10.1007/PL00009337
A. S. Besicovitch. Sur deux questions de l'intégrabilité. Journal de la Société des Math. et de Phys., II, 1920.
A. S. Besicovitch. On Kakeya’s problem and a similar one. Math. Zeitschrift, 27:312-320, 1928.
J. Bourgain. Harmonic analysis and combinatorics: How much may they contribute to each other? In V. I. Arnold, M. Atiyah, P. Lax, and B. Mazur, editors, Mathematics: Frontiers and Perspectives, pages 13-32. American Math. Society, 2000.
A. DePano, Yan Ke, and J. O’Rourke. Finding largest inscribed equilateral triangles and squares. In Proc. 25th Allerton Conf. Commun. Control Comput., 1987.
S. Kakeya. Some problems on maxima and minima regarding ovals. The Science Report of the Tohoku Imperial University, Series 1, Mathematics, Physics, Chemistry, 6:71-88, 1917.
I. Laba. From harmonic analysis to arithmetic combinatorics. Bulletin (New Series) of the AMS, 45:77-115, 2008.
G. Pál. Ein Minimumproblem für Ovale. Math. Ann., 83:311-319, 1921.
T. Tao. From rotating needles to stability of waves: Emerging connections between combinatorics, analysis and PDE. Notices of the AMS, 48:297-303, 2001.
T. Wolff. Recent work connected with the Kakeya problem. In H. Rossi, editor, Prospects in Mathematics. American Math. Society, 1999.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Capacitated Covering Problems in Geometric Spaces
In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B' subseteq B of balls and assign each point in P to some ball in B' that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B'.
We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for this problem. In fact, in the Euclidean setting, only (1+epsilon) factor expansion is sufficient for any epsilon > 0, with the approximation factor being a polynomial in 1/epsilon. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems.
Capacitated covering
Geometric set cover
LP rounding
Bi-criteria approximation
7:1-7:15
Regular Paper
Sayan
Bandyapadhyay
Sayan Bandyapadhyay
Santanu
Bhowmick
Santanu Bhowmick
Tanmay
Inamdar
Tanmay Inamdar
Kasturi
Varadarajan
Kasturi Varadarajan
10.4230/LIPIcs.SoCG.2018.7
Anshul Aggarwal, Venkatesan T. Chakaravarthy, Neelima Gupta, Yogish Sabharwal, Sachin Sharma, and Sonika Thakral. Replica placement on bounded treewidth graphs. In Algorithms and Data Structures - 15th International Symposium, WADS 2017, St. John’s, NL, Canada, July 31 - August 2, 2017, Proceedings, pages 13-24, 2017. URL: http://dx.doi.org/10.1007/978-3-319-62127-2_2.
http://dx.doi.org/10.1007/978-3-319-62127-2_2
Hyung-Chan An, Aditya Bhaskara, Chandra Chekuri, Shalmoli Gupta, Vivek Madan, and Ola Svensson. Centrality of trees for capacitated k-center. Math. Program., 154(1-2):29-53, 2015. URL: http://dx.doi.org/10.1007/s10107-014-0857-y.
http://dx.doi.org/10.1007/s10107-014-0857-y
Hyung-Chan An, Mohit Singh, and Ola Svensson. Lp-based algorithms for capacitated facility location. In FOCS, pages 256-265, 2014.
Boris Aronov, Esther Ezra, and Micha Sharir. Small-size ε-nets for axis-parallel rectangles and boxes. SIAM J. Comput., 39(7):3248-3282, 2010. URL: http://dx.doi.org/10.1137/090762968.
http://dx.doi.org/10.1137/090762968
Judit Bar-Ilan, Guy Kortsarz, and David Peleg. How to allocate network centers. J. Algorithms, 15(3):385-415, 1993. URL: http://dblp.uni-trier.de/db/journals/jal/jal15.html#Bar-IlanKP93.
http://dblp.uni-trier.de/db/journals/jal/jal15.html#Bar-IlanKP93
Amariah Becker. Capacitated dominating set on planar graphs. In Approximation and Online Algorithms - 15th International Workshop, WAOA 2017, Vienna, Austria, September 7-8, 2017.
Piotr Berman, Marek Karpinski, and Andrzej Lingas. Exact and approximation algorithms for geometric and capacitated set cover problems. Algorithmica, 64(2):295-310, 2012.
Prosenjit Bose, Paz Carmi, Mirela Damian, Robin Y. Flatland, Matthew J. Katz, and Anil Maheshwari. Switching to directional antennas with constant increase in radius and hop distance. Algorithmica, 69(2):397-409, 2014.
Hervé Brönnimann and Michael T. Goodrich. Almost optimal set covers in finite vc-dimension. Discrete & Computational Geometry, 14(4):463-479, 1995. URL: http://dx.doi.org/10.1007/BF02570718.
http://dx.doi.org/10.1007/BF02570718
Timothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1576-1585, 2012. URL: https://dl.acm.org/citation.cfm?id=2095241.
https://dl.acm.org/citation.cfm?id=2095241
Julia Chuzhoy and Joseph Naor. Covering problems with hard capacities. SIAM J. Comput., 36(2):498-515, 2006.
Kenneth L. Clarkson and Kasturi R. Varadarajan. Improved approximation algorithms for geometric set cover. Discrete & Computational Geometry, 37(1):43-58, 2007.
Marek Cygan, MohammadTaghi Hajiaghayi, and Samir Khuller. LP rounding for k-centers with non-uniform hard capacities. In FOCS, pages 273-282, 2012.
Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 624-633, 2014. URL: http://dx.doi.org/10.1145/2591796.2591884.
http://dx.doi.org/10.1145/2591796.2591884
Rajiv Gandhi, Eran Halperin, Samir Khuller, Guy Kortsarz, and Srinivasan Aravind. An improved approximation algorithm for vertex cover with hard capacities. J. Comput. Syst. Sci., 72(1):16-33, 2006.
Taha Ghasemi and Mohammadreza Razzazi. A PTAS for the cardinality constrained covering with unit balls. Theor. Comput. Sci., 527:50-60, 2014.
Sathish Govindarajan, Rajiv Raman, Saurabh Ray, and Aniket Basu Roy. Packing and covering with non-piercing regions. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 47:1-47:17, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2016.47.
http://dx.doi.org/10.4230/LIPIcs.ESA.2016.47
Sariel Har-Peled and Mira Lee. Weighted geometric set cover problems revisited. JoCG, 3(1):65-85, 2012.
Dorit S. Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM, 32(1):130-136, 1985.
Mong-Jen Kao. Iterative partial rounding for vertex cover with hard capacities. In SODA, pages 2638-2653, 2017.
Samir Khuller and Yoram J. Sussmann. The capacitated K-center problem. SIAM J. Discrete Math., 13(3):403-418, 2000.
Nissan Lev-Tov and David Peleg. Polynomial time approximation schemes for base station coverage with minimum total radii. Computer Networks, 47(4):489-501, 2005.
Retsef Levi, David B. Shmoys, and Chaitanya Swamy. Lp-based approximation algorithms for capacitated facility location. Math. Program., 131(1-2):365-379, 2012. URL: http://dx.doi.org/10.1007/s10107-010-0380-8.
http://dx.doi.org/10.1007/s10107-010-0380-8
Shi Li. On uniform capacitated k-median beyond the natural LP relaxation. In SODA, pages 696-707, 2015.
Robert Lupton, F. Miller Maley, and Neal E. Young. Data collection for the sloan digital sky survey - A network-flow heuristic. J. Algorithms, 27(2):339-356, 1998. URL: http://dx.doi.org/10.1006/jagm.1997.0922.
http://dx.doi.org/10.1006/jagm.1997.0922
Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete &Computational Geometry, 44(4):883-895, 2010. URL: http://dx.doi.org/10.1007/s00454-010-9285-9.
http://dx.doi.org/10.1007/s00454-010-9285-9
Kasturi R. Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 641-648, 2010. URL: http://dx.doi.org/10.1145/1806689.1806777.
http://dx.doi.org/10.1145/1806689.1806777
Laurence A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2(4):385-393, 1982.
Sam Chiu-wai Wong. Tight algorithms for vertex cover with hard capacities on multigraphs and hypergraphs. In SODA, pages 2626-2637, 2017.
Creative Commons Attribution 3.0 Unported license
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Faster Algorithms for some Optimization Problems on Collinear Points
We propose faster algorithms for the following three optimization problems on n collinear points, i.e., points in dimension one. The first two problems are known to be NP-hard in higher dimensions.
1) Maximizing total area of disjoint disks: In this problem the goal is to maximize the total area of nonoverlapping disks centered at the points. Acharyya, De, and Nandy (2017) presented an O(n^2)-time algorithm for this problem. We present an optimal Theta(n)-time algorithm.
2) Minimizing sum of the radii of client-server coverage: The n points are partitioned into two sets, namely clients and servers. The goal is to minimize the sum of the radii of disks centered at servers such that every client is in some disk, i.e., in the coverage range of some server. Lev-Tov and Peleg (2005) presented an O(n^3)-time algorithm for this problem. We present an O(n^2)-time algorithm, thereby improving the running time by a factor of Theta(n).
3) Minimizing total area of point-interval coverage: The n input points belong to an interval I. The goal is to find a set of disks of minimum total area, covering I, such that every disk contains at least one input point. We present an algorithm that solves this problem in O(n^2) time.
collinear points
range assignment
8:1-8:14
Regular Paper
Ahmad
Biniaz
Ahmad Biniaz
Prosenjit
Bose
Prosenjit Bose
Paz
Carmi
Paz Carmi
Anil
Maheshwari
Anil Maheshwari
Ian
Munro
Ian Munro
Michiel
Smid
Michiel Smid
10.4230/LIPIcs.SoCG.2018.8
Ankush Acharyya, Minati De, and Subhas C. Nandy. Range assignment of base-stations maximizing coverage area without interference. In Proceedings of the 29th Canadian Conference on Computational Geometry (CCCG), pages 126-131, 2017.
Ankush Acharyya, Minati De, Subhas C. Nandy, and Bodhayan Roy. Range assignment of base-stations maximizing coverage area without interference. CoRR, abs/1705.09346, 2017.
Helmut Alt, Esther M. Arkin, Hervé Brönnimann, Jeff Erickson, Sándor P. Fekete, Christian Knauer, Jonathan Lenchner, Joseph S. B. Mitchell, and Kim Whittlesey. Minimum-cost coverage of point sets by disks. In Proceedings of the 22nd ACM Symposium on Computational Geometry, (SoCG), pages 449-458, 2006.
Vittorio Bilò, Ioannis Caragiannis, Christos Kaklamanis, and Panagiotis Kanellopoulos. Geometric clustering to minimize the sum of cluster sizes. In Proceedings of the 13th European Symposium on Algorithms, (ESA), pages 460-471, 2005.
Ahmad Biniaz, Prosenjit Bose, Paz Carmi, Anil Maheshwari, Ian Munro, and Michiel Smid. Faster algorithms for some optimization problems on collinear points. CoRR, abs/1802.09505, 2018.
Paz Carmi, Matthew J. Katz, and Joseph S. B. Mitchell. The minimum-area spanning tree problem. Computational Geometry: Theory and Applications, 35(3):218-225, 2006.
Erin W. Chambers, Sándor P. Fekete, Hella-Franziska Hoffmann, Dimitri Marinakis, Joseph S. B. Mitchell, Srinivasan Venkatesh, Ulrike Stege, and Sue Whitesides. Connecting a set of circles with minimum sum of radii. Computational Geometry: Theory and Applications, 68:62-76, 2018.
David Eppstein. Maximizing the sum of radii of disjoint balls or disks. In Proceedings of the 28th Canadian Conference on Computational Geometry (CCCG), pages 260-265, 2016.
Ju Yuan Hsiao, Chuan Yi Tang, and Ruay Shiung Chang. An efficient algorithm for finding a maximum weight 2-independent set on interval graphs. Information Processing Letters, 43(5):229-235, 1992.
Nissan Lev-Tov and David Peleg. Polynomial time approximation schemes for base station coverage with minimum total radii. Computer Networks, 47(4):489-501, 2005.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Local Criteria for Triangulation of Manifolds
We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.
manifold
simplicial complex
homeomorphism
triangulation
9:1-9:14
Regular Paper
Jean-Daniel
Boissonnat
Jean-Daniel Boissonnat
Ramsay
Dyer
Ramsay Dyer
Arijit
Ghosh
Arijit Ghosh
Mathijs
Wintraecken
Mathijs Wintraecken
10.4230/LIPIcs.SoCG.2018.9
N. Amenta and M. Bern. Surface reconstruction by Voronoi filtering. Discrete and Computational Geometry, 22(4):481-504, 1999.
N. Amenta, S. Choi, T. K. Dey, and N. Leekha. A simple algorithm for homeomorphic surface reconstruction. Int. J. Computational Geometry and Applications, 12(2):125-141, 2002.
J.-D. Boissonnat, R. Dyer, and A. Ghosh. The stability of Delaunay triangulations. International Journal of Computational Geometry &Applications, 23(4-5):303-333, 2013. (arXiv:1304.2947).
J.-D. Boissonnat, R. Dyer, and A. Ghosh. Delaunay triangulation of manifolds. Foundations of Computational Mathematics, 2017. (arXiv:1311.0117).
J.-D. Boissonnat, R. Dyer, A. Ghosh, and M. Wintraecken. Local criteria for triangulation of manifolds. Technical Report 1803.07642, arXiv, 2017. URL: http://arxiv.org/abs/1803.07642.
http://arxiv.org/abs/1803.07642
J.-D. Boissonnat and A. Ghosh. Manifold reconstruction using tangential Delaunay complexes. Discrete and Computational Geometry, 51(1):221-267, 2014.
J.-D. Boissonnat, A. Lieutier, and M. Wintraecken. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. Technical Report hal-01661227, Inria, Sophia-Antipolis, 2017. Accepted for SoCG 2018. URL: https://hal.inria.fr/hal-01661227.
https://hal.inria.fr/hal-01661227
J.-D. Boissonnat and S. Oudot. Provably good sampling and meshing of surfaces. Graphical Models, 67(5):405-451, 2005.
S. S. Cairns. On the triangulation of regular loci. Annals of Mathematics. Second Series, 35(3):579-587, 1934.
S.-W. Cheng, T. K. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In SODA, pages 1018-1027, 2005.
R. Dyer, G. Vegter, and M. Wintraecken. Riemannian simplices and triangulations. Geometriae Dedicata, 179:91-138, 2015.
R. Dyer, H. Zhang, and T. Möller. Surface sampling and the intrinsic Voronoi diagram. Computer Graphics Forum (Special Issue of Symp. Geometry Processing), 27(5):1393-1402, 2008.
H. Edelsbrunner and N. R. Shah. Triangulating topological spaces. Int. J. Comput. Geometry Appl., 7(4):365-378, 1997.
J. R. Munkres. Elementary differential topology. Princton University press, second edition, 1968.
J. H. C. Whitehead. On C¹-complexes. Ann. of Math, 41(4):809-824, 1940.
H. Whitney. Geometric Integration Theory. Princeton University Press, 1957.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces
In this paper we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the distance between the points and the reach.
Reach
Metric distortion
Manifolds
Convexity
10:1-10:14
Regular Paper
Jean-Daniel
Boissonnat
Jean-Daniel Boissonnat
André
Lieutier
André Lieutier
Mathijs
Wintraecken
Mathijs Wintraecken
10.4230/LIPIcs.SoCG.2018.10
N. Amenta and M. W. Bern. Surface reconstruction by Voronoi filtering. In SoCG, pages 39-48, 1998. URL: http://dx.doi.org/10.1145/276884.276889.
http://dx.doi.org/10.1145/276884.276889
D. Attali, H. Edelsbrunner, and Yu. Mileyko. Weak Witnesses for Delaunay triangulations of Submanifold. In ACM Symposium on Solid and Physical Modeling, pages 143-150, Beijing, China, 2007. URL: https://hal.archives-ouvertes.fr/hal-00201055.
https://hal.archives-ouvertes.fr/hal-00201055
D. Attali and A. Lieutier. Geometry-driven collapses for converting a Čech complex into a triangulation of a nicely triangulable shape. Discrete &Computational Geometry, 54(4):798-825, 2015.
M. Belkin, J. Sun, and Y. Wang. Constructing laplace operator from point clouds in ℝ^d. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1031-1040, 2009. URL: http://dx.doi.org/10.1137/1.9781611973068.112.
http://dx.doi.org/10.1137/1.9781611973068.112
J.-D. Boissonnat and F. Cazals. Natural neighbor coordinates of points on a surface. Computational Geometry Theory &Applications, 19(2-3):155-173, Jul 2001. URL: http://www.sciencedirect.com/science/article/pii/S0925772101000189.
http://www.sciencedirect.com/science/article/pii/S0925772101000189
J.-D. Boissonnat, R. Dyer, and A. Ghosh. Constructing intrinsic Delaunay triangulations of submanifolds. Research Report RR-8273, INRIA, 2013. arXiv:1303.6493. URL: http://hal.inria.fr/hal-00804878.
http://hal.inria.fr/hal-00804878
J.-D. Boissonnat, R. Dyer, A. Ghosh, and M.H.M.J. Wintraecken. Local criteria for triangulation of manifolds. Accepted for SoCG 2018, 2018. URL: https://hal.inria.fr/hal-01661230.
https://hal.inria.fr/hal-01661230
J.-D. Boissonnat and A. Ghosh. Triangulating smooth submanifolds with light scaffolding. Mathematics in Computer Science, 4(4):431-461, 2010.
J.-D. Boissonnat and S. Oudot. Provably good surface sampling and approximation. In Symp. Geometry Processing, pages 9-18, 2003.
Jean-Daniel Boissonnat, André Lieutier, and Mathijs Wintraecken. The reach, metric distortion, geodesic convexity and the variation of tangent spaces. full version, 2017. URL: https://hal.inria.fr/hal-01661227.
https://hal.inria.fr/hal-01661227
S.-W. Cheng, T. K. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In SODA, pages 1018-1027, 2005.
T. K. Dey. Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics). Cambridge University Press, New York, NY, USA, 2006.
T.K. Dey, J. Giesen, E.A. Ramos, and B. Sadri. Critical points of distance to an ε-sampling of a surface and flow-complex-based surface reconstruction. International Journal of Computational Geometry &Applications, 18(01n02):29-61, 2008. URL: http://dx.doi.org/10.1142/S0218195908002532.
http://dx.doi.org/10.1142/S0218195908002532
M. P. do Carmo. Riemannian Geometry. Birkhäuser, 1992.
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https://creativecommons.org/licenses/by/3.0/legalcode
Orthogonal Terrain Guarding is NP-complete
A terrain is an x-monotone polygonal curve, i.e., successive vertices have increasing x-coordinates. Terrain Guarding can be seen as a special case of the famous art gallery problem where one has to place at most k guards on a terrain made of n vertices in order to fully see it. In 2010, King and Krohn showed that Terrain Guarding is NP-complete [SODA '10, SIAM J. Comput. '11] thereby solving a long-standing open question. They observe that their proof does not settle the complexity of Orthogonal Terrain Guarding where the terrain only consists of horizontal or vertical segments; those terrains are called rectilinear or orthogonal. Recently, Ashok et al. [SoCG'17] presented an FPT algorithm running in time k^{O(k)}n^{O(1)} for Dominating Set in the visibility graphs of rectilinear terrains without 180-degree vertices. They ask if Orthogonal Terrain Guarding is in P or NP-hard. In the same paper, they give a subexponential-time algorithm running in n^{O(sqrt n)} (actually even n^{O(sqrt k)}) for the general Terrain Guarding and notice that the hardness proof of King and Krohn only disproves a running time 2^{o(n^{1/4})} under the ETH. Hence, there is a significant gap between their 2^{O(n^{1/2} log n)}-algorithm and the no 2^{o(n^{1/4})} ETH-hardness implied by King and Krohn's result.
In this paper, we answer those two remaining questions. We adapt the gadgets of King and Krohn to rectilinear terrains in order to prove that even Orthogonal Terrain Guarding is NP-complete. Then, we show how their reduction from Planar 3-SAT (as well as our adaptation for rectilinear terrains) can actually be made linear (instead of quadratic).
terrain guarding
rectilinear terrain
computational complexity
11:1-11:15
Regular Paper
Édouard
Bonnet
Édouard Bonnet
Panos
Giannopoulos
Panos Giannopoulos
10.4230/LIPIcs.SoCG.2018.11
Pradeesha Ashok, Fedor V. Fomin, Sudeshna Kolay, Saket Saurabh, and Meirav Zehavi. Exact algorithms for terrain guarding. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 11:1-11:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.11.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.11
Boaz Ben-Moshe, Matthew J. Katz, and Joseph S. B. Mitchell. A constant-factor approximation algorithm for optimal 1.5d terrain guarding. SIAM J. Comput., 36(6):1631-1647, 2007. URL: http://dx.doi.org/10.1137/S0097539704446384.
http://dx.doi.org/10.1137/S0097539704446384
Édouard Bonnet and Panos Giannopoulos. Orthogonal terrain guarding is NP-complete. CoRR, abs/1710.00386, 2017. URL: http://arxiv.org/abs/1710.00386.
http://arxiv.org/abs/1710.00386
Édouard Bonnet and Tillmann Miltzow. Parameterized hardness of art gallery problems. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 19:1-19:17, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2016.19.
http://dx.doi.org/10.4230/LIPIcs.ESA.2016.19
Édouard Bonnet and Tillmann Miltzow. An approximation algorithm for the art gallery problem. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 20:1-20:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.20.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.20
Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9417-5.
http://dx.doi.org/10.1007/s00454-012-9417-5
Kenneth L. Clarkson and Kasturi R. Varadarajan. Improved approximation algorithms for geometric set cover. Discrete & Computational Geometry, 37(1):43-58, 2007. URL: http://dx.doi.org/10.1007/s00454-006-1273-8.
http://dx.doi.org/10.1007/s00454-006-1273-8
Ajay Deshpande, Taejung Kim, Erik D. Demaine, and Sanjay E. Sarma. A pseudopolynomial time O (log n )-approximation algorithm for art gallery problems. In Algorithms and Data Structures, 10th International Workshop, WADS 2007, Halifax, Canada, August 15-17, 2007, Proceedings, pages 163-174, 2007. URL: http://dx.doi.org/10.1007/978-3-540-73951-7_15.
http://dx.doi.org/10.1007/978-3-540-73951-7_15
Alon Efrat and Sariel Har-Peled. Guarding galleries and terrains. Inf. Process. Lett., 100(6):238-245, 2006. URL: http://dx.doi.org/10.1016/j.ipl.2006.05.014.
http://dx.doi.org/10.1016/j.ipl.2006.05.014
Stephan Eidenbenz. Inapproximability results for guarding polygons without holes. In Algorithms and Computation, 9th International Symposium, ISAAC '98, Taejon, Korea, December 14-16, 1998, Proceedings, pages 427-436, 1998. URL: http://dx.doi.org/10.1007/3-540-49381-6_45.
http://dx.doi.org/10.1007/3-540-49381-6_45
Stephan Eidenbenz, Christoph Stamm, and Peter Widmayer. Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1):79-113, 2001. URL: http://dx.doi.org/10.1007/s00453-001-0040-8.
http://dx.doi.org/10.1007/s00453-001-0040-8
Khaled Elbassioni. Finding small hitting sets in infinite range spaces of bounded vc-dimension. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 40:1-40:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.40.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.40
Khaled M. Elbassioni, Erik Krohn, Domagoj Matijevic, Julián Mestre, and Domagoj Severdija. Improved approximations for guarding 1.5-dimensional terrains. Algorithmica, 60(2):451-463, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9358-4.
http://dx.doi.org/10.1007/s00453-009-9358-4
Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
http://dx.doi.org/10.1006/jcss.2000.1727
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
http://dx.doi.org/10.1006/jcss.2001.1774
James King. A 4-approximation algorithm for guarding 1.5-dimensional terrains. In LATIN 2006: Theoretical Informatics, 7th Latin American Symposium, Valdivia, Chile, March 20-24, 2006, Proceedings, pages 629-640, 2006. URL: http://dx.doi.org/10.1007/11682462_58.
http://dx.doi.org/10.1007/11682462_58
James King and David G. Kirkpatrick. Improved approximation for guarding simple galleries from the perimeter. Discrete & Computational Geometry, 46(2):252-269, 2011. URL: http://dx.doi.org/10.1007/s00454-011-9352-x.
http://dx.doi.org/10.1007/s00454-011-9352-x
James King and Erik Krohn. Terrain guarding is NP-hard. SIAM J. Comput., 40(5):1316-1339, 2011. URL: http://dx.doi.org/10.1137/100791506.
http://dx.doi.org/10.1137/100791506
David G. Kirkpatrick. An o(lg lg opt)-approximation algorithm for multi-guarding galleries. Discrete & Computational Geometry, 53(2):327-343, 2015. URL: http://dx.doi.org/10.1007/s00454-014-9656-8.
http://dx.doi.org/10.1007/s00454-014-9656-8
Erik Krohn, Matt Gibson, Gaurav Kanade, and Kasturi R. Varadarajan. Guarding terrains via local search. JoCG, 5(1):168-178, 2014. URL: http://jocg.org/index.php/jocg/article/view/128.
http://jocg.org/index.php/jocg/article/view/128
Erik Krohn and Bengt J. Nilsson. Approximate guarding of monotone and rectilinear polygons. Algorithmica, 66(3):564-594, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9653-3.
http://dx.doi.org/10.1007/s00453-012-9653-3
David Lichtenstein. Planar formulae and their uses. SIAM J. Comput., 11(2):329-343, 1982. URL: http://dx.doi.org/10.1137/0211025.
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Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS, 105:41-72, 2011. URL: http://albcom.lsi.upc.edu/ojs/index.php/beatcs/article/view/96.
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Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010. URL: http://dx.doi.org/10.1007/s00454-010-9285-9.
http://dx.doi.org/10.1007/s00454-010-9285-9
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QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails.
disk graph
maximum clique
computational complexity
12:1-12:15
Regular Paper
Édouard
Bonnet
Édouard Bonnet
Panos
Giannopoulos
Panos Giannopoulos
Eun Jung
Kim
Eun Jung Kim
Pawel
Rzazewski
Pawel Rzazewski
Florian
Sikora
Florian Sikora
10.4230/LIPIcs.SoCG.2018.12
Pankaj K Agarwal, János Pach, and Micha Sharir. State of the Union (of Geometric Objects). Surveys in Discrete and Computational Geometry: Twenty Years Later. Contemporary Mathematics, 453:9-48, 2008.
Jochen Alber and Jirí Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms, 52(2):134-151, 2004. URL: http://dx.doi.org/10.1016/j.jalgor.2003.10.001.
http://dx.doi.org/10.1016/j.jalgor.2003.10.001
Noga Alon, Raphael Yuster, and Uri Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997. URL: http://dx.doi.org/10.1007/BF02523189.
http://dx.doi.org/10.1007/BF02523189
Christoph Ambühl and Uli Wagner. The Clique Problem in Intersection Graphs of Ellipses and Triangles. Theory Comput. Syst., 38(3):279-292, 2005. URL: http://dx.doi.org/10.1007/s00224-005-1141-6.
http://dx.doi.org/10.1007/s00224-005-1141-6
Aistis Atminas and Viktor Zamaraev. On forbidden induced subgraphs for unit disk graphs. arXiv preprint arXiv:1602.08148, 2016.
Jørgen Bang-Jensen, Bruce Reed, Mathias Schacht, Robert Šámal, Bjarne Toft, and Uli Wagner. On six problems posed by Jarik Nešetřil. Topics in Discrete Mathematics, pages 613-627, 2006.
Adrian Bock, Yuri Faenza, Carsten Moldenhauer, and Andres J. Ruiz-Vargas. Solving the Stable Set Problem in Terms of the Odd Cycle Packing Number. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2014, December 15-17, 2014, New Delhi, India, pages 187-198, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2014.187.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2014.187
Édouard Bonnet, Bruno Escoffier, Eun Jung Kim, and Vangelis Th. Paschos. On Subexponential and FPT-Time Inapproximability. Algorithmica, 71(3):541-565, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9889-1.
http://dx.doi.org/10.1007/s00453-014-9889-1
Édouard Bonnet, Panos Giannopoulos, Eun Jung Kim, Paweł Rzążewski, and Florian Sikora. QPTAS and subexponential algorithm for maximum clique on disk graphs. CoRR, abs/1712.05010, 2017. URL: http://arxiv.org/abs/1712.05010.
http://arxiv.org/abs/1712.05010
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http://dx.doi.org/10.1016/S0925-7721(97)00014-X
Valentin E. Brimkov, Konstanty Junosza-Szaniawski, Sean Kafer, Jan Kratochvíl, Martin Pergel, Paweł Rzążewski, Matthew Szczepankiewicz, and Joshua Terhaar. Homothetic polygons and beyond: Intersection graphs, recognition, and maximum clique. CoRR, abs/1411.2928, 2014. URL: http://arxiv.org/abs/1411.2928.
http://arxiv.org/abs/1411.2928
Sergio Cabello. Maximum clique for disks of two sizes. Open problems from Geometric Intersection Graphs: Problems and Directions CG Week Workshop, Eindhoven, June 25, 2015 (http://cgweek15.tcs.uj.edu.pl/problems.pdf), 2015. [Online; accessed 07-December-2017].
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Sergio Cabello. Open problems presented at the Algorithmic Graph Theory on the Adriatic Coast workshop, Koper, Slovenia (https://conferences.matheo.si/event/6/picture/35.pdf), June 16-19 2015.
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Sergio Cabello, Jean Cardinal, and Stefan Langerman. The clique problem in ray intersection graphs. Discrete & Computational Geometry, 50(3):771-783, 2013. URL: http://dx.doi.org/10.1007/s00454-013-9538-5.
http://dx.doi.org/10.1007/s00454-013-9538-5
Stephan Ceroi. The clique number of unit quasi-disk graphs. Technical Report RR-4419, INRIA, 2002. URL: https://hal.inria.fr/inria-00072169.
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Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178-189, 2003. URL: http://dx.doi.org/10.1016/S0196-6774(02)00294-8.
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Miroslav Chlebík and Janka Chlebíková. Complexity of approximating bounded variants of optimization problems. Theor. Comput. Sci., 354(3):320-338, 2006. URL: http://dx.doi.org/10.1016/j.tcs.2005.11.029.
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Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Mathematics, 86(1-3):165-177, 1990. URL: http://dx.doi.org/10.1016/0012-365X(90)90358-O.
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Aleksei V. Fishkin. Disk graphs: A short survey. In Klaus Jansen and Roberto Solis-Oba, editors, Approximation and Online Algorithms, First International Workshop, WAOA 2003, Budapest, Hungary, September 16-18, 2003, Revised Papers, volume 2909 of Lecture Notes in Computer Science, pages 260-264. Springer, 2003. URL: http://dx.doi.org/10.1007/978-3-540-24592-6_23.
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Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001.
Ross J. Kang and Tobias Müller. Sphere and Dot Product Representations of Graphs. Discrete & Computational Geometry, 47(3):548-568, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9394-8.
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Dániel Marx. Parameterized Complexity and Approximation Algorithms. Comput. J., 51(1):60-78, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm048.
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Vijay Raghavan and Jeremy P. Spinrad. Robust algorithms for restricted domains. J. Algorithms, 48(1):160-172, 2003. URL: http://dx.doi.org/10.1016/S0196-6774(03)00048-8.
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Erik Jan van Leeuwen. Optimization and Approximation on Systems of Geometric Objects. PhD thesis, Utrecht University, 2009.
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Computational Complexity of the Interleaving Distance
The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show that the interleaving distance is NP-hard to compute for persistence modules valued in the category of vector spaces. In the specific setting of multidimensional persistent homology we show that the problem is at least as hard as a matrix invertibility problem. Furthermore, this allows us to conclude that the interleaving distance of interval decomposable modules depends on the characteristic of the field. Persistence modules valued in the category of sets are also studied. As a corollary, we obtain that the isomorphism problem for Reeb graphs is graph isomorphism complete.
Persistent Homology
Interleavings
NP-hard
13:1-13:15
Regular Paper
Håvard Bakke
Bjerkevik
Håvard Bakke Bjerkevik
Magnus Bakke
Botnan
Magnus Bakke Botnan
10.4230/LIPIcs.SoCG.2018.13
Pankaj K Agarwal, Kyle Fox, Abhinandan Nath, Anastasios Sidiropoulos, and Yusu Wang. Computing the gromov-hausdorff distance for metric trees. In International Symposium on Algorithms and Computation, pages 529-540. Springer, 2015.
Gorô Azumaya. Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem. Nagoya Mathematical Journal, 1:117-124, 1950.
Michael Barot. Introduction to the representation theory of algebras. Springer, 2014.
Ulrich Bauer and Michael Lesnick. Induced matchings and the algebraic stability of persistence barcodes. Journal of Computational Geometry, 6(2):162-191, 2015.
Silvia Biasotti, Andrea Cerri, Patrizio Frosini, Daniela Giorgi, and Claudia Landi. Multidimensional size functions for shape comparison. Journal of Mathematical Imaging and Vision, 32(2):161-179, 2008.
Håvard Bakke Bjerkevik. Stability of higher-dimensional interval decomposable persistence modules. arXiv preprint arXiv:1609.02086, 2016.
Håvard Bakke Bjerkevik and Magnus Bakke Botnan. Computational complexity of the interleaving distance. arXiv preprint arXiv:1712.04281, 2017.
Magnus Bakke Botnan. Interval decomposition of infinite zigzag persistence modules. Proceedings of the American Mathematical Society, 145(8):3571–-3577, 2017.
Magnus Bakke Botnan and Michael Lesnick. Algebraic stability of zigzag persistence modules. arXiv preprint arXiv:1604.00655, 2016.
Peter A Brooksbank and Eugene M Luks. Testing isomorphism of modules. Journal of Algebra, 320(11):4020-4029, 2008.
Peter Bubenik, Vin de Silva, and Vidit Nanda. Higher interpolation and extension for persistence modules. SIAM Journal on Applied Algebra and Geometry, 1(1):272-284, 2017.
Peter Bubenik, Vin de Silva, and Jonathan Scott. Metrics for generalized persistence modules. Foundations of Computational Mathematics, 15(6):1501-1531, 2015.
Peter Bubenik and Jonathan A Scott. Categorification of persistent homology. Discrete &Computational Geometry, 51(3):600-627, 2014.
G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete &Computational Geometry, 42(1):71-93, 2009.
Gunnar Carlsson and Vin de Silva. Zigzag persistence. Foundations of computational mathematics, 10(4):367-405, 2010.
Gunnar Carlsson, Vin de Silva, and Dmitriy Morozov. Zigzag persistent homology and real-valued functions. In Proceedings of the twenty-fifth annual symposium on Computational geometry, pages 247-256. ACM, 2009.
Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J Guibas, and Steve Y Oudot. Proximity of persistence modules and their diagrams. In Proceedings of the twenty-fifth annual symposium on Computational geometry, pages 237-246. ACM, 2009.
Jérémy Cochoy and Steve Oudot. Decomposition of exact pfd persistence bimodules. arXiv preprint arXiv:1605.09726, 2016.
William Crawley-Boevey. Decomposition of pointwise finite-dimensional persistence modules. Journal of Algebra and Its Applications, 14(05):1550066, 2015.
Vin de Silva, Elizabeth Munch, and Amit Patel. Categorified Reeb graphs. arXiv preprint arXiv:1501.04147, 2015.
Tamal K Dey and Cheng Xin. Computing bottleneck distance for 2-d interval decomposable modules. arXiv preprint arXiv:1803.02869, 2018.
Michael Kerber, Dmitriy Morozov, and Arnur Nigmetov. Geometry helps to compare persistence diagrams. Journal of Experimental Algorithmics (JEA), 22(1):1-4, 2017.
Michael Lesnick. Multidimensional interleavings and applications to topological inference. arXiv preprint arXiv:1206.1365, 2012.
Michael Lesnick. The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics, 15(3):613-650, 2015.
Nikola Milosavljević, Dmitriy Morozov, and Primoz Skraba. Zigzag persistent homology in matrix multiplication time. In Proceedings of the twenty-seventh annual symposium on Computational geometry, pages 216-225. ACM, 2011.
Martina Scolamiero, Wojciech Chachólski, Anders Lundman, Ryan Ramanujam, and Sebastian Öberg. Multidimensional persistence and noise. Foundations of Computational Mathematics, 17(6):1367-1406, 2017.
Cary Webb. Decomposition of graded modules. Proceedings of the American Mathematical Society, 94(4):565-571, 1985.
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Sheaf-Theoretic Stratification Learning
In this paper, we investigate a sheaf-theoretic interpretation of stratification learning. Motivated by the work of Alexandroff (1937) and McCord (1978), we aim to redirect efforts in the computational topology of triangulated compact polyhedra to the much more computable realm of sheaves on partially ordered sets. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (2012), and the cohomology stratification algorithm given in Nanda (2017). We envision that our sheaf-theoretic algorithm could give rise to a larger class of stratification beyond homology-based stratification. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.
Sheaf theory
stratification learning
topological data analysis
stratification
14:1-14:14
Regular Paper
Adam
Brown
Adam Brown
Bei
Wang
Bei Wang
10.4230/LIPIcs.SoCG.2018.14
Pavel Sergeyevich Alexandroff. Diskrete Räume. Mathematiceskii Sbornik, 2:501-518, 1937.
Paul Bendich. Analyzing Stratified Spaces Using Persistent Versions of Intersection and Local Homology. PhD thesis, Duke University, 2008.
Paul Bendich, David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Dmitriy Morozov. Inferring local homology from sampled stratified spaces. IEEE Symposium on Foundations of Computer Science, pages 536-546, 2007.
Paul Bendich and John Harer. Persistent intersection homology. Foundations of Computational Mathematics, 11:305-336, 2011.
Paul Bendich, Bei Wang, and Sayan Mukherjee. Local homology transfer and stratification learning. ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1355-1370, 2012.
Adam Brown and Bei Wang. Sheaf-theoretic stratification learning. arXiv:1712.07734, 2017.
Nicolás Cianci and Miguel Ottina. A new spectral sequence for homology of posets. Topology and its Applications, 217:1-19, 2017.
Justin Curry. Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania, 2014.
Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Society, 2010.
Mark Goresky and Robert MacPherson. Intersection homology I. Topology, 19:135-162, 1982.
Mark Goresky and Robert MacPherson. Intersection homology II. Inventiones Mathematicae, 71:77-129, 1983.
Mark Goresky and Robert MacPherson. Stratified Morse Theory. Springer-Verlag, 1988.
Gloria Haro, Gregory Randall, and Guillermo Sapiro. Stratification learning: Detecting mixed density and dimensionality in high dimensional point clouds. Advances in Neural Information Processing Systems (NIPS), 17, 2005.
Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.
Frances Clare Kirwan. An introduction to intersection homology theory. Chapman &Hall/CRC, 2006.
Gilad Lerman and Teng Zhang. Probabilistic recovery of multiple subspaces in point clouds by geometric lp minimization. Annals of Statistics, 39(5):2686-2715, 2010.
Michael C. McCord. Singular homology groups and homotopy groups of finite topological spaces. Duke Mathematical Journal, 33:465-474, 1978.
Washington Mio. Homology manifolds. Annals of Mathematics Studies (AM-145), 1:323-343, 2000.
James R. Munkres. Elements of algebraic topology. Addison-Wesley, Redwood City, CA, USA, 1984.
Vidit Nanda. Local cohomology and stratification. ArXiv:1707.00354, 2017.
Colin Rourke and Brian Sanderson. Homology stratifications and intersection homology. Geometry and Topology Monographs, 2:455-472, 1999.
Allen Dudley Shepard. A Cellular Description Of The Derived Category Of A Stratified Space. PhD thesis, Brown University, 1985.
Primoz Skraba and Bei Wang. Approximating local homology from samples. ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 174-192, 2014.
René Vidal, Yi Ma, and Shankar Sastry. Generalized principal component analysis (GPCA). IEEE Transactions on Pattern Analysis and Machine Intelligence, 27:1945-1959, 2005.
Shmuel Weinberger. The topological classification of stratified spaces. University of Chicago Press, Chicago, IL, 1994.
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Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension
While persistent homology has taken strides towards becoming a widespread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.
persistent homology
multi-persistence
representation theory
quivers
commutative ladders
Vietoris-Rips filtration
15:1-15:13
Regular Paper
Mickaël
Buchet
Mickaël Buchet
Emerson G.
Escolar
Emerson G. Escolar
10.4230/LIPIcs.SoCG.2018.15
Hideto Asashiba, Emerson G. Escolar, Yasuaki Hiraoka, and Hiroshi Takeuchi. Matrix method for persistence modules on commutative ladders of finite type. arXiv preprint arXiv:1706.10027, 2017.
Ibrahim Assem, Andrzej Skowronski, and Daniel Simson. Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory, volume 65. Cambridge University Press, 2006.
Michael Barot. Introduction to the representation theory of algebras. Springer, 2014.
Gunnar Carlsson and Vin de Silva. Zigzag persistence. Foundations of computational mathematics, 10(4):367-405, 2010.
Gunnar Carlsson and Afra Zomorodian. The theory of multidimensional persistence. Discrete &Computational Geometry, 42(1):71-93, 2009.
Lorin Crawford, Anthea Monod, Andrew X. Chen, Sayan Mukherjee, and Raúl Rabadán. Topological summaries of tumor images improve prediction of disease free survival in glioblastoma multiforme. arXiv preprint arXiv:1611.06818, 2016.
Mary-Lee Dequeant, Sebastian Ahnert, Herbert Edelsbrunner, Thomas M. A. Fink, Earl F. Glynn, Gaye Hattem, Andrzej Kudlicki, Yuriy Mileyko, Jason Morton, Arcady R. Mushegian, et al. Comparison of pattern detection methods in microarray time series of the segmentation clock. PLoS One, 3(8):e2856, 2008.
Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. Discrete Comput Geom, 28:511-533, 2002.
Emerson G. Escolar and Yasuaki Hiraoka. Persistence modules on commutative ladders of finite type. Discrete &Computational Geometry, 55(1):100-157, 2016.
Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov. Clique topology reveals intrinsic geometric structure in neural correlations. Proceedings of the National Academy of Sciences, 112(44):13455-13460, 2015.
Heather A Harrington, Nina Otter, Hal Schenck, and Ulrike Tillmann. Stratifying multiparameter persistent homology. arXiv preprint arXiv:1708.07390, 2017.
Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue, and Yasumasa Nishiura. Hierarchical structures of amorphous solids characterized by persistent homology. Proceedings of the National Academy of Sciences, 113(26):7035-7040, 2016.
Lida Kanari, Paweł Dłotko, Martina Scolamiero, Ran Levi, Julian Shillcock, Kathryn Hess, and Henry Markram. A topological representation of branching neuronal morphologies. Neuroinformatics, pages 1-11, 2017.
Yongjin Lee, Senja D. Barthel, Paweł Dłotko, S. Mohamad Moosavi, Kathryn Hess, and Berend Smit. Quantifying similarity of pore-geometry in nanoporous materials. Nature Communications, 8, 2017.
Michael Lesnick and Matthew Wright. Interactive visualization of 2-d persistence modules. arXiv preprint arXiv:1512.00180, 2015.
James R. Munkres. Elements of algebraic topology, volume 2. Addison-Wesley Menlo Park, 1984.
Monica Nicolau, Arnold J. Levine, and Gunnar Carlsson. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings of the National Academy of Sciences, 108(17):7265-7270, 2011.
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Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data
Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has a discrete probability distribution describing its location. The probabilistic nature of uncertain data makes it challenging to compute such estimators, since the true value of the estimator is now described by a distribution rather than a single point. We show how to construct and estimate the distribution of the median of a point set. Building the approximate support of the distribution takes near-linear time, and assigning probability to that support takes quadratic time. We also develop a general approximation technique for distributions of robust estimators with respect to ranges with bounded VC dimension. This includes the geometric median for high dimensions and the Siegel estimator for linear regression.
Uncertain Data
Robust Estimators
Geometric Median
Tukey Median
16:1-16:14
Regular Paper
Kevin
Buchin
Kevin Buchin
Jeff M.
Phillips
Jeff M. Phillips
Pingfan
Tang
Pingfan Tang
10.4230/LIPIcs.SoCG.2018.16
Pankaj K. Agarwal, Boris Aronov, Sariel Har-Peled, Jeff M. Phillips, Ke Yi, and Wuzhou Zhang. Nearest-neighbor searching under uncertainty II. In PODS, 2013.
Pankaj K. Agarwal, Siu-Wing Cheng, Yufei Tao, and Ke Yi. Indexing uncertain data. In PODS, 2009.
Pankaj K. Agarwal, Alon Efrat, Swaminathan Sankararaman, and Wuzhou Zhang. Nearest-neighbor searching under uncertainty. In PODS, 2012.
Pankaj K. Agarwal, Sariel Har-Peled, Subhash Suri, Hakan Yildiz, and Wuzhou Zhang. Convex hulls under uncertainty. In ESA, 2014.
Greg Aloupis. Geometric measures of data depth. In Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications. AMS, 2006.
Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation schemes for Euclidean k-medians and related problems. In STOC, 1998.
Prosenjit Bose, Anil Maheshwari, and Pat Morin. Fast approximations for sums of distances clustering and the Fermet-Weber problem. CGTA, 24:135-146, 2003.
Kevin Buchin, Jeff M. Phillips, and Pingfan Tang. Approximating the distribution of the median and other robust estimators on uncertain data. ArXiv e-prints, 2018. URL: http://arxiv.org/abs/1601.00630.
http://arxiv.org/abs/1601.00630
R. Chandrasekaran and A. Tamir. Algebraic optimization: The Fermet-Weber location problem. Mathematical Programming, 46:219-224, 1990.
Graham Cormode and Andrew McGregor. Approximation algorithms for clustering uncertain data. In PODS, 2008.
David Donoho and Peter J. Huber. The notion of a breakdown point. In P. Bickel, K. Doksum, and J. Hodges, editors, A Festschrift for Erich L. Lehmann, pages 157-184. 1983.
Lingxiao Huang and Jian Li. Approximating the expected values for combinatorial optimization problems over stochastic points. In ICALP, 2015.
Allan G. Jørgensen, Maarten Löffler, and Jeff M. Phillips. Geometric computation on indecisive points. In WADS, 2011.
Jian Li, Barna Saha, and Amol Deshpande. A unified approach to ranking in probabilistic databases. In VLDB, 2009.
Yi Li, Philip M. Long, and Aravind Srinivasan. Improved bounds on the samples complexity of learning. Journal of Computer and System Science, 62:516-527, 2001.
Maarten Löffler and Jeff Phillips. Shape fitting on point sets with probability distributions. In ESA, 2009.
Hendrik P. Lopuhaa and Peter J. Rousseeuw. Breakdown points of affine equivaniant estimators of multivariate location and converiance matrices. The Annals of Statistics, 19:229-248, 1991.
Peter J. Rousseeuw. Multivariate estimation with high breakdown point. Mathematical Statistics and Applications, pages 283-297, 1985.
Andrew F. Siegel. Robust regression using repeated medians. Biometrika, 82:242-244, 1982.
J. W. Tukey. Mathematics and the picturing of data. In Proceedings of the 1974 International Congress of Mathematics, Vancouver, volume 2, pages 523-531, 1975.
Vladimir Vapnik and Alexey Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Th. Probability and Applications, 16:264-280, 1971.
Endre Weiszfeld. Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Mathematical Journal, First Series, 43:355-386, 1937.
Ying Zhang, Xuemin Lin, Yufei Tao, and Wenjie Zhang. Uncertain location based range aggregates in a multi-dimensional space. In Proceedings 25th IEEE International Conference on Data Engineering, 2009.
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Consistent Sets of Lines with no Colorful Incidence
We consider incidences among colored sets of lines in {R}^d and examine whether the existence of certain concurrences between lines of k colors force the existence of at least one concurrence between lines of k+1 colors. This question is relevant for problems in 3D reconstruction in computer vision.
Incidence geometry
image consistency
probabilistic construction
algebraic construction
projective configuration
17:1-17:14
Regular Paper
Boris
Bukh
Boris Bukh
Xavier
Goaoc
Xavier Goaoc
Alfredo
Hubard
Alfredo Hubard
Matthew
Trager
Matthew Trager
10.4230/LIPIcs.SoCG.2018.17
Kalle Åström, Roberto Cipolla, and Peter J Giblin. Generalised epipolar constraints. In European Conference on Computer Vision, pages 95-108. Springer, 1996.
Guillaume Batog, Xavier Goaoc, and Jean Ponce. Admissible linear map models of linear cameras. In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, pages 1578-1585. IEEE, 2010.
Edmond Boyer. On using silhouettes for camera calibration. Computer Vision-ACCV 2006, pages 1-10, 2006.
Boris Bukh, Xavier Goaoc, Alfredo Hubard, and Matthew Trager. Consistent sets of lines with no colorful incidence. Preprint arXiv:1803.06267, 2018.
Paul Erdős and George Purdy. Some extremal problems in geometry. Discrete Math, pages 305-315, 1974.
Olivier Faugeras and Bernard Mourrain. On the geometry and algebra of the point and line correspondences between n images. In Computer Vision, 1995. Proceedings., Fifth International Conference on, pages 951-956. IEEE, 1995.
Jacob Fox, János Pach, Adam Sheffer, Andrew Suk, and Joshua Zahl. A semi-algebraic version of Zarankiewicz’s problem. Journal of the European Mathematical Society, 19:1785-1810, 2017.
Ben Green and Terence Tao. On sets defining few ordinary lines. Discrete &Computational Geometry, 50(2):409-468, 2013.
Harald Gropp. Configurations and their realization. Discrete Mathematics, 174(1-3):137-151, 1997.
Larry Guth. Ruled surface theory and incidence geometry. In A Journey Through Discrete Mathematics, pages 449-466. Springer, 2017.
Larry Guth and Nets Hawk Katz. Algebraic methods in discrete analogs of the Kakeya problem. Advances in Mathematics, 225(5):2828-2839, 2010.
Richard Hartley and Andrew Zisserman. Multiple view geometry in computer vision. Cambridge university press, 2003.
Carlos Hernández, Francis Schmitt, and Roberto Cipolla. Silhouette coherence for camera calibration under circular motion. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(2), 2007.
Onur Özyeşil, Vladislav Voroninski, Ronen Basri, and Amit Singer. A survey of structure from motion. Acta Numerica, 26:305-364, 2017.
József Solymosi and Miloš Stojaković. Many collinear k-tuples with no k+1 collinear points. Discrete &Computational Geometry, 50(3):811-820, 2013.
Matthew Trager, Martial Hebert, and Jean Ponce. Consistency of silhouettes and their duals. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3346-3354, 2016.
Matthew Trager, Bernd Sturmfels, John Canny, Martial Hebert, and Jean Ponce. General models for rational cameras and the case of two-slit projections. In CVPR 2017 - IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, United States, 2017. URL: https://hal.archives-ouvertes.fr/hal-01506996.
https://hal.archives-ouvertes.fr/hal-01506996
Oswald Veblen and John Wesley Young. Projective geometry, volume 1. Ginn, 1918.
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The HOMFLY-PT Polynomial is Fixed-Parameter Tractable
Many polynomial invariants of knots and links, including the Jones and HOMFLY-PT polynomials, are widely used in practice but #P-hard to compute. It was shown by Makowsky in 2001 that computing the Jones polynomial is fixed-parameter tractable in the treewidth of the link diagram, but the parameterised complexity of the more powerful HOMFLY-PT polynomial remained an open problem. Here we show that computing HOMFLY-PT is fixed-parameter tractable in the treewidth, and we give the first sub-exponential time algorithm to compute it for arbitrary links.
Knot theory
knot invariants
parameterised complexity
18:1-18:14
Regular Paper
Benjamin A.
Burton
Benjamin A. Burton
10.4230/LIPIcs.SoCG.2018.18
Benjamin A. Burton. Introducing Regina, the 3-manifold topology software. Experiment. Math., 13(3):267-272, 2004.
Benjamin A. Burton, Ryan Budney, William Pettersson, et al. Regina: Software for low-dimensional topology. http://regina-normal.github.io/, 1999-2018.
Federico Comoglio and Maurizio Rinaldi. A topological framework for the computation of the HOMFLY polynomial and its application to proteins. PLoS ONE, 6(4):e18693, 2011.
Bruno Courcelle. On context-free sets of graphs and their monadic second-order theory. In Graph-Grammars and their Application to Computer Science (Warrenton, VA, 1986), volume 291 of Lecture Notes in Comput. Sci., pages 133-146. Springer, Berlin, 1987.
Bruno Courcelle. Graph rewriting: An algebraic and logic approach. In Handbook of Theoretical Computer Science, Vol. B, pages 193-242. Elsevier, Amsterdam, 1990.
Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, Cham, 2015.
P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu. A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. (N.S.), 12(2):239-246, 1985.
F. Jaeger, D. L. Vertigan, and D. J. A. Welsh. On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc., 108(1):35-53, 1990.
Robert J. Jenkins, Jr. A Dynamic Programming Approach to Calculating the HOMFLY Polynomial for Directed Knots and Links. Master’s thesis, 1989.
Vaughan F. R. Jones. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. (N.S.), 12(1):103-111, 1985.
Louis H. Kauffman. State models for link polynomials. Enseign. Math. (2), 36(1-2):1-37, 1990.
Ton Kloks. Treewidth: Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1994.
W. B. Raymond Lickorish. An Introduction to Knot Theory. Number 175 in Graduate Texts in Mathematics. Springer, New York, 1997.
Richard J. Lipton and Robert Endre Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math., 36(2):177-189, 1979.
J. A. Makowsky. Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width. Discrete Appl. Math., 145(2):276-290, 2005.
J. A. Makowsky and J. P. Mariño. The parameterized complexity of knot polynomials. J. Comput. Syst. Sci., 67(4):742-756, 2003.
H. R. Morton and H. B. Short. Calculating the 2-variable polynomial for knots presented as closed braids. J. Algorithms, 11(1):117-131, 1990.
Masahiko Murakami, Fumio Takeshita, and Seiichi Tani. Computing HOMFLY polynomials of 2-bridge links from 4-plat representation. Discrete Appl. Math., 162:271-284, 2014.
Józef H. Przytycki. The first coefficient of Homflypt and Kauffman polynomials: Vertigan proof of polynomial complexity using dynamic programming. Preprint, arXiv:\allowbreak 1707.07733, 2017.
Józef H. Przytycki and Paweł Traczyk. Invariants of links of Conway type. Kobe J. Math., 4(2):115-139, 1988.
L. Traldi. On the colored Tutte polynomial of a graph of bounded tree-width. Discrete Applied Mathematics, 154.6:1032-1036, 2006.
D. J. A. Welsh. Complexity: Knots, Colourings and Counting, volume 186 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 1993.
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Practical Volume Computation of Structured Convex Bodies, and an Application to Modeling Portfolio Dependencies and Financial Crises
We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be constructed by computing volumes of convex bodies.
We design and implement practical algorithms in the exact and approximate setting, we experimentally juxtapose them and study the tradeoff of exactness and accuracy for speed. We analyze the following methods in order of increasing generality: rejection sampling relying on uniformly sampling the simplex, which is the fastest approach, but inaccurate for small volumes; exact formulae based on the computation of integrals of probability distribution functions, which are the method of choice for intersections with a single hyperplane; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple with additional structure; Markov chain Monte Carlo algorithms using random walks based on the hit-and-run paradigm generalized to nonlinear convex bodies and relying on new methods for computing a ball enclosed in the given body, such as a second-order cone program; the latter is experimentally extended to non-convex bodies with very encouraging results. Our C++ software, based on CGAL and Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.
Polytope volume
convex body
simplex
sampling
financial portfolio
19:1-19:15
Regular Paper
Ludovic
Calès
Ludovic Calès
Apostolos
Chalkis
Apostolos Chalkis
Ioannis Z.
Emiris
Ioannis Z. Emiris
Vissarion
Fisikopoulos
Vissarion Fisikopoulos
10.4230/LIPIcs.SoCG.2018.19
Y. Abbasi-Yadkori, P.L. Bartlett, V. Gabillon, and A. Malek. Hit-and-run for sampling and planning in non-convex spaces. In Proc. 20th Intern. Conf. Artificial Intelligence &Stat. (AISTATS), pages 888-895, 2017. URL: http://proceedings.mlr.press/v54/abbasi-yadkori17a.html.
http://proceedings.mlr.press/v54/abbasi-yadkori17a.html
M. Maswood Ali. Content of the frustum oa a simplex. Pacific J. Math., 48(2):313-322, 1973.
A. Banerjee and C-H. Hung. Informed momentum trading versus uninformed "naive" investors strategies. J. Banking &Finance, 35(11):3077-3089, 2011.
M. Billio, L. Calès, and D. Guéguan. A cross-sectional score for the relative performance of an allocation. Intern. Review Appl. Financial Issues &Economics, 3(4):700-710, 2011.
M. Billio, M. Getmansky, and L. Pelizzon. Dynamic risk exposures in hedge funds. Comput. Stat. &Data Analysis, 56(11):3517-3532, 2012. URL: http://dx.doi.org/10.1016/j.csda.2010.08.015.
http://dx.doi.org/10.1016/j.csda.2010.08.015
S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, UK, 2004.
B. Büeler, A. Enge, and K. Fukuda. Exact volume computation for polytopes: A practical study. In G. Kalai and G.M. Ziegler, editors, Polytopes: Combinatorics and Computation, volume 29 of Math. &Statistics, pages 131-154. Birkhäuser, Basel, 2000.
L. Cal\a`es, A. Chalkis, I.Z. Emiris, and V. Fisikopoulos. Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises. CoRR, arXiv:1803.05861, March 2018. URL: https://arxiv.org/abs/1803.05861.
https://arxiv.org/abs/1803.05861
K. Chandrasekaran, D. Dadush, and S. Vempala. Thin partitions: Isoperimetric inequalities and sampling algorithms for some nonconvex families. CoRR, abs/0904.0583, 2009. URL: http://arxiv.org/abs/0904.0583.
http://arxiv.org/abs/0904.0583
B. Cousins and S. Vempala. A cubic algorithm for computing Gaussian volume. In Proc. Symp. on Discrete Algorithms, pages 1215-1228. SIAM/ACM, 2014.
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M. Lo Duca, A. Koban, M. Basten, E. Bengtsson, B. Klaus, P. Kusmierczyk, J.H. Lang, C. Detken, and T. Peltonen. A new database for financial crises in european countries. Technical Report 13, European Central Bank and European Systemic Risk Board, Frankfurt am Main, Germany, 2017.
M. Dyer, A. Frieze, and R. Kannan. A random polynomial-time algorithm for approximating the volume of convex bodies. J. ACM, 38(1):1-17, 1991. URL: http://dx.doi.org/10.1145/102782.102783.
http://dx.doi.org/10.1145/102782.102783
M.E. Dyer and A.M. Frieze. On the complexity of computing the volume of a polyhedron. SIAM J. Comput., 17(5):967-974, 1988. URL: http://dx.doi.org/10.1137/0217060.
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G. Elekes. A geometric inequality and the complexity of computing volume. Discrete &Computational Geometry, 1:289-292, 1986.
I.Z. Emiris and V. Fisikopoulos. Practical polytope volume approximation. ACM Trans. Math. Soft., 2018. To appear. Prelim. version: Proc. Sympos. on Comput. Geometry, 2014.
E. Erdougan and G. Iyengar. An active set method for single-cone second-order cone programs. SIAM J. Optimization, 17(2):459-484, 2006. URL: http://dx.doi.org/10.1137/040612592.
http://dx.doi.org/10.1137/040612592
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O. Ledoit and M. Wolf. Honey, I shrunk the sample covariance matrix. J. Portfolio Management, 30(4):110-119, 2004. URL: http://dx.doi.org/10.3905/jpm.2004.110.
http://dx.doi.org/10.3905/jpm.2004.110
Y.T. Lee and S.S. Vempala. Convergence rate of Riemannian Hamiltonian Monte Carlo and faster polytope volume computation. CoRR, abs/1710.06261, 2017. URL: http://arxiv.org/abs/1710.06261.
http://arxiv.org/abs/1710.06261
Y.T. Lee and S.S. Vempala. Geodesic walks in polytopes. In Proc. ACM Symp. on Theory of Computing, pages 927-940. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055416.
http://dx.doi.org/10.1145/3055399.3055416
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Subquadratic Encodings for Point Configurations
For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model with word size w >= log n. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 {(log log n)}^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/log log n) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.
point configuration
order type
chirotope
succinct data structure
20:1-20:14
Regular Paper
Jean
Cardinal
Jean Cardinal
Timothy M.
Chan
Timothy M. Chan
John
Iacono
John Iacono
Stefan
Langerman
Stefan Langerman
Aurélien
Ooms
Aurélien Ooms
10.4230/LIPIcs.SoCG.2018.20
Oswin Aichholzer, Franz Aurenhammer, and Hannes Krasser. Enumerating order types for small point sets with applications. Order, 19(3):265-281, 2002.
Oswin Aichholzer, Franz Aurenhammer, and Hannes Krasser. On the crossing number of complete graphs. In SOCG, pages 19-24. ACM, 2002.
Oswin Aichholzer, Jean Cardinal, Vincent Kusters, Stefan Langerman, and Pavel Valtr. Reconstructing point set order types from radial orderings. International Journal of Computational Geometry &Applications, 26(3-4):167-184, 2016.
Oswin Aichholzer, Matias Korman, Alexander Pilz, and Birgit Vogtenhuber. Geodesic order types. Algorithmica, 70(1):112-128, 2014.
Oswin Aichholzer and Hannes Krasser. The point set order type data base: A collection of applications and results. In CCCG, pages 17-20, 2001.
Oswin Aichholzer and Hannes Krasser. Abstract order type extension and new results on the rectilinear crossing number. In SOCG, pages 91-98. ACM, 2005.
Oswin Aichholzer, Vincent Kusters, Wolfgang Mulzer, Alexander Pilz, and Manuel Wettstein. An optimal algorithm for reconstructing point set order types from radial orderings. In ISAAC, pages 505-516. Springer, 2015.
Oswin Aichholzer, Tillmann Miltzow, and Alexander Pilz. Extreme point and halving edge search in abstract order types. Computational Geometry, 46(8):970-978, 2013.
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Jean Cardinal, Timothy M. Chan, John Iacono, Stefan Langerman, and Aurélien Ooms. Subquadratic encodings for point configurations. ArXiv e-prints, 2018. URL: https://arxiv.org/abs/1801.01767.
https://arxiv.org/abs/1801.01767
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Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces
We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric X. Computing such an embedding (exactly or approximately) is a non-trivial task even when X is the metric induced by a path, or, equivalently, the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph H, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs G, H and integer c, is it possible to embed G with distortion c into a graph homeomorphic to H? Then embedding into the line is the special case H=K_2, and embedding into the cycle is the case H=K_3, where K_k denotes the complete graph on k vertices. For this problem we give
- an approximation algorithm, which in time f(H)* poly (n), for some function f, either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion poly(c);
- an exact algorithm, which in time f'(H, c)* poly (n), for some function f', either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion c. Prior to our work, poly(OPT)-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.
Metric embeddings
minimum-distortion embeddings
1-dimensional simplicial complex
Fixed-parameter tractable algorithms
Approximation algorithms
21:1-21:14
Regular Paper
Timothy
Carpenter
Timothy Carpenter
Fedor V.
Fomin
Fedor V. Fomin
Daniel
Lokshtanov
Daniel Lokshtanov
Saket
Saurabh
Saket Saurabh
Anastasios
Sidiropoulos
Anastasios Sidiropoulos
10.4230/LIPIcs.SoCG.2018.21
Sanjeev Arora, James Lee, and Assaf Naor. Euclidean distortion and the sparsest cut. Journal of the American Mathematical Society, 21(1):1-21, 2008.
Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56(2):5, 2009.
Mihai Bădoiu, Julia Chuzhoy, Piotr Indyk, and Anastasios Sidiropoulos. Low-distortion embeddings of general metrics into the line. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), pages 225-233. ACM, 2005.
Mihai Bădoiu, Julia Chuzhoy, Piotr Indyk, and Anastasios Sidiropoulos. Embedding ultrametrics into low-dimensional spaces. In Proceedings of the 22nd Annual Symposium on Computational Geometry (SoCG), pages 187-196. ACM, 2006.
Mihai Bădoiu, Kedar Dhamdhere, Anupam Gupta, Yuri Rabinovich, Harald Räcke, R. Ravi, and Anastasios Sidiropoulos. Approximation algorithms for low-distortion embeddings into low-dimensional spaces. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 119-128. SIAM, 2005.
Mihai Bădoiu, Piotr Indyk, and Anastasios Sidiropoulos. Approximation algorithms for embedding general metrics into trees. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 512-521. ACM and SIAM, 2007.
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http://dx.doi.org/10.1145/1383369.1383376
Mark de Berg, Krzysztof Onak, and Anastasios Sidiropoulos. Fat polygonal partitions with applications to visualization and embeddings. Journal of Computational Geometry, 4(1):212–239, 2013.
Jeff Edmonds, Anastasios Sidiropoulos, and Anastasios Zouzias. Inapproximability for planar embedding problems. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 222-235. Society for Industrial and Applied Mathematics, 2010.
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http://dx.doi.org/10.1145/2489789
Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Elena Losievskaja, Frances A. Rosamond, and Saket Saurabh. Distortion is fixed parameter tractable. In Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP), volume 5555 of Lecture Notes in Computer Science, pages 463-474. Springer, 2009.
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Claire Kenyon, Yuval Rabani, and Alistair Sinclair. Low distortion maps between point sets. SIAM J. Comput., 39(4):1617-1636, 2009. URL: http://dx.doi.org/10.1137/080712921.
http://dx.doi.org/10.1137/080712921
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Jiří Matoušek and Anastasios Sidiropoulos. Inapproximability for metric embeddings into ℝ^d. Transactions of the American Mathematical Society, 362(12):6341-6365, 2010.
Amir Nayyeri and Benjamin Raichel. Reality distortion: Exact and approximate algorithms for embedding into the line. In Proceedings of the 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 729-747. IEEE, 2015.
Amir Nayyeri and Benjamin Raichel. A treehouse with custom windows: Minimum distortion embeddings into bounded treewidth graphs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 724-736, Philadelphia, PA, USA, 2017. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=3039686.3039732.
http://dl.acm.org/citation.cfm?id=3039686.3039732
Christos Papadimitriou and Shmuel Safra. The complexity of low-distortion embeddings between point sets. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), volume 5, pages 112-118. SIAM, 2005.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Fast Approximation and Exact Computation of Negative Curvature Parameters of Graphs
In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. Our main contribution in this note is a new characterization of hyperbolicity for graphs (and for complete geodesic metric spaces). This characterization has algorithmic implications in the field of large-scale network analysis, which was one of our initial motivations. A sharp estimate of graph hyperbolicity is useful, {e.g.}, in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in polynomial-time, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm for computing the hyperbolicity of an n-vertex graph G=(V,E) in optimal time O(n^2) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space.
Gromov hyperbolicity
Graphs
Geodesic metric spaces
Approximation algorithms
22:1-22:15
Regular Paper
Jérémie
Chalopin
Jérémie Chalopin
Victor
Chepoi
Victor Chepoi
Feodor F.
Dragan
Feodor F. Dragan
Guillaume
Ducoffe
Guillaume Ducoffe
Abdulhakeem
Mohammed
Abdulhakeem Mohammed
Yann
Vaxès
Yann Vaxès
10.4230/LIPIcs.SoCG.2018.22
M. Abu-Ata and F.F. Dragan. Metric tree-like structures in real-world networks: an empirical study. Networks, 67(1):49-68, 2016.
A.B. Adcock, B.D. Sullivan, and M.W. Mahoney. Tree-like structure in large social and information networks. In ICDM, pages 1-10. IEEE Computer Society, 2013.
J.M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short. Notes on word hyperbolic groups. In E. Ghys, A. Haefliger, and A. Verjovsky, editors, Group Theory from a Geometrical Viewpoint, ICTP Trieste 1990, pages 3-63. World Scientific, 1991.
A.M. Ben-Amram. The Euler path to static level-ancestors. CoRR, abs/0909.1030, 2009.
M.A. Bender and M Farach-Colton. The level ancestor problem simplified. Theor. Comput. Sci., 321(1):5-12, 2004.
M. Borassi, D. Coudert, P. Crescenzi, and A. Marino. On computing the hyperbolicity of real-world graphs. In ESA, volume 9294 of Lecture Notes in Computer Science, pages 215-226. Springer, 2015.
M. Borassi, P. Crescenzi, and M. Habib. Into the square: On the complexity of some quadratic-time solvable problems. Electr. Notes Theor. Comput. Sci., 322:51-67, 2016.
B.H. Bowditch. Notes on Gromov’s hyperbolicity criterion for path-metric spaces. In E. Ghys, A. Haefliger, and A. Verjovsky, editors, Group Theory from a Geometrical Viewpoint, ICTP Trieste 1990, pages 64-167. World Scientific, 1991.
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J. Chalopin, V. Chepoi, F.F. Dragan, G. Ducoffe, A. Mohammed, and Y. Vaxès. Fast approximation and exact computation of negative curvature parameters of graphs. CoRR, abs/1803.06324, 2018.
J. Chalopin, V. Chepoi, P. Papasoglu, and T. Pecatte. Cop and robber game and hyperbolicity. SIAM J. Discrete Math., 28(4):1987-2007, 2014.
V. Chepoi, F.F. Dragan, B. Estellon, M. Habib, and Y. Vaxès. Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs. In Symposium on Computational Geometry, pages 59-68. ACM, 2008.
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V. Chepoi, F.F. Dragan, and Y. Vaxès. Core congestion is inherent in hyperbolic networks. In SODA, pages 2264-2279. SIAM, 2017.
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N. Cohen, D. Coudert, and A. Lancin. On computing the Gromov hyperbolicity. ACM Journal of Experimental Algorithmics, 20:1.6:1-1.6:18, 2015.
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D. Coudert, G. Ducoffe, and A. Popa. Fully polynomial FPT algorithms for some classes of bounded clique-width graphs. In SODA, pages 2765-2784. SIAM, 2018.
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O. Narayan and I. Saniee. Large-scale curvature of networks. Phys. Rev. E, 84:066108, 2011.
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M. Soto. Quelques propriétés topologiques des graphes et applications à Internet et aux réseaux. PhD thesis, Université Paris Diderot, 2011.
K. Verbeek and S. Suri. Metric embedding, hyperbolic space, and social networks. In Symposium on Computational Geometry, pages 501-510. ACM, 2014.
H. Yu. An improved combinatorial algorithm for boolean matrix multiplication. In ICALP(1), volume 9134 of Lecture Notes in Computer Science, pages 1094-1105. Springer, 2015.
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Tree Drawings Revisited
We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that
1) every tree of size n (with arbitrarily large degree) has a straight-line drawing with area n2^{O(sqrt{log log n log log log n})}, improving the longstanding O(n log n) bound;
2) every tree of size n (with arbitrarily large degree) has a straight-line upward drawing with area n sqrt{log n}(log log n)^{O(1)}, improving the longstanding O(n log n) bound;
3) every binary tree of size n has a straight-line orthogonal drawing with area n2^{O(log^*n)}, improving the previous O(n log log n) bound by Shin, Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996);
4) every binary tree of size n has a straight-line order-preserving drawing with area n2^{O(log^*n)}, improving the previous O(n log log n) bound by Garg and Rusu (2003);
5) every binary tree of size n has a straight-line orthogonal order-preserving drawing with area n2^{O(sqrt{log n})}, improving the O(n^{3/2}) previous bound by Frati (2007).
graph drawing
trees
recursion
23:1-23:15
Regular Paper
Timothy M.
Chan
Timothy M. Chan
10.4230/LIPIcs.SoCG.2018.23
Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. J. ACM, 51(4):606-635, 2004. URL: http://dx.doi.org/10.1145/1008731.1008736.
http://dx.doi.org/10.1145/1008731.1008736
Christian Bachmaier, Franz-Josef Brandenburg, Wolfgang Brunner, Andreas Hofmeier, Marco Matzeder, and Thomas Unfried. Tree drawings on the hexagonal grid. In Proc. 16th International Symposium on Graph Drawing (GD), pages 372-383, 2008. URL: http://dx.doi.org/10.1007/978-3-642-00219-9_36.
http://dx.doi.org/10.1007/978-3-642-00219-9_36
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Timothy M. Chan. A near-linear area bound for drawing binary trees. Algorithmica, 34(1):1-13, 2002. URL: http://dx.doi.org/10.1007/s00453-002-0937-x.
http://dx.doi.org/10.1007/s00453-002-0937-x
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http://dx.doi.org/10.1007/978-3-540-77537-9_11
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https://www.cc.gatech.edu/fac/Vijay.Vazirani/book.pdf
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Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs
We present the first near-linear-time (1 + epsilon)-approximation algorithm for the diameter of a weighted unit-disk graph of n vertices, running in O(n log^2 n) time, for any constant epsilon>0, improving the near-O(n^{3/2})-time algorithm of Gao and Zhang [STOC 2003]. Using similar ideas, we can construct a (1+epsilon)-approximate distance oracle for weighted unit-disk graphs with O(1) query time, with a similar improvement in the preprocessing time, from near O(n^{3/2}) to O(n log^3 n). We also obtain new results for a number of other related problems in the weighted unit-disk graph metric, such as the radius and bichromatic closest pair.
As a further application, we use our new distance oracle, along with additional ideas, to solve the (1 + epsilon)-approximate all-pairs bounded-leg shortest paths problem for a set of n planar points, with near O(n^{2.579}) preprocessing time, O(n^2 log n) space, and O(log{log n}) query time, improving thus the near-cubic preprocessing bound by Roditty and Segal [SODA 2007].
shortest paths
distance oracles
unit-disk graphs
planar graphs
24:1-24:13
Regular Paper
Timothy M.
Chan
Timothy M. Chan
Dimitrios
Skrepetos
Dimitrios Skrepetos
10.4230/LIPIcs.SoCG.2018.24
Srinivasa Arikati, Danny Z. Chen, L. Paul Chew, Gautam Das, Michiel Smid, and Christos D. Zaroliagis. Planar spanners and approximate shortest path queries among obstacles in the plane. In Proceedings of the 4th Annual European Symposium on Algorithms (ESA), pages 514-528, 1996.
Prosenjit Bose, Anil Maheshwari, Giri Narasimhan, Michiel Smid, and Norbert Zeh. Approximating geometric bottleneck shortest paths. Computational Geometry, 29(3):233-249, 2004.
Costas Busch, Ryan LaFortune, and Srikanta Tirthapura. Improved sparse covers for graphs excluding a fixed minor. In Proceedings of the 26th Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 61-70, 2007.
Sergio Cabello. Many distances in planar graphs. Algorithmica, 62(1-2):361-381, 2012. URL: http://dx.doi.org/10.1007/s00453-010-9459-0.
http://dx.doi.org/10.1007/s00453-010-9459-0
Sergio Cabello. Subquadratic algorithms for the diameter and the sum of pairwise distances in planar graphs. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2143-2152, 2017.
Sergio Cabello and Miha Jejčič. Shortest paths in intersection graphs of unit disks. Computational Geometry, 48(4):360-367, 2015.
Paul B. Callahan and S. Rao Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Journal of the ACM, 42(1):67-90, 1995. URL: http://dx.doi.org/10.1145/200836.200853.
http://dx.doi.org/10.1145/200836.200853
Timothy M. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. Journal of the ACM, 57(3):16:1-16:15, 2010. URL: http://dx.doi.org/10.1145/1706591.1706596.
http://dx.doi.org/10.1145/1706591.1706596
Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in unit-disk graphs in slightly subquadratic time. In Proceedings of the 27th Annual International Symposium on Algorithms and Computation (ISAAC), pages 24:1-24:13, 2016.
Timothy M. Chan and Dimitrios Skrepetos. Faster approximate diameter and distance oracles in planar graphs. In Proceedings of the 25th European Symposium on Algorithms (ESA), pages 25:1-25:13, 2017.
Vincent Cohen-Addad, Søren Dahlgaard, and Christian Wulff-Nilsen. Fast and compact exact distance oracle for planar graphs. In Proceedings of the 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 962-973, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.93.
http://dx.doi.org/10.1109/FOCS.2017.93
David Eppstein, Gary L. Miller, and Shang-Hua Teng. A deterministic linear time algorithm for geometric separators and its applications. Fundamenta Informaticae, 22(4):309-329, 1995.
Jie Gao and Li Zhang. Well-separated pair decomposition for the unit-disk graph metric and its applications. SIAM Journal on Computing, 35(1):151-169, 2005. Preliminary version in STOC 2003.
Pawel Gawrychowski, Shay Mozes, Oren Weimann, and Christian Wulff-Nilsen. Better tradeoffs for exact distance oracles in planar graphs. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 515-529, 2018.
Pawel‚ Gawrychowski, Haim Kaplan, Shay Mozes, Micha Sharir, and Oren Weimann. Voronoi diagrams on planar graphs, and computing the diameter in deterministic Õ(n^5/3) time. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 495-514, 2018.
Qian-Ping Gu and Gengchun Xu. Constant query time (1+ε)-approximate distance oracle for planar graphs. In Proceedings of the 26th International Symposium on Algorithms and Computation (ISAAC), pages 625-636, 2015.
Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2495-2504, 2017.
Ken-ichi Kawarabayashi, Christian Sommer, and Mikkel Thorup. More compact oracles for approximate distances in undirected planar graphs. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 550-563, 2013.
Philip Klein. Preprocessing an undirected planar network to enable fast approximate distance queries. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 820-827, 2002.
François Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC), pages 296-303, 2014. URL: http://dx.doi.org/10.1145/2608628.2608664.
http://dx.doi.org/10.1145/2608628.2608664
Xiang-Yang Li, Gruia Calinescu, and Peng-Jun Wan. Distributed construction of a planar spanner and routing for ad hoc wireless networks. In Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), volume 3, pages 1268-1277, 2002.
Gary L Miller, S-H Teng, and Stephen A Vavasis. A unified geometric approach to graph separators. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 538-547, 1991.
Franco P. Preparata and Michael Ian Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985.
Liam Roditty and Michael Segal. On bounded leg shortest paths problems. Algorithmica, 59(4):583-600, 2011. Preliminary version in SODA 2007.
Mikkel Thorup. Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM, 51(6):993-1024, 2004.
Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the 44th ACM Symposium on Theory of Computing (STOC), pages 887-898, 2012. URL: http://dx.doi.org/10.1145/2213977.2214056.
http://dx.doi.org/10.1145/2213977.2214056
Oren Weimann and Raphael Yuster. Approximating the diameter of planar graphs in near linear time. ACM Transactions on Algorithms, 12(1):1-13, 2016.
Chenyu Yan, Yang Xiang, and Feodor F. Dragan. Compact and low delay routing labeling scheme for unit disk graphs. Computational Geometry, 45(7):305-325, 2012. URL: http://dx.doi.org/10.1016/j.comgeo.2012.01.015.
http://dx.doi.org/10.1016/j.comgeo.2012.01.015
Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM, 49(3):289-317, 2002.
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Dynamic Planar Orthogonal Point Location in Sublogarithmic Time
We study a longstanding problem in computational geometry: dynamic 2-d orthogonal point location, i.e., vertical ray shooting among n horizontal line segments. We present a data structure achieving O(log n / log log n) optimal expected query time and O(log^{1/2+epsilon} n) update time (amortized) in the word-RAM model for any constant epsilon>0, under the assumption that the x-coordinates are integers bounded polynomially in n. This substantially improves previous results of Giyora and Kaplan [SODA 2007] and Blelloch [SODA 2008] with O(log n) query and update time, and of Nekrich (2010) with O(log n / log log n) query time and O(log^{1+epsilon} n) update time. Our result matches the best known upper bound for simpler problems such as dynamic 2-d dominance range searching.
We also obtain similar bounds for orthogonal line segment intersection reporting queries, vertical ray stabbing, and vertical stabbing-max, improving previous bounds, respectively, of Blelloch [SODA 2008] and Mortensen [SODA 2003], of Tao (2014), and of Agarwal, Arge, and Yi [SODA 2005] and Nekrich [ISAAC 2011].
dynamic data structures
point location
word RAM
25:1-25:15
Regular Paper
Timothy M.
Chan
Timothy M. Chan
Konstantinos
Tsakalidis
Konstantinos Tsakalidis
10.4230/LIPIcs.SoCG.2018.25
Pankaj K. Agarwal, Lars Arge, Haim Kaplan, Eyal Molad, Robert E. Tarjan, and Ke Yi. An optimal dynamic data structure for stabbing-semigroup queries. SIAM Journal on Computing, 41(1):104-127, 2012. Preliminary version in SODA 2005. URL: http://dx.doi.org/10.1137/10078791X.
http://dx.doi.org/10.1137/10078791X
Stephen Alstrup, Thore Husfeldt, and Theis Rauhe. Marked ancestor problems. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 534-543, 1998. URL: http://dx.doi.org/10.1109/SFCS.1998.743504.
http://dx.doi.org/10.1109/SFCS.1998.743504
Arne Andersson and Mikkel Thorup. Dynamic ordered sets with exponential search trees. Journal of the ACM, 54(3), 2007. URL: http://dx.doi.org/10.1145/1236457.1236460.
http://dx.doi.org/10.1145/1236457.1236460
Lars Arge. The buffer tree: A technique for designing batched external data structures. Algorithmica, 37(1):1-24, 2003. URL: http://dx.doi.org/10.1007/s00453-003-1021-x.
http://dx.doi.org/10.1007/s00453-003-1021-x
Lars Arge and Jeffrey Scott Vitter. Optimal external memory interval management. SIAM Journal on Computing, 32(6):1488-1508, 2003. URL: http://dx.doi.org/10.1137/S009753970240481X.
http://dx.doi.org/10.1137/S009753970240481X
Guy E. Blelloch. Space-efficient dynamic orthogonal point location, segment intersection, and range reporting. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 894-903, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347180.
http://dl.acm.org/citation.cfm?id=1347082.1347180
Timothy M. Chan. Geometric applications of a randomized optimization technique. Discrete & Computational Geometry, 22(4):547-567, 1999. URL: http://dx.doi.org/10.1007/PL00009478.
http://dx.doi.org/10.1007/PL00009478
Timothy M. Chan. Persistent predecessor search and orthogonal point location on the word RAM. ACM Transactions on Algorithms, 9(3):22:1-22:22, 2013. Preliminary version in SODA 2011. URL: http://dx.doi.org/10.1145/2483699.2483702.
http://dx.doi.org/10.1145/2483699.2483702
Timothy M. Chan and Yakov Nekrich. Towards an optimal method for dynamic planar point location. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 390-409, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.31.
http://dx.doi.org/10.1109/FOCS.2015.31
Timothy M. Chan and Mihai Pătraşcu. Counting inversions, offline orthogonal range counting, and related problems. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 161-173, 2010. URL: http://dx.doi.org/10.1137/1.9781611973075.15.
http://dx.doi.org/10.1137/1.9781611973075.15
Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in unit-disk graphs in slightly subquadratic time. In Proceedings of the 27th International Symposium on Algorithms and Computation (ISAAC), pages 24:1-24:13, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2016.24.
http://dx.doi.org/10.4230/LIPIcs.ISAAC.2016.24
Timothy M. Chan and Konstantinos Tsakalidis. Dynamic orthogonal range searching on the RAM, revisited. In Proceedings of the 33rd International Symposium on Computational Geometry (SoCG), pages 28:1-28:13, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.28.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.28
Richard Cole and Ramesh Hariharan. Dynamic LCA queries on trees. SIAM Journal on Computing, 34(4):894-923, 2005. URL: http://dx.doi.org/10.1137/S0097539700370539.
http://dx.doi.org/10.1137/S0097539700370539
Paul F. Dietz. Fully persistent arrays. In Proceedings of the 1st Workshop for Algorithms and Data Structures (WADS), pages 67-74, 1989. URL: http://dx.doi.org/10.1007/3-540-51542-9_8.
http://dx.doi.org/10.1007/3-540-51542-9_8
Michael Fredman and Michael Saks. The cell probe complexity of dynamic data structures. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing (STOC), pages 345-354, 1989. URL: http://dx.doi.org/10.1145/73007.73040.
http://dx.doi.org/10.1145/73007.73040
Yoav Giyora and Haim Kaplan. Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. ACM Transactions on Algorithms, 5(3):28:1-28:51, 2009. Preliminary version in SODA 2007. URL: http://dx.doi.org/10.1145/1541885.1541889.
http://dx.doi.org/10.1145/1541885.1541889
Katherine Jane Lai. Complexity of union-split-find problems. Master’s thesis, MIT, 2008. URL: http://hdl.handle.net/1721.1/45638.
http://hdl.handle.net/1721.1/45638
Kurt Mehlhorn and Stefan Näher. Dynamic fractional cascading. Algorithmica, 5(1):215-241, 1990. URL: http://dx.doi.org/10.1007/BF01840386.
http://dx.doi.org/10.1007/BF01840386
Christian Worm Mortensen. Fully-dynamic two dimensional orthogonal range and line segment intersection reporting in logarithmic time. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 618-627, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644210.
http://dl.acm.org/citation.cfm?id=644108.644210
Christian Worm Mortensen. Data structures for orthogonal intersection searching and other problems. PhD thesis, IT University of Copenhagen, 2006. URL: http://www.epust.dk/main.pdf?attredirects=0.
http://www.epust.dk/main.pdf?attredirects=0
Christian Worm Mortensen. Fully dynamic orthogonal range reporting on RAM. SIAM Journal on Computing, 35(6):1494-1525, 2006. URL: http://dx.doi.org/10.1137/S0097539703436722.
http://dx.doi.org/10.1137/S0097539703436722
Christian Worm Mortensen, Rasmus Pagh, and Mihai Pǎtraşcu. On dynamic range reporting in one dimension. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), pages 104-111, 2005. URL: http://dx.doi.org/10.1145/1060590.1060606.
http://dx.doi.org/10.1145/1060590.1060606
Yakov Nekrich. Searching in dynamic catalogs on a tree. CoRR, abs/1007.3415, 2010. URL: http://arxiv.org/abs/1007.3415.
http://arxiv.org/abs/1007.3415
Yakov Nekrich. A dynamic stabbing-max data structure with sub-logarithmic query time. In Proceedings of the 22nd International Symposium on Algorithms and Computation (ISAAC), pages 170-179, 2011. URL: http://dx.doi.org/10.1007/978-3-642-25591-5_19.
http://dx.doi.org/10.1007/978-3-642-25591-5_19
Mark H. Overmars. The Design of Dynamic Data Structures, volume 156 of Lecture Notes in Computer Science. Springer, 1983. URL: http://dx.doi.org/10.1007/BFb0014927.
http://dx.doi.org/10.1007/BFb0014927
Yufei Tao. Dynamic ray stabbing. ACM Transactions on Algorithms, 11(2):11:1-11:19, 2014. URL: http://dx.doi.org/10.1145/2559153.
http://dx.doi.org/10.1145/2559153
Peter van Emde Boas. Preserving order in a forest in less than logarithmic time and linear space. Information Processing Letters, 6(3):80-82, 1977. URL: http://dx.doi.org/10.1016/0020-0190(77)90031-X.
http://dx.doi.org/10.1016/0020-0190(77)90031-X
Peter van Emde Boas, Robert Kaas, and Erik Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10(1):99-127, 1976.
Bryan T. Wilkinson. Amortized bounds for dynamic orthogonal range reporting. In Proceedings of the 22nd Annual European Symposium on Algorithms (ESA), pages 842-856, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_69.
http://dx.doi.org/10.1007/978-3-662-44777-2_69
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The Density of Expected Persistence Diagrams and its Kernel Based Estimation
Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane R^2 that can equivalently be seen as discrete measures in R^2. When the data come as a random point cloud, these discrete measures become random measures whose expectation is studied in this paper. First, we show that for a wide class of filtrations, including the Cech and Rips-Vietoris filtrations, the expected persistence diagram, that is a deterministic measure on R^2, has a density with respect to the Lebesgue measure. Second, building on the previous result we show that the persistence surface recently introduced in [Adams et al., 2017] can be seen as a kernel estimator of this density. We propose a cross-validation scheme for selecting an optimal bandwidth, which is proven to be a consistent procedure to estimate the density.
topological data analysis
persistence diagrams
subanalytic geometry
26:1-26:15
Regular Paper
Frédéric
Chazal
Frédéric Chazal
Vincent
Divol
Vincent Divol
10.4230/LIPIcs.SoCG.2018.26
Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: a stable vector representation of persistent homology. Journal of Machine Learning Research, 18(8):1-35, 2017.
Christophe Biscio and Jesper Møller. The accumulated persistence function, a new useful functional summary statistic for topological data analysis, with a view to brain artery trees and spatial point process applications. arXiv preprint arXiv:1611.00630, 2016.
Omer Bobrowski, Matthew Kahle, Primoz Skraba, et al. Maximally persistent cycles in random geometric complexes. The Annals of Applied Probability, 27(4):2032-2060, 2017.
Peter Bubenik. Statistical topological data analysis using persistence landscapes. The Journal of Machine Learning Research, 16(1):77-102, 2015.
Mickaël Buchet, Frédéric Chazal, Steve Y Oudot, and Donald R Sheehy. Efficient and robust persistent homology for measures. Computational Geometry, 58:70-96, 2016.
F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Memoli, and S. Y. Oudot. Gromov-hausdorff stable signatures for shapes using persistence. Computer Graphics Forum (proc. SGP 2009), pages 1393-1403, 2009.
F. Chazal, V. de Silva, and S. Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193-214, 2014.
Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules. SpringerBriefs in Mathematics. Springer, 2016.
Frédéric Chazal and Vincent Divol. The density of expected persistence diagrams and its kernel based estimation. Extended version of a paper to appear in the proceedings of the Symposium of Computational Geometry 2018, 2018. URL: https://hal.archives-ouvertes.fr/hal-01716181.
https://hal.archives-ouvertes.fr/hal-01716181
Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman. Stochastic convergence of persistence landscapes and silhouettes. In Proceedings of the thirtieth annual symposium on Computational geometry, page 474. ACM, 2014.
Frédéric Chazal, Marc Glisse, Catherine Labruère, and Bertrand Michel. Convergence rates for persistence diagram estimation in topological data analysis. Journal of Machine Learning Research, 16:3603-3635, 2015. URL: http://jmlr.org/papers/v16/chazal15a.html.
http://jmlr.org/papers/v16/chazal15a.html
Yen-Chi Chen, Daren Wang, Alessandro Rinaldo, and Larry Wasserman. Statistical analysis of persistence intensity functions. arXiv preprint arXiv:1510.02502, 2015.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete &Computational Geometry, 37(1):103-120, 2007.
HSM Coxeter. The circumradius of the general simplex. The Mathematical Gazette, pages 229-231, 1930.
Trinh Khanh Duy, Yasuaki Hiraoka, and Tomoyuki Shirai. Limit theorems for persistence diagrams. arXiv preprint arXiv:1612.08371, 2016.
B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan, A. Singh, et al. Confidence sets for persistence diagrams. The Annals of Statistics, 42(6):2301-2339, 2014.
D. Morozov H. Edelsbrunner. Persistent homology. In Handbook of Discrete and Computational Geometry (3rd Ed - To appear). CRC Press (to appear), 2017.
Matthew Kahle, Elizabeth Meckes, et al. Limit the theorems for betti numbers of random simplicial complexes. Homology, Homotopy and Applications, 15(1):343-374, 2013.
Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Kernel method for persistence diagrams via kernel embedding and weight factor. arXiv preprint arXiv:1706.03472, 2017.
Genki Kusano, Yasuaki Hiraoka, and Kenji Fukumizu. Persistence weighted gaussian kernel for topological data analysis. In International Conference on Machine Learning, pages 2004-2013, 2016.
Claire Lacour, Pascal Massart, and Vincent Rivoirard. Estimator selection: a new method with applications to kernel density estimation. arXiv preprint arXiv:1607.05091, 2016.
Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: isoperimetry and processes. Springer Science &Business Media, 2013.
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Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt. A stable multi-scale kernel for topological machine learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4741-4748, 2015.
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Katharine Turner, Yuriy Mileyko, Sayan Mukherjee, and John Harer. Fréchet means for distributions of persistence diagrams. Discrete &Computational Geometry, 52(1):44-70, 2014.
Yuhei Umeda. Time series classification via topological data analysis. Transactions of the Japanese Society for Artificial Intelligence, 32(3):D-G72_1, 2017.
D Yogeshwaran, Robert J Adler, et al. On the topology of random complexes built over stationary point processes. The Annals of Applied Probability, 25(6):3338-3380, 2015.
D. Yogeshwaran, Eliran Subag, and Robert J. Adler. Random geometric complexes in the thermodynamic regime. Probability Theory and Related Fields, 167(1):107-142, Feb 2017. URL: http://dx.doi.org/10.1007/s00440-015-0678-9.
http://dx.doi.org/10.1007/s00440-015-0678-9
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Embedding Graphs into Two-Dimensional Simplicial Complexes
We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the topological crossing number of a graph, but is more general.
The problem is NP-complete when C is part of the input, and we give a polynomial-time algorithm if the complex C is fixed.
Our strategy is to reduce the problem to an embedding extension problem on a surface, which has the following form: Given a subgraph H' of a graph G', and an embedding of H' on a surface S, can that embedding be extended to an embedding of G' on S? Such problems can be solved, in turn, using a key component in Mohar's algorithm to decide the embeddability of a graph on a fixed surface (STOC 1996, SIAM J. Discr. Math. 1999).
computational topology
embedding
simplicial complex
graph
surface
27:1-27:14
Regular Paper
Éric Colin de
Verdière
Éric Colin de Verdière
Thomas
Magnard
Thomas Magnard
Bojan
Mohar
Bojan Mohar
10.4230/LIPIcs.SoCG.2018.27
Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati, Vít Jelínek, Jan Kratochvíl, Maurizio Patrignani, and Ignaz Rutter. Testing planarity of partially embedded graphs. ACM Transactions on Algorithms (TALG), 11(4):32, 2015.
Dan Archdeacon. Topological graph theory. A survey. Congressus Numerantium, 115:5-54, 1996.
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Martin Čadek, Marek Krčál, Jiří Matoušek, Lukáš Vokřínek, and Uli Wagner. Polynomial-time computation of homotopy groups and Postnikov systems in fixed dimension. SIAM Journal on Computing, 43(5):1728-1780, 2014.
Éric Colin de Verdière. Computational topology of graphs on surfaces. In Jacob E. Goodman, Joseph O'Rourke, and Csaba Toth, editors, Handbook of Discrete and Computational Geometry, chapter 23. CRC Press LLC, third edition, 2018. See URL: http://www.arxiv.org/abs/1702.05358.
http://www.arxiv.org/abs/1702.05358
Éric Colin de Verdière and Arnaud de Mesmay. Testing graph isotopy on surfaces. Discrete &Computational Geometry, 51(1):171-206, 2014.
Erik Demaine, Shay Mozes, Christian Sommer, and Siamak Tazari. Algorithms for planar graphs and beyond, 2011. Course notes available at URL: http://courses.csail.mit.edu/6.889/fall11/lectures/.
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Bojan Mohar. On the minimal genus of 2-complexes. Journal of Graph Theory, 24(3):281-290, 1997.
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On the Complexity of Closest Pair via Polar-Pair of Point-Sets
Every graph G can be represented by a collection of equi-radii spheres in a d-dimensional metric Delta such that there is an edge uv in G if and only if the spheres corresponding to u and v intersect. The smallest integer d such that G can be represented by a collection of spheres (all of the same radius) in Delta is called the sphericity of G, and if the collection of spheres are non-overlapping, then the value d is called the contact-dimension of G. In this paper, we study the sphericity and contact dimension of the complete bipartite graph K_{n,n} in various L^p-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.
Contact dimension
Sphericity
Closest Pair
Fine-Grained Complexity
28:1-28:15
Regular Paper
Roee
David
Roee David
Karthik
C. S.
Karthik C. S.
Bundit
Laekhanukit
Bundit Laekhanukit
10.4230/LIPIcs.SoCG.2018.28
Amir Abboud, Aviad Rubinstein, and R. Ryan Williams. Distributed PCP theorems for hardness of approximation in P. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 25-36, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.12.
http://dx.doi.org/10.1109/FOCS.2017.12
Amir Abboud, Aviad Rubinstein, and Ryan Williams. Distributed PCP theorems for hardness of approximation in P. CoRR, abs/1706.06407, 2017. Preliminary version in FOCS'17. URL: http://arxiv.org/abs/1706.06407,
http://arxiv.org/abs/1706.06407
Josh Alman and Ryan Williams. Probabilistic polynomials and hamming nearest neighbors. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 136-150, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.18.
http://dx.doi.org/10.1109/FOCS.2015.18
Noga Alon and Pavel Pudlák. Equilateral sets in 𝓁_pⁿ. Geometric & Functional Analysis GAFA, 13(3):467-482, 2003. URL: http://dx.doi.org/10.1007/s00039-003-0418-7.
http://dx.doi.org/10.1007/s00039-003-0418-7
Hans-Jürgen Bandelt, Victor Chepoi, and Monique Laurent. Embedding into rectilinear spaces. Discrete & Computational Geometry, 19(4):595-604, 1998. URL: http://dx.doi.org/10.1007/PL00009370.
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Peter Frankl and Hiroshi Maehara. On the contact dimensions of graphs. Discrete & Computational Geometry, 3:89-96, 1988. URL: http://dx.doi.org/10.1007/BF02187899.
http://dx.doi.org/10.1007/BF02187899
Omer Gold and Micha Sharir. Dominance products and faster algorithms for high-dimensional closest pair under dollarl_backslashinftydollar. CoRR, abs/1605.08107, 2016. URL: http://arxiv.org/abs/1605.08107,
http://arxiv.org/abs/1605.08107
Oded Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press, New York, NY, USA, 1 edition, 2008.
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http://dx.doi.org/10.1016/0020-0190(88)90150-0
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Hiroshi Maehara. Contact patterns of equal nonoverlapping spheres. Graphs and Combinatorics, 1(1):271-282, 1985. URL: http://dx.doi.org/10.1007/BF02582952.
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Aviad Rubinstein. Hardness of approximate nearest neighbor search. CoRR, abs/1803.00904, 2018. URL: http://arxiv.org/abs/1803.00904.
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Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005. Preliminary version in ICALP'04. URL: http://dx.doi.org/10.1016/j.tcs.2005.09.023.
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Virginia Vassilevska Williams. Hardness of easy problems: Basing hardness on popular conjectures such as the strong exponential time hypothesis (invited talk). In 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, September 16-18, 2015, Patras, Greece, pages 17-29, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.17.
http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.17
Virginia Vassilevska Williams. Fine-grained algorithms and complexity (invited talk). In 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016, February 17-20, 2016, Orléans, France, pages 3:1-3:1, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2016.3.
http://dx.doi.org/10.4230/LIPIcs.STACS.2016.3
Andrew Chi-Chih Yao. Lower bounds for algebraic computation trees with integer inputs. SIAM J. Comput., 20(4):655-668, 1991. Preliminary version in FOCS'89. URL: http://dx.doi.org/10.1137/0220041.
http://dx.doi.org/10.1137/0220041
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Coordinated Motion Planning: Reconfiguring a Swarm of Labeled Robots with Bounded Stretch
We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Despite a broad range of other non-trivial results for multi-object motion planning, previous work has largely focused on sequential schedules, in which one robot moves at a time, with objectives such as the number of moves; attempts to minimize the overall makespan of a coordinated parallel motion schedule (with many robots moving simultaneously) have defied all attempts at establishing the complexity in the absence of obstacles, as well as the existence of efficient approximation methods.
We resolve these open problems by developing a framework that provides constant-factor approximation algorithms for minimizing the execution time of a coordinated, parallel motion plan for a swarm of robots in the absence of obstacles, provided their arrangement entails some amount of separability. In fact, our algorithm achieves constant stretch factor: If all robots want to move at most d units from their respective starting positions, then the total duration of the overall schedule (and hence the distance traveled by each robot) is O(d). Various extensions include unlabeled robots and different classes of robots. We also resolve the complexity of finding a reconfiguration plan with minimal execution time by proving that this is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Omega(N^{1/4}) may be required. On the positive side, we establish a stretch factor of O(N^{1/2}) even in this case. The intricate difficulties of computing precise optimal solutions are demonstrated by the seemingly simple case of just two disks, which is shown to be excruciatingly difficult to solve to optimality.
Robot swarms
coordinated motion planning
parallel motion
makespan
bounded stretch
complexity
approximation
29:1-29:15
Regular Paper
Erik D.
Demaine
Erik D. Demaine
Sándor P.
Fekete
Sándor P. Fekete
Phillip
Keldenich
Phillip Keldenich
Christian
Scheffer
Christian Scheffer
Henk
Meijer
Henk Meijer
10.4230/LIPIcs.SoCG.2018.29
M. Abellanas, S. Bereg, F. Hurtado, A. G. Olaverri, D. Rappaport, and J. Tejel. Moving coins. Computational Geometry: Theory and Applications, 34(1):35-48, 2006.
Aviv Adler, Mark de Berg, Dan Halperin, and Kiril Solovey. Efficient multi-robot motion planning for unlabeled discs in simple polygons. IEEE Transactions on Automation Science and Engineering, 12(4):1309-1317, 2015.
B. Aronov, M. de Berg, A. F. van der Stappen, P. Švestka, and J. Vleugels. Motion planning for multiple robots. Discrete &Computational Geometry, 22(4):505-525, 1999.
Aaron T. Becker, Sándor P. Fekete, Phillip Keldenich, Lillian Lin, and Christian Scheffer. Coordinated motion planning: The video. In Csaba Tóth and Bettina Speckmann, editors, 34th International Symposium on Computational Geometry (SoCG 2018, these proceedings), volume 99 of Leibniz International Proceedings in Informatics (LIPIcs), pages 74:1-74:6, 2018. Video available via https://youtu.be/0OrG72sX5gk.
S. Bereg, A. Dumitrescu, and J. Pach. Sliding disks in the plane. International Journal of Computational Geometry & Applications, 18(5):373-387, 2008.
Erik D. Demaine, Sándor P. Fekete, Phillip Keldenich, Henk Meijer, and Christian Scheffer. Coordinated motion planning: Reconfiguring a swarm of labeled robots with bounded stretch. Computing Research Repository (CoRR), 1801:1-32, 2018. Available at https://arxiv.org/abs/1801.01689.
A. Dumitrescu and M. Jiang. On reconfiguration of disks in the plane and related problems. Computational Geometry: Theory and Applications, 46:191-202, 2013.
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K. Solovey and D. Halperin. k-color multi-robot motion planning. International Journal of Robotics Research, 33(1):82-97, 2014.
K. Solovey, J. Yu, O. Zamir, and D. Halperin. Motion planning for unlabeled discs with optimality guarantees. In Robotics: Science and Systems (RSS), 2015.
Kiril Solovey and Dan Halperin. On the hardness of unlabeled multi-robot motion planning. International Journal of Robotics Research, 35(14):1750-1759, 2016.
P. Spirakis and C. K. Yap. Strong NP-hardness of moving many discs. Information Processing Letters, 19(1):55-59, 1984.
M. Turpin, N. Michael, and V. Kumar. Trajectory planning and assignment in multirobot systems. In Algorithmic Foundations of Robotics X, pages 175-190. Springer, 2013.
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J. Yu. Constant factor optimal multi-robot path planning in well-connected environments. arXiv. URL: https://arxiv.org/abs/1706.07255.
https://arxiv.org/abs/1706.07255
J. Yu. Constant factor time optimal multi-robot routing on high-dimensional grids in mostly sub-quadratic time. arXiv. URL: https://arxiv.org/abs/1801.10465.
https://arxiv.org/abs/1801.10465
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3D Snap Rounding
Let P be a set of n polygons in R^3, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps P to a simplicial complex Q whose vertices have integer coordinates. Every face of P is mapped to a set of faces (or edges or vertices) of Q and the mapping from P to Q can be done through a continuous motion of the faces such that (i) the L_infty Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case, the size of Q is O(n^{15}) and the time complexity of the algorithm is O(n^{19}) but, under reasonable hypotheses, these complexities decrease to O(n^{5}) and O(n^{6}sqrt{n}).
Geometric algorithms
Robustness
Fixed-precision computations
30:1-30:14
Regular Paper
Olivier
Devillers
Olivier Devillers
Sylvain
Lazard
Sylvain Lazard
William J.
Lenhart
William J. Lenhart
10.4230/LIPIcs.SoCG.2018.30
Mark de Berg, Dan Halperin, and Mark Overmars. An intersection-sensitive algorithm for snap rounding. Computational Geometry, 36(3):159-165, 2007. URL: http://dx.doi.org/10.1016/j.comgeo.2006.03.002.
http://dx.doi.org/10.1016/j.comgeo.2006.03.002
Olivier Devillers, Menelaos Karavelas, and Monique Teillaud. Qualitative Symbolic Perturbation: Two Applications of a New Geometry-based Perturbation Framework. Journal of Computational Geometry, 8(1):282-315, 2017. URL: http://dx.doi.org/10.20382/jocg.v8i1a11.
http://dx.doi.org/10.20382/jocg.v8i1a11
Steven Fortune. Polyhedral modelling with multiprecision integer arithmetic. Computer-Aided Design, 29(2):123-133, 1997. Solid Modelling. URL: http://dx.doi.org/10.1016/S0010-4485(96)00041-3.
http://dx.doi.org/10.1016/S0010-4485(96)00041-3
Steven Fortune. Vertex-rounding a three-dimensional polyhedral subdivision. Discrete &Computational Geometry, 22(4):593-618, 1999. URL: http://dx.doi.org/10.1007/PL00009480.
http://dx.doi.org/10.1007/PL00009480
Michael T. Goodrich, Leonidas J. Guibas, John Hershberger, and Paul J. Tanenbaum. Snap rounding line segments efficiently in two and three dimensions. In Proceedings of the thirteenth annual symposium on Computational geometry, pages 284-293. ACM, 1997. URL: http://dx.doi.org/10.1145/262839.262985.
http://dx.doi.org/10.1145/262839.262985
Daniel H. Greene and F. Frances Yao. Finite-resolution computational geometry. In Foundations of Computer Science, 1986., 27th Annual Symposium on, pages 143-152. IEEE, 1986. URL: http://dx.doi.org/10.1109/SFCS.1986.19.
http://dx.doi.org/10.1109/SFCS.1986.19
Leonidas J. Guibas and David H. Marimont. Rounding arrangements dynamically. International Journal of Computational Geometry &Applications, 8(02):157-178, 1998. URL: http://dx.doi.org/10.1142/S0218195998000096.
http://dx.doi.org/10.1142/S0218195998000096
Dan Halperin and Eli Packer. Iterated snap rounding. Computational Geometry, 23(2):209-225, 2002. URL: http://dx.doi.org/10.1016/S0925-7721(01)00064-5.
http://dx.doi.org/10.1016/S0925-7721(01)00064-5
John Hershberger. Improved output-sensitive snap rounding. Discrete & Computational Geometry, 39(1):298-318, Mar 2008. URL: http://dx.doi.org/10.1007/s00454-007-9015-0.
http://dx.doi.org/10.1007/s00454-007-9015-0
John Hershberger. Stable snap rounding. Computational Geometry, 46(4):403-416, 2013. URL: http://dx.doi.org/10.1016/j.comgeo.2012.02.011.
http://dx.doi.org/10.1016/j.comgeo.2012.02.011
John D. Hobby. Practical segment intersection with finite precision output. Computational Geometry, 13(4):199-214, 1999. URL: http://dx.doi.org/10.1016/S0925-7721(99)00021-8.
http://dx.doi.org/10.1016/S0925-7721(99)00021-8
V. Milenkovic and L. R. Nackman. Finding compact coordinate representations for polygons and polyhedra. IBM Journal of Research and Development, 34(35):753-769, 1990.
Victor Milenkovic. Rounding face lattices in d dimensions. In Proceedings of the 2nd Canadian Conference on Computational geometry, pages 40-45, 1990.
Eli Packer. Iterated snap rounding with bounded drift. Computational Geometry, 40(3):231-251, 2008. URL: http://dx.doi.org/10.1016/j.comgeo.2007.09.002.
http://dx.doi.org/10.1016/j.comgeo.2007.09.002
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Graph Reconstruction by Discrete Morse Theory
Recovering hidden graph-like structures from potentially noisy data is a fundamental task in modern data analysis. Recently, a persistence-guided discrete Morse-based framework to extract a geometric graph from low-dimensional data has become popular. However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same loop structure as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm.
graph reconstruction
discrete Morse theory
persistence
31:1-31:15
Regular Paper
Tamal K.
Dey
Tamal K. Dey
Jiayuan
Wang
Jiayuan Wang
Yusu
Wang
Yusu Wang
10.4230/LIPIcs.SoCG.2018.31
M. Aanjaneya, F. Chazal, D. Chen, M. Glisse, L. Guibas, and D. Morozov. Metric graph reconstruction from noisy data. International Journal of Computational Geometry &Applications, 22(04):305-325, 2012.
D. Attali, M. Glisse, S. Hornus, F. Lazarus, and D. Morozov. Persistence-sensitive simplification of functions on surfaces in linear time. Presented at TOPOINVIS, 9:23-24, 2009.
U. Bauer, C. Lange, and M. Wardetzky. Optimal topological simplification of discrete functions on surfaces. Discr. Comput. Geom., 47(2):347-377, 2012.
S. Biasotti, D. Giorgi, M. Spagnuolo, and B. Falcidieno. Reeb graphs for shape analysis and applications. Theoretical Computer Science, 392(1-3):5-22, 2008.
F. Chazal, R. Huang, and J. Sun. Gromov-hausdorff approximation of filamentary structures using reeb-type graphs. Discr. Comput. Geom., 53(3):621-649, 2015.
O. Delgado-Friedrichs, V. Robins, and A. Sheppard. Skeletonization and partitioning of digital images using discrete morse theory. IEEE Trans. Pattern Anal. Machine Intelligence, 37(3):654-666, March 2015.
T. Dey and J. Sun. Defining and computing curve-skeletons with medial geodesic function. In Sympos. Geom. Proc., volume 6, pages 143-152, 2006.
T. Dey, J. Wang, and Y. Wang. Improved road network reconstruction using discrete morse theory. In Proc. 25th ACM SIGSPATIAL. ACM, 2017.
T. Dey, J. Wang, and Y. Wang. Graph reconstruction by discrete morse theory. arXiv preprint arXiv:1803.05093, 2018.
H. Edelsbrunner and J. Harer. Computational Topology - an Introduction. American Mathematical Soc., 2010. URL: http://www.ams.org/bookstore-getitem/item=MBK-69.
http://www.ams.org/bookstore-getitem/item=MBK-69
H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discr. Comput. Geom., 28:511-533, 2002.
The ENZO project. URL: http://enzo-project.org.
http://enzo-project.org
The Center for Integrative Biomedical Computing (CIBC). Micro-CT Dataset Archive. URL: https://www.sci.utah.edu/cibc-software/cibc-datasets.html.
https://www.sci.utah.edu/cibc-software/cibc-datasets.html
R. Forman. Morse theory for cell complexes. Advances in mathematics, 134(1):90-145, 1998.
X. Ge, I. I Safa, M. Belkin, and Y. Wang. Data skeletonization via reeb graphs. In Advances in Neural Info. Proc. Sys., pages 837-845, 2011.
A. Gyulassy, M. Duchaineau, V. Natarajan, V. Pascucci, E. Bringa, A. Higginbotham, and B. Hamann. Topologically clean distance fields. IEEE Trans. Visualization Computer Graphics, 13(6):1432-1439, Nov 2007.
T. Hastie and W. Stuetzle. Principal curves. Journal of the American Statistical Association, 84(406):502-516, 1989.
B. Kégl and A. Krzyzak. Piecewise linear skeletonization using principal curves. IEEE Trans. Pattern Anal. Machine Intelligence, 24(1):59-74, 2002.
L. Liu, E. W Chambers, D. Letscher, and T. Ju. Extended grassfire transform on medial axes of 2d shapes. Computer-Aided Design, 43(11):1496-1505, 2011.
J. Milnor. Morse Theory. Princeton Univ. Press, New Jersey, 1963.
M. Natali, S. Biasotti, G. Patanè, and B. Falcidieno. Graph-based representations of point clouds. Graphical Models, 73(5):151-164, 2011.
U. Ozertem and D. Erdogmus. Locally defined principal curves and surfaces. Journal of Machine learning research, 12(Apr):1249-1286, 2011.
V. Robins, P. J. Wood, and A. P. Sheppard. Theory and algorithms for constructing discrete morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Machine Intelligence, 33(8):1646-1658, Aug 2011.
T. Sousbie. The persistent cosmic web and its filamentary structure-i. theory and implementation. Monthly Notices of the Royal Astronomical Society, 414(1):350-383, 2011.
S. Wang, Y. Wang, and Y. Li. Efficient map reconstruction and augmentation via topological methods. In Proc. 23rd ACM SIGSPATIAL, page 25. ACM, 2015.
Y. Yan, K. Sykes, E. Chambers, D. Letscher, and T. Ju. Erosion thickness on medial axes of 3d shapes. ACM Trans. on Graphics, 35(4):38, 2016.
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Computing Bottleneck Distance for 2-D Interval Decomposable Modules
Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For 1-D persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck distance with known efficient algorithms. The question is open for most n-D persistence modules, n>1, because of the well recognized complications of the indecomposables. Here, we consider a reasonably complicated class called 2-D interval decomposable modules whose indecomposables may have a description of non-constant complexity. We present a polynomial time algorithm to compute the bottleneck distance for these modules from indecomposables, which bounds the interleaving distance from above, and give another algorithm to compute a new distance called dimension distance that bounds it from below.
Persistence modules
bottleneck distance
interleaving distance
32:1-32:15
Regular Paper
Tamal K.
Dey
Tamal K. Dey
Cheng
Xin
Cheng Xin
10.4230/LIPIcs.SoCG.2018.32
Michael Atiyah. On the krull-schmidt theorem with application to sheaves. Bulletin de la Société Mathématique de France, 84:307-317, 1956. URL: http://eudml.org/doc/86907.
http://eudml.org/doc/86907
Ulrich Bauer and Michael Lesnick. Induced matchings of barcodes and the algebraic stability of persistence. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 355:355-355:364, 2014.
Silvia Biasotti, Andrea Cerri, Patrizio Frosini, and Daniela Giorgi. A new algorithm for computing the 2-dimensional matching distance between size functions. Pattern Recognition Letters, 32(14):1735-1746, 2011.
Håvard Bjerkevik. Stability of higher-dimensional interval decomposable persistence modules. arXiv preprint arXiv:1609.02086, 2016.
Håvard Bjerkevik and Magnus Botnan. Computational complexity of the interleaving distance. arXiv preprint arXiv:1712.04281, 2017.
Magnus Botnan, Justin Curry, and Elizabeth Munch. The poset interleaving distance, 2016. URL: https://jointmathematicsmeetings.org/amsmtgs/2180_abstracts/1125-55-1151.pdf.
https://jointmathematicsmeetings.org/amsmtgs/2180_abstracts/1125-55-1151.pdf
Magnus Botnan and Michael Lesnick. Algebraic stability of zigzag persistence modules. arXiv preprint arXiv:1604.00655, 2016.
Peter Bubenik and Jonathan Scott. Categorification of persistent homology. Discrete & Computational Geometry, 51(3):600-627, 2014.
Gunnar Carlsson and Afra Zomorodian. The theory of multidimensional persistence. Discrete & Computational Geometry, 42(1):71-93, Jul 2009.
Andrea Cerri, Barbara Di Fabio, Massimo Ferri, Patrizio Frosini, and Claudia Landi. Betti numbers in multidimensional persistent homology are stable functions. Mathematical Methods in the Applied Sciences, 36(12):1543-1557, 2013.
Andrea Cerri and Patrizio Frosini. A new approximation algorithm for the matching distance in multidimensional persistence. Technical report, February 2011. URL: http://amsacta.unibo.it/2971.
http://amsacta.unibo.it/2971
Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas Guibas, and Steve Oudot. Proximity of persistence modules and their diagrams. In Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry, SCG '09, pages 237-246, 2009.
Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules. arXiv preprint arXiv:1207.3674, 2012.
Jérémy Cochoy and Steve Oudot. Decomposition of exact pfd persistence bimodules. arXiv preprint arXiv:1605.09726, 2016.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete &Computational Geometry, 37(1):103-120, 2007.
William Crawley-Boevey. Decomposition of pointwise finite-dimensional persistence modules. Journal of Algebra and Its Applications, 14(05):1550066, 2015. URL: http://dx.doi.org/10.1142/S0219498815500668.
http://dx.doi.org/10.1142/S0219498815500668
Vin de Silva, Elizabeth Munch, and Amit Patel. Categorified reeb graphs. Discrete & Computational Geometry, 55(4):854-906, Jun 2016.
Tamal Dey and Cheng Xin. Computing bottleneck distance for 2-d interval decomposable modules, 2018. URL: http://arxiv.org/abs/1803.02869.
http://arxiv.org/abs/1803.02869
Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. Applied Mathematics. American Mathematical Society, 2010.
Michael Kerber, Dmitriy Morozov, and Arnur Nigmetov. Geometry helps to compare persistence diagrams. Journal of Experimental Algorithmics (JEA), 22(1):1-4, 2017.
Claudia Landi. The rank invariant stability via interleavings. arXiv preprint arXiv:1412.3374, 2014.
Michael Lesnick. The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics, 15(3):613-650, 2015.
Klaus Lux and Magdolna Sźoke. Computing decompositions of modules over finite-dimensional algebras. Experimental Mathematics, 16(1):1-6, 2007.
Steve Oudot. Persistence theory: from quiver representations to data analysis, volume 209. American Mathematical Society, 2015.
Amit Patel. Generalized persistence diagrams. arXiv preprint arXiv:1601.03107, 2016.
Carry Webb. Decomposition of graded modules. Proc. American Math. Soc., 94(4):565-571, 1985.
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Structure and Generation of Crossing-Critical Graphs
We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For c=1 there are only two such graphs without degree-2 vertices, K_5 and K_{3,3}, but for any fixed c>1 there exist infinitely many c-crossing-critical graphs. It has been previously shown that c-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every c-crossing-critical graph can be obtained from a c-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the c-crossing-critical graphs of at most given order n in polynomial time per each generated graph.
crossing number
crossing-critical
path-width
exhaustive generation
33:1-33:14
Regular Paper
Zdenek
Dvorák
Zdenek Dvorák
Petr
Hlinený
Petr Hlinený
Bojan
Mohar
Bojan Mohar
10.4230/LIPIcs.SoCG.2018.33
M. Ajtai, V. Chvátal, M.M. Newborn, and E. Szemerédi. Crossing-free subgraphs. In Theory and Practice of Combinatorics, volume 60 of North-Holland Mathematics Studies, pages 9-12. North-Holland, 1982. URL: http://dx.doi.org/10.1016/S0304-0208(08)73484-4.
http://dx.doi.org/10.1016/S0304-0208(08)73484-4
Drago Bokal. Infinite families of crossing-critical graphs with prescribed average degree and crossing number. Journal of Graph Theory, 65(2):139-162, 2010. URL: http://dx.doi.org/10.1002/jgt.20470.
http://dx.doi.org/10.1002/jgt.20470
Drago Bokal, Bogdan Oporowski, R. Bruce Richter, and Gelasio Salazar. Characterizing 2-crossing-critical graphs. Advances in Applied Mathematics, 74:23-208, 2016. URL: http://dx.doi.org/10.1016/j.aam.2015.10.003.
http://dx.doi.org/10.1016/j.aam.2015.10.003
Thomas Colcombet. Factorization forests for infinite words and applications to countable scattered linear orderings. Theor. Comput. Sci., 411(4-5):751-764, 2010. URL: http://dx.doi.org/10.1016/j.tcs.2009.10.013.
http://dx.doi.org/10.1016/j.tcs.2009.10.013
Z. Dvořák and B. Mohar. Crossing-critical graphs with large maximum degree. J. Combin. Theory, Ser. B, 100:413-417, 2010. URL: http://dx.doi.org/10.1016/j.jctb.2009.11.003.
http://dx.doi.org/10.1016/j.jctb.2009.11.003
C. Hernandez-Velez, G. Salazar, and R. Thomas. Nested cycles in large triangulations and crossing-critical graphs. Journal of Combinatorial Theory, Series B, 102:86-92, 2012. URL: http://dx.doi.org/10.1016/j.jctb.2011.04.006.
http://dx.doi.org/10.1016/j.jctb.2011.04.006
P. Hliněný. Crossing-number critical graphs have bounded path-width. J. Combin. Theory, Ser. B, 88:347-367, 2003. URL: http://dx.doi.org/10.1016/S0095-8956(03)00037-6.
http://dx.doi.org/10.1016/S0095-8956(03)00037-6
P. Hliněný and G. Salazar. Stars and bonds in crossing-critical graphs. J. Graph Theory, 65:198-215, 2010. URL: http://dx.doi.org/10.1002/jgt.20473.
http://dx.doi.org/10.1002/jgt.20473
Petr Hliněný. Crossing-critical graphs and path-width. In Petra Mutzel, Michael Jünger, and Sebastian Leipert, editors, Graph Drawing, 9th International Symposium, GD 2001, Revised Papers, volume LNCS 2265 of Lecture Notes in Computer Science, pages 102-114. Springer, 2002. URL: http://dx.doi.org/10.1007/3-540-45848-4_9.
http://dx.doi.org/10.1007/3-540-45848-4_9
Petr Hliněný and Marek Dernár. Crossing number is hard for kernelization. In 32nd International Symposium on Computational Geometry, SoCG 2016, June 14-18, 2016, Boston, MA, USA, volume 51 of LIPIcs, pages 42:1-42:10. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.42.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.42
Martin Kochol. Construction of crossing-critical graphs. Discrete Mathematics, 66(3):311-313, 1987. URL: http://dx.doi.org/10.1016/0012-365X(87)90108-7.
http://dx.doi.org/10.1016/0012-365X(87)90108-7
Tom Leighton. Complexity Issues in VLSI. Foundations of Computing Series. MIT Press, Cambridge, MA, 1983.
B. Mohar and C. Thomassen. Graphs on Surfaces. The Johns Hopkins University Press, Baltimore and London, 2001.
Benny Pinontoan and R. Bruce Richter. Crossing numbers of sequences of graphs II: Planar tiles. Journal of Graph Theory, 42(4):332-341, 2003. URL: http://dx.doi.org/10.1002/jgt.10097.
http://dx.doi.org/10.1002/jgt.10097
R. Bruce Richter and Carsten Thomassen. Minimal graphs with crossing number at least k. J. Comb. Theory, Ser. B, 58(2):217-224, 1993. URL: http://dx.doi.org/10.1006/jctb.1993.1038.
http://dx.doi.org/10.1006/jctb.1993.1038
Gelasio Salazar. Infinite families of crossing-critical graphs with given average degree. Discrete Mathematics, 271(1-3):343-350, 2003. URL: http://dx.doi.org/10.1016/S0012-365X(03)00136-5.
http://dx.doi.org/10.1016/S0012-365X(03)00136-5
Imre Simon. Factorization forests of finite height. Theor. Comput. Sci., 72(1):65-94, 1990. URL: http://dx.doi.org/10.1016/0304-3975(90)90047-L.
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László A. Székely. Crossing numbers and hard Erdös problems in discrete geometry. Combinatorics, Probability & Computing, 6(3):353-358, 1997. URL: http://journals.cambridge.org/action/displayAbstract?aid=46513.
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Jozef Širáň. Infinite families of crossing-critical graphs with a given crossing number. Discrete Mathematics, 48(1):129-132, 1984. URL: http://dx.doi.org/10.1016/0012-365X(84)90140-7.
http://dx.doi.org/10.1016/0012-365X(84)90140-7
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The Multi-cover Persistence of Euclidean Balls
Given a locally finite X subseteq R^d and a radius r >= 0, the k-fold cover of X and r consists of all points in R^d that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in R^{d+1} whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.
Delaunay mosaics
hyperplane arrangements
discrete Morse theory
zigzag modules
persistent homology
34:1-34:14
Regular Paper
Herbert
Edelsbrunner
Herbert Edelsbrunner
Georg
Osang
Georg Osang
10.4230/LIPIcs.SoCG.2018.34
Franz Aurenhammer. A new duality result concerning Voronoi diagrams. Discrete &Computational Geometry, 5(3):243-254, 1990.
Franz Aurenhammer and Otfried Schwarzkopf. A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams. International Journal of Computational Geometry &Applications, 2(04):363-381, 1992.
Ulrich Bauer and Herbert Edelsbrunner. The Morse theory of Čech and Delaunay complexes. Trans. Amer. Math. Soc., 369(369):3741-3762, 2017.
Gunnar Carlsson and Vin de Silva. Zigzag persistence. Found. Comput. Math., 10(4):367-405, 2010.
Gunnar Carlsson, Vin De Silva, and Dmitriy Morozov. Zigzag persistent homology and real-valued functions. In Proceedings of the twenty-fifth annual symposium on Computational geometry, pages 247-256. ACM, 2009.
Frédéric Chazal, David Cohen-Steiner, and Quentin Mérigot. Geometric inference for probability measures. Found. Comput. Math., 11(6):733-751, 2011.
Herbert Edelsbrunner. Algorithms in Combinatorial Geometry. Springer-Verlag, Heidelberg, Germany, 1987.
Herbert Edelsbrunner and John L. Harer. Computational Topology. An Introduction. American Mathematical Society, Providence, RI, 2010.
Herbert Edelsbrunner, Anton Nikitenko, and Georg Osang. A step in the weighted Delaunay mosaic of order k., 2017. Manuscript, IST Austria, Klosterneuburg, Austria.
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Robin Forman. Morse theory for cell complexes. Adv. Math., 134(1):90-145, 1998.
Ragnar Freij. Equivariant discrete Morse theory. Discrete Math., 309(12):3821-3829, 2009.
Leonidas Guibas, Dmitriy Morozov, and Quentin Mérigot. Witnessed k-distance. Discrete Comput. Geom., 49(1):22-45, 2013.
Dmitry Krasnoshchekov and Valentin Polishchuk. Order-k α-hulls and α-shapes. Inform. Process. Lett., 114(1-2):76-83, 2014.
Der-Tsai Lee et al. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., 31(6):478-487, 1982.
Jean Leray. Sur la forme des espaces topologiques et sur les points fixes des représentations. J. Math. Pures Appl. (9), 24:95-167, 1945.
Michael Lesnick and Matthew Wright. Interactive visualization of 2-D persistence modules. arXiv preprint arXiv:1512.00180, 2015.
Ketan Mulmuley. Output sensitive construction of levels and Voronoi diagrams in ℝ^d of order 1 to k. In Proceedings of the twenty-second annual ACM symposium on Theory of computing, pages 322-330. ACM, 1990.
Michael Ian Shamos and Dan Hoey. Closest-point problems. In 16th Annual Symposium on Foundations of Computer Science (Berkeley, Calif., 1975), pages 151-162. IEEE Computer Society, Long Beach, Calif., 1975.
Donald R. Sheehy. A multi-cover nerve for geometric inference. In Proc. Canadian Conf. Comput. Geom., 2012.
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Smallest Enclosing Spheres and Chernoff Points in BregmanGeometry
Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex.
Bregman divergence
smallest enclosing spheres
Chernoff points
convexity
barycenter polytopes
35:1-35:13
Regular Paper
Herbert
Edelsbrunner
Herbert Edelsbrunner
Ziga
Virk
Ziga Virk
Hubert
Wagner
Hubert Wagner
10.4230/LIPIcs.SoCG.2018.35
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Near Isometric Terminal Embeddings for Doubling Metrics
Given a metric space (X,d), a set of terminals K subseteq X, and a parameter t >= 1, we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in K x X up to a factor of t, and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist in several settings, no terminal spanner or embedding with distortion close to 1, i.e., t=1+epsilon for some small 0<epsilon<1, is currently known.
Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion 1+epsilon and size s(|X|) has its terminal counterpart, with distortion 1+O(epsilon) and size s(|K|)+1. In particular, for any doubling metric on n points, a set of k=o(n) terminals, and constant 0<epsilon<1, there exists
- A spanner with stretch 1+epsilon for pairs in K x X, with n+o(n) edges.
- A labeling scheme with stretch 1+epsilon for pairs in K x X, with label size ~~ log k.
- An embedding into l_infty^d with distortion 1+epsilon for pairs in K x X, where d=O(log k). Moreover, surprisingly, the last two results apply if only K is a doubling metric, while X can be arbitrary.
metric embedding
spanners
doubling metrics
36:1-36:15
Regular Paper
Michael
Elkin
Michael Elkin
Ofer
Neiman
Ofer Neiman
10.4230/LIPIcs.SoCG.2018.36
Amir Abboud and Greg Bodwin. Reachability preservers: New extremal bounds and approximation algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1865-1883, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.122.
http://dx.doi.org/10.1137/1.9781611975031.122
I. Althöfer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9:81-100, 1993.
P. Assouad. Plongements lipschitziens dans ℝⁿ. Bull. Soc. Math. France, 111(4):429-448, 1983.
Y. Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Proceedings of the 37th IEEE Symp. on Foundations of Computer Science, pages 184-193, 1996.
J. Bourgain. On lipschitz embedding of finite metric spaces in hilbert space. Israel Journal of Mathematics, 52(1-2):46-52, 1985. URL: http://dx.doi.org/10.1007/BF02776078.
http://dx.doi.org/10.1007/BF02776078
T-H. Hubert Chan and Anupam Gupta. Small hop-diameter sparse spanners for doubling metrics. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, SODA '06, pages 70-78, Philadelphia, PA, USA, 2006. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=1109557.1109566.
http://dl.acm.org/citation.cfm?id=1109557.1109566
T.-H. Hubert Chan, Anupam Gupta, Bruce M. Maggs, and Shuheng Zhou. On hierarchical routing in doubling metrics. ACM Trans. Algorithms, 12(4):55:1-55:22, aug 2016. URL: http://dx.doi.org/10.1145/2915183.
http://dx.doi.org/10.1145/2915183
T.-H. Hubert Chan, Mingfei Li, Li Ning, and Shay Solomon. New doubling spanners: Better and simpler. SIAM J. Comput., 44(1):37-53, 2015. URL: http://dx.doi.org/10.1137/130930984.
http://dx.doi.org/10.1137/130930984
Barun Chandra, Gautam Das, Giri Narasimhan, and José Soares. New sparseness results on graph spanners. In Proc. of 8th SOCG, pages 192-201, 1992.
D. Coppersmith and M. Elkin. Sparse source-wise and pair-wise distance preservers. In SODA: ACM-SIAM Symposium on Discrete Algorithms, pages 660-669, 2005.
Marek Cygan, Fabrizio Grandoni, and Telikepalli Kavitha. On pairwise spanners. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, pages 209-220, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.209.
http://dx.doi.org/10.4230/LIPIcs.STACS.2013.209
Gautam Das, Paul J. Heffernan, and Giri Narasimhan. Optimally sparse spanners in 3-dimensional euclidean space. In Proceedings of the Ninth Annual Symposium on Computational GeometrySan Diego, CA, USA, May 19-21, 1993, pages 53-62, 1993. URL: http://dx.doi.org/10.1145/160985.160998.
http://dx.doi.org/10.1145/160985.160998
Michael Elkin, Arnold Filtser, and Ofer Neiman. Prioritized metric structures and embedding. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 489-498, 2015. URL: http://dx.doi.org/10.1145/2746539.2746623.
http://dx.doi.org/10.1145/2746539.2746623
Michael Elkin, Arnold Filtser, and Ofer Neiman. Terminal embeddings. Theor. Comput. Sci., 697:1-36, 2017. URL: http://dx.doi.org/10.1016/j.tcs.2017.06.021.
http://dx.doi.org/10.1016/j.tcs.2017.06.021
Michael Elkin and Shay Solomon. Optimal euclidean spanners: Really short, thin, and lanky. J. ACM, 62(5):35:1-35:45, 2015. URL: http://dx.doi.org/10.1145/2819008.
http://dx.doi.org/10.1145/2819008
Jie Gao, Leonidas J. Guibas, and An Nguyen. Deformable spanners and applications. Comput. Geom. Theory Appl., 35(1-2):2-19, 2006. URL: http://dx.doi.org/10.1016/j.comgeo.2005.10.001.
http://dx.doi.org/10.1016/j.comgeo.2005.10.001
Lee-Ad Gottlieb. A light metric spanner. In Proc. of 56th FOCS, pages 759-772, 2015.
Lee-Ad Gottlieb and Liam Roditty. An optimal dynamic spanner for doubling metric spaces. In Algorithms - ESA 2008, 16th Annual European Symposium, Karlsruhe, Germany, September 15-17, 2008. Proceedings, pages 478-489, 2008. URL: http://dx.doi.org/10.1007/978-3-540-87744-8_40.
http://dx.doi.org/10.1007/978-3-540-87744-8_40
Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS '03, pages 534-, Washington, DC, USA, 2003. IEEE Computer Society. URL: http://portal.acm.org/citation.cfm?id=946243.946308.
http://portal.acm.org/citation.cfm?id=946243.946308
Sariel Har-Peled and Manor Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM J. Comput., 35(5):1148-1184, 2006. URL: http://dx.doi.org/10.1137/S0097539704446281.
http://dx.doi.org/10.1137/S0097539704446281
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Telikepalli Kavitha. New pairwise spanners. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, pages 513-526, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.513.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.513
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J. Matoušek. On the distortion required for embeding finite metric spaces into normed spaces. Israel Journal of Math, 93:333-344, 1996.
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Manor Mendel and Assaf Naor. Ramsey partitions and proximity data structures. Journal of the European Mathematical Society, 9(2):253-275, 2007.
Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, New York, NY, USA, 2007.
Ofer Neiman. Low dimensional embeddings of doubling metrics. Theory Comput. Syst., 58(1):133-152, 2016. URL: http://dx.doi.org/10.1007/s00224-014-9567-3.
http://dx.doi.org/10.1007/s00224-014-9567-3
Merav Parter. Bypassing erdős' girth conjecture: Hybrid stretch and sourcewise spanners. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, pages 608-619, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43951-7_49.
http://dx.doi.org/10.1007/978-3-662-43951-7_49
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http://dx.doi.org/10.1007/s00446-006-0015-8
Kunal Talwar. Bypassing the embedding: Algorithms for low dimensional metrics. In Proceedings of the Thirty-sixth Annual ACM Symposium on Theory of Computing, STOC '04, pages 281-290, New York, NY, USA, 2004. ACM. URL: http://dx.doi.org/10.1145/1007352.1007399.
http://dx.doi.org/10.1145/1007352.1007399
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Products of Euclidean Metrics and Applications to Proximity Questions among Curves
The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments and molecular backbones to time-series in general dimension. For l_p-products of Euclidean metrics, for any p >= 1, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fréchet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms; our approach is especially efficient when the length of the curves is bounded.
Approximate nearest neighbor
polygonal curves
Fréchet distance
dynamic time warping
37:1-37:13
Regular Paper
Ioannis Z.
Emiris
Ioannis Z. Emiris
Ioannis
Psarros
Ioannis Psarros
10.4230/LIPIcs.SoCG.2018.37
E. Anagnostopoulos, I. Z. Emiris, and I. Psarros. Low-quality dimension reduction and high-dimensional approximate nearest neighbor. In Proc. 31st Intern. Symp. on Computational Geometry (SoCG), pages 436-450, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.436.
http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.436
E. Anagnostopoulos, I. Z. Emiris, and I. Psarros. Randomized embeddings with slack, and high-dimensional approximate nearest neighbor. In ACM Transactions on Algorithm, 2018, To appear.
A. Andoni. NN search: the old, the new, and the impossible. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2009. URL: http://hdl.handle.net/1721.1/55090.
http://hdl.handle.net/1721.1/55090
A. Andoni, T. Laarhoven, I. Razenshteyn, and E. Waingarten. Optimal hashing-based time-space trade-offs for approximate near neighbors. In Proc. Symp. Discrete Algorithms (SODA), 2017. Also as arxiv.org/abs/1608.03580.
S. Arya, T. Malamatos, and D. M. Mount. Space-time tradeoffs for approximate nearest neighbor searching. J. ACM, 57(1):1:1-1:54, 2009. URL: http://dx.doi.org/10.1145/1613676.1613677.
http://dx.doi.org/10.1145/1613676.1613677
Y. Bartal and L. -A. Gottlieb. Dimension reduction techniques for l_p (1lesspless2), with applications. In 32nd International Symposium on Computational Geometry, SoCG 2016, June 14-18, 2016, Boston, MA, USA, pages 16:1-16:15, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.16.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.16
A. Driemel and F. Silvestri. Locality-sensitive hashing of curves. In Proc. 33rd Intern. Symposium on Computational Geometry, pages 37:1-37:16, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.37.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.37
S. Har-Peled, P. Indyk, and R. Motwani. Approximate nearest neighbor: Towards removing the curse of dimensionality. Theory of Computing, 8(1):321-350, 2012. URL: http://dx.doi.org/10.4086/toc.2012.v008a014.
http://dx.doi.org/10.4086/toc.2012.v008a014
S. Har-Peled and M. Mendel. Fast construction of nets in low dimensional metrics, and their applications. In Proc. 21st Annual Symp. Computational Geometry, SCG'05, pages 150-158, 2005. URL: http://dx.doi.org/10.1145/1064092.1064117.
http://dx.doi.org/10.1145/1064092.1064117
P. Indyk. Approximate nearest neighbor algorithms for frechet distance via product metrics. In Proc. 18th Annual Symp. on Computational Geometry, SCG '02, pages 102-106, New York, NY, USA, 2002. ACM. URL: http://dx.doi.org/10.1145/513400.513414.
http://dx.doi.org/10.1145/513400.513414
J. Matoušek. On variants of the Johnson-Lindenstrauss lemma. Random Struct. Algorithms, 33(2):142-156, 2008. URL: http://dx.doi.org/10.1002/rsa.v33:2.
http://dx.doi.org/10.1002/rsa.v33:2
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Rainbow Cycles in Flip Graphs
The flip graph of triangulations has as vertices all triangulations of a convex n-gon, and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of r-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex n-gon, the flip graph of plane spanning trees on an arbitrary set of n points, and the flip graph of non-crossing perfect matchings on a set of n points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of {1,2,...,n } and the flip graph of k-element subsets of {1,2,...,n }. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of r, n and k.
flip graph
cycle
rainbow
Gray code
triangulation
spanning tree
matching
permutation
subset
combination
38:1-38:14
Regular Paper
Stefan
Felsner
Stefan Felsner
Linda
Kleist
Linda Kleist
Torsten
Mütze
Torsten Mütze
Leon
Sering
Leon Sering
10.4230/LIPIcs.SoCG.2018.38
O. Aichholzer, F. Aurenhammer, C. Huemer, and B. Vogtenhuber. Gray code enumeration of plane straight-line graphs. Graphs Combin., 23(5):467-479, 2007. URL: http://dx.doi.org/10.1007/s00373-007-0750-z.
http://dx.doi.org/10.1007/s00373-007-0750-z
N. Alon, A. Pokrovskiy, and B. Sudakov. Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles. arXiv:1608.07028, Aug 2016.
B. Alspach. The wonderful Walecki construction. Bull. Inst. Combin. Appl., 52:7-20, 2008.
L. D. Andersen. Hamilton circuits with many colours in properly edge-coloured complete graphs. Mathematica Scandinavica, 64:5-14, 1989.
J. Balogh and T. Molla. Long rainbow cycles and Hamiltonian cycles using many colors in properly edge-colored complete graphs. arXiv:1706.04950, Jun 2017.
G. S. Bhat and C. D. Savage. Balanced Gray codes. Electron. J. Combin., 3(1):Research Paper 25, approx. 11 pp., 1996. URL: http://www.combinatorics.org/Volume_3/Abstracts/v3i1r25.html.
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P. Bose and F. Hurtado. Flips in planar graphs. Comput. Geom., 42(1):60-80, 2009. URL: http://dx.doi.org/10.1016/j.comgeo.2008.04.001.
http://dx.doi.org/10.1016/j.comgeo.2008.04.001
M. Buck and D. Wiedemann. Gray codes with restricted density. Discrete Math., 48(2-3):163-171, 1984. URL: http://dx.doi.org/10.1016/0012-365X(84)90179-1.
http://dx.doi.org/10.1016/0012-365X(84)90179-1
C. Ceballos, F. Santos, and G. M. Ziegler. Many non-equivalent realizations of the associahedron. Combinatorica, 35(5):513-551, 2015. URL: http://dx.doi.org/10.1007/s00493-014-2959-9.
http://dx.doi.org/10.1007/s00493-014-2959-9
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http://dx.doi.org/10.1145/2422.322413
D. Eppstein. Happy endings for flip graphs. J. Comput. Geom., 1(1):3-28, 2010.
R. Fabila-Monroy, D. Flores-Peñaloza, C. Huemer, F. Hurtado, J. Urrutia, and D. R. Wood. On the chromatic number of some flip graphs. Discrete Math. Theor. Comput. Sci., 11(2):47-56, 2009.
S. Felsner, L. Kleist, T. Mütze, and L. Sering. Rainbow cycles in flip graphs. arXiv:1712. 07421, Dec 2017.
P. Gregor, T. Mütze, and J. Nummenpalo. A short proof of the middle levels theorem. arXiv:1710.08249, Oct 2017.
S. Hanke, T. Ottmann, and S. Schuierer. The edge-flipping distance of triangulations. Journal of Universal Computer Science, 2(8):570-579, 1996.
C. Hernando, F. Hurtado, and M. Noy. Graphs of non-crossing perfect matchings. Graphs Combin., 18(3):517-532, 2002. URL: http://dx.doi.org/10.1007/s003730200038.
http://dx.doi.org/10.1007/s003730200038
C. Huemer, F. Hurtado, M. Noy, and E. Omaña-Pulido. Gray codes for non-crossing partitions and dissections of a convex polygon. Discrete Appl. Math., 157(7):1509-1520, 2009. URL: http://dx.doi.org/10.1016/j.dam.2008.06.018.
http://dx.doi.org/10.1016/j.dam.2008.06.018
F. Hurtado and M. Noy. Graph of triangulations of a convex polygon and tree of triangulations. Comput. Geom., 13(3):179-188, 1999. URL: http://dx.doi.org/10.1016/S0925-7721(99)00016-4.
http://dx.doi.org/10.1016/S0925-7721(99)00016-4
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http://dx.doi.org/10.1007/PL00009464
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C. W. Lee. The associahedron and triangulations of the n-gon. European J. Combin., 10(6):551-560, 1989. URL: http://dx.doi.org/10.1016/S0195-6698(89)80072-1.
http://dx.doi.org/10.1016/S0195-6698(89)80072-1
M. Li and L. Zhang. Better approximation of diagonal-flip transformation and rotation transformation. In Computing and Combinatorics, 4th Annual International Conference, COCOON '98, Taipei, Taiwan, R.o.C., August 12-14, 1998, Proceedings, pages 85-94, 1998. URL: http://dx.doi.org/10.1007/3-540-68535-9_12.
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T. Mütze. Proof of the middle levels conjecture. Proc. Lond. Math. Soc. (3), 112(4):677-713, 2016. URL: http://dx.doi.org/10.1112/plms/pdw004.
http://dx.doi.org/10.1112/plms/pdw004
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http://dx.doi.org/10.1016/j.aim.2014.02.035
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http://dx.doi.org/10.1137/1.9781611970166
Creative Commons Attribution 3.0 Unported license
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Hanani-Tutte for Approximating Maps of Graphs
We resolve in the affirmative conjectures of A. Skopenkov and Repovs (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing whether a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise disjoint "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.
Hanani-Tutte theorem
graph embedding
map approximation
weak embedding
clustered planarity
39:1-39:15
Regular Paper
Radoslav
Fulek
Radoslav Fulek
Jan
Kyncl
Jan Kyncl
10.4230/LIPIcs.SoCG.2018.39
Hugo A. Akitaya, Greg Aloupis, Jeff Erickson, and Csaba Tóth. Recognizing weakly simple polygons. In 32nd International Symposium on Computational Geometry (SoCG 2016), volume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:16, 2016.
Hugo A. Akitaya, Radoslav Fulek, and Csaba D. Tóth. Recognizing weak embeddings of graphs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 274-292, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.20.
http://dx.doi.org/10.1137/1.9781611975031.20
Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, and Fabrizio Frati. Strip planarity testing for embedded planar graphs. Algorithmica, 77(4):1022-1059, 2017.
Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Ignaz Rutter. Beyond Level Planarity, pages 482-495. Springer International Publishing, Cham, 2016.
Patrizio Angelini and Giordano Da Lozzo. Clustered Planarity with Pipes. In Seok-Hee Hong, editor, 27th International Symposium on Algorithms and Computation (ISAAC 2016), volume 64 of Leibniz International Proceedings in Informatics (LIPIcs), pages 13:1-13:13, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2016.13.
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Therese C. Biedl. Drawing planar partitions III: Two constrained embedding problems, 1998.
Hsien-Chih Chang, Jeff Erickson, and Chao Xu. Detecting weakly simple polygons. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1655-1670. SIAM, Philadelphia, PA, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.110.
http://dx.doi.org/10.1137/1.9781611973730.110
F. Cortese and G. Di Battista. Clustered planarity (invited lecture). In Twenty-first annual symposium on Computational Geometry (proc. SoCG 05), pages 30-32, 2005.
Pier Francesco Cortese, Giuseppe Di Battista, Maurizio Patrignani, and Maurizio Pizzonia. On embedding a cycle in a plane graph. Discrete Math., 309(7):1856-1869, 2009. URL: http://dx.doi.org/10.1016/j.disc.2007.12.090.
http://dx.doi.org/10.1016/j.disc.2007.12.090
Qing-Wen Feng, Robert F. Cohen, and Peter Eades. How to draw a planar clustered graph. In Computing and combinatorics (Xi'an, 1995), volume 959 of Lecture Notes in Comput. Sci., pages 21-30. Springer, Berlin, 1995. URL: http://dx.doi.org/10.1007/BFb0030816.
http://dx.doi.org/10.1007/BFb0030816
Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. In Algorithms - ESA '95, volume 979 of Lecture Notes in Comput. Sci., pages 213-226. Springer Berlin Heidelberg, 1995.
Radoslav Fulek. Embedding graphs into embedded graphs. In 28th International Symposium on Algorithms and Computation, ISAAC 2017, December 9-12, 2017, Phuket, Thailand, pages 34:1-34:12, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.34.
http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.34
Radoslav Fulek, Jan Kynčl, Igor Malinović, and Dömötör Pálvölgyi. Clustered planarity testing revisited. Electron. J. Combin., 22(4):Paper 4.24, 29 pp., 2015.
Radoslav Fulek, Jan Kynčl, and Dömötör Pálvölgyi. Unified Hanani-Tutte theorem. Electr. J. Comb., 24(3):P3.18, 2017. URL: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p18.
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i3p18
Radoslav Fulek, Michael Pelsmajer, and Marcus Schaefer. Hanani-Tutte for radial planarity II. In Yifan Hu and Martin Nöllenburg, editors, Graph Drawing and Network Visualization: 24th International Symposium, GD 2016, Athens, Greece, September 19-21, 2016, Revised Selected Papers, pages 468-481, Cham, 2016. Springer International Publishing.
Radoslav Fulek, Michael Pelsmajer, and Marcus Schaefer. Hanani-Tutte for radial planarity. Journal of Graph Algorithms and Applications, 21(1):135-154, 2017.
Radoslav Fulek, Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Hanani-Tutte, monotone drawings, and level-planarity. In Thirty essays on geometric graph theory, pages 263-287. Springer, New York, 2013. URL: http://dx.doi.org/10.1007/978-1-4614-0110-0_14.
http://dx.doi.org/10.1007/978-1-4614-0110-0_14
François Le Gall. Powers of tensors and fast matrix multiplication. http://arxiv.org/abs/1401.7714, 2014.
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Carsten Gutwenger, Michael Jünger, Sebastian Leipert, Petra Mutzel, Merijam Percan, and René Weiskircher. Advances in c-planarity testing of clustered graphs. In Michael T. Goodrich and Stephen G. Kobourov, editors, Graph Drawing: 10th International Symposium, GD 2002 Irvine, CA, USA, August 26-28, 2002 Revised Papers, pages 220-236, Berlin, Heidelberg, 2002. Springer Berlin Heidelberg. URL: http://dx.doi.org/10.1007/3-540-36151-0_21.
http://dx.doi.org/10.1007/3-540-36151-0_21
Haim Hanani. Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae, 23:135-142, 1934.
Seok hee Hong and Hiroshi Nagamochi. Two-page book embedding and clustered graph planarity. Theoretical Computing Science, 2016.
John Hopcroft and Robert Tarjan. Efficient planarity testing. J. ACM, 21(4):549-568, 1974.
Piotr Minc. Embedding of simplicial arcs into the plane. Topology Proc., 22(Summer):305-340, 1997.
Bojan Mohar. A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Discrete Math., 12(1):6-26, 1999.
Bojan Mohar and Carsten Thomassen. Graphs on surfaces. Johns Hopkins University Pres, 2001.
Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Removing even crossings. J. Combin. Theory Ser. B, 97(4):489-500, 2007.
Dušan Repovš and Arkadij B. Skopenkov. A deleted product criterion for approximability of maps by embeddings. Topology Appl., 87(1):1-19, 1998. URL: http://dx.doi.org/10.1016/S0166-8641(97)00121-1.
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Marcus Schaefer. Hanani-Tutte and related results. In Geometry - intuitive, discrete, and convex, volume 24 of Bolyai Soc. Math. Stud., pages 259-299. János Bolyai Math. Soc., Budapest, 2013.
Marcus Schaefer. Toward a theory of planarity: Hanani-Tutte and planarity variants. J. Graph Algorithms Appl., 17(4):367-440, 2013. URL: http://dx.doi.org/10.7155/jgaa.00298.
http://dx.doi.org/10.7155/jgaa.00298
Mikhail Skopenkov. On approximability by embeddings of cycles in the plane. Topology Appl., 134(1):1-22, 2003. URL: http://dx.doi.org/10.1016/S0166-8641(03)00069-5.
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The Z_2-Genus of Kuratowski Minors
A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G is the minimum g such that G has an independently even drawing on the orientable surface of genus g. An unpublished result by Robertson and Seymour implies that for every t, every graph of sufficiently large genus contains as a minor a projective t x t grid or one of the following so-called t-Kuratowski graphs: K_{3,t}, or t copies of K_5 or K_{3,3} sharing at most 2 common vertices. We show that the Z_2-genus of graphs in these families is unbounded in t; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its Z_2-genus, solving a problem posed by Schaefer and Stefankovic, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces.
Hanani-Tutte theorem
genus of a graph
Z_2-genus of a graph
Kuratowski graph
40:1-40:14
Regular Paper
Radoslav
Fulek
Radoslav Fulek
Jan
Kyncl
Jan Kyncl
10.4230/LIPIcs.SoCG.2018.40
Joseph Battle, Frank Harary, Yukihiro Kodama, and J. W. T. Youngs. Additivity of the genus of a graph. Bull. Amer. Math. Soc., 68:565-568, 1962. URL: http://dx.doi.org/10.1090/S0002-9904-1962-10847-7.
http://dx.doi.org/10.1090/S0002-9904-1962-10847-7
Thomas Böhme, Ken-ichi Kawarabayashi, John Maharry, and Bojan Mohar. K_3,k-minors in large 7-connected graphs. Preprint available at http://preprinti.imfm.si/PDF/01051.pdf, 2008.
http://preprinti.imfm.si/PDF/01051.pdf
Thomas Böhme, Ken-ichi Kawarabayashi, John Maharry, and Bojan Mohar. Linear connectivity forces large complete bipartite minors. J. Combin. Theory Ser. B, 99(3):557-582, 2009. URL: http://dx.doi.org/10.1016/j.jctb.2008.07.006.
http://dx.doi.org/10.1016/j.jctb.2008.07.006
André Bouchet. Orientable and nonorientable genus of the complete bipartite graph. J. Combin. Theory Ser. B, 24(1):24-33, 1978. URL: http://dx.doi.org/10.1016/0095-8956(78)90073-4.
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G. Cairns and Y. Nikolayevsky. Bounds for generalized thrackles. Discrete Comput. Geom., 23(2):191-206, 2000. URL: http://dx.doi.org/10.1007/PL00009495.
http://dx.doi.org/10.1007/PL00009495
R. Christian, R. B. Richter, and G. Salazar. Embedding a graph-like continuum in some surface. J. Graph Theory, 79(2):159-165, 2015. URL: http://dx.doi.org/10.1002/jgt.21823.
http://dx.doi.org/10.1002/jgt.21823
Éric Colin de Verdière, Vojtěch Kaluža, Pavel Paták, Zuzana Patáková, and Martin Tancer. A direct proof of the strong Hanani-Tutte theorem on the projective plane. J. Graph Algorithms Appl., 21(5):939-981, 2017. URL: http://dx.doi.org/10.7155/jgaa.00445.
http://dx.doi.org/10.7155/jgaa.00445
D. de Caen. The ranks of tournament matrices. Amer. Math. Monthly, 98(9):829-831, 1991. URL: http://dx.doi.org/10.2307/2324270.
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R. W. Decker, H. H. Glover, and J. P. Huneke. Computing the genus of the 2-amalgamations of graphs. Combinatorica, 5(4):271-282, 1985. URL: http://dx.doi.org/10.1007/BF02579241.
http://dx.doi.org/10.1007/BF02579241
J. R. Fiedler, J. P. Huneke, R. B. Richter, and N. Robertson. Computing the orientable genus of projective graphs. J. Graph Theory, 20(3):297-308, 1995. URL: http://dx.doi.org/10.1002/jgt.3190200305.
http://dx.doi.org/10.1002/jgt.3190200305
R. Fulek and J. Kynčl. Counterexample to an extension of the hanani-tutte theorem on the surface of genus 4. Submitted, https://arxiv.org/abs/1709.00508, 2017.
https://arxiv.org/abs/1709.00508
R. Fulek and J. Kynčl. The ℤ₂-genus of kuratowski minors. Manuscript, , 2018.
Haim Hanani. Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae, 23:135-142, 1934.
Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.
D. J. Kleitman. A note on the parity of the number of crossings of a graph. J. Combinatorial Theory Ser. B, 21(1):88-89, 1976.
Bojan Mohar and Carsten Thomassen. Graphs on surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, 2001.
Michael J. Pelsmajer, Marcus Schaefer, and Despina Stasi. Strong Hanani-Tutte on the projective plane. SIAM J. Discrete Math., 23(3):1317-1323, 2009. URL: http://dx.doi.org/10.1137/08072485X.
http://dx.doi.org/10.1137/08072485X
Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Removing even crossings on surfaces. European J. Combin., 30(7):1704-1717, 2009. URL: http://dx.doi.org/10.1016/j.ejc.2009.03.002.
http://dx.doi.org/10.1016/j.ejc.2009.03.002
Gerhard Ringel. Das Geschlecht des vollständigen paaren Graphen. Abh. Math. Sem. Univ. Hamburg, 28:139-150, 1965. URL: http://dx.doi.org/10.1007/BF02993245.
http://dx.doi.org/10.1007/BF02993245
Neil Robertson and Richard Vitray. Representativity of surface embeddings. In Paths, flows, and VLSI-layout (Bonn, 1988), volume 9 of Algorithms Combin., pages 293-328. Springer, Berlin, 1990.
Marcus Schaefer. Hanani-Tutte and related results. In Geometry - intuitive, discrete, and convex, volume 24 of Bolyai Soc. Math. Stud., pages 259-299. János Bolyai Math. Soc., Budapest, 2013. URL: http://dx.doi.org/10.1007/978-3-642-41498-5_10.
http://dx.doi.org/10.1007/978-3-642-41498-5_10
Marcus Schaefer and Daniel Štefankovič. Block additivity of ℤ₂-embeddings. In Graph drawing, volume 8242 of Lecture Notes in Comput. Sci., pages 185-195. Springer, Cham, 2013. URL: http://dx.doi.org/10.1007/978-3-319-03841-4_17.
http://dx.doi.org/10.1007/978-3-319-03841-4_17
Paul Seymour, 2017. Personal communication.
W. T. Tutte. Toward a theory of crossing numbers. J. Combinatorial Theory, 8:45-53, 1970.
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Shellability is NP-Complete
We prove that for every d >= 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d >= 2 and k >= 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d >= 3, both problems remain NP-hard when restricted to contractible pure d-dimensional complexes.
Shellability
simplicial complexes
NP-completeness
collapsibility
41:1-41:15
Regular Paper
Xavier
Goaoc
Xavier Goaoc
Pavel
Paták
Pavel Paták
Zuzana
Patáková
Zuzana Patáková
Martin
Tancer
Martin Tancer
Uli
Wagner
Uli Wagner
10.4230/LIPIcs.SoCG.2018.41
S. Arora and B. Barak. Complexity Theory: A Modern Approach. Cambridge University Press, Cambridge, 2009. URL: http://www.cs.princeton.edu/theory/complexity/.
http://www.cs.princeton.edu/theory/complexity/
D. Attali, O. Devillers, M. Glisse, and S. Lazard. Recognizing shrinkable complexes is NP-complete. Journal of Computational Geometry, 7(1):430-443, 2016.
R. H. Bing. The geometric topology of 3-manifolds, volume 40 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1983. URL: http://dx.doi.org/10.1090/coll/040.
http://dx.doi.org/10.1090/coll/040
A. Björner. Shellable and Cohen-Macaulay partially ordered sets. Trans. Amer. Math. Soc., 260(1):159-183, 1980. URL: http://dx.doi.org/10.2307/1999881.
http://dx.doi.org/10.2307/1999881
A. Björner. Topological methods. In Handbook of combinatorics, Vol. 1, 2, pages 1819-1872. Elsevier Sci. B. V., Amsterdam, 1995.
A. Björner and M. Wachs. Bruhat order of Coxeter groups and shellability. Advances in Mathematics, 43(1):87-100, 1982. URL: http://dx.doi.org/10.1016/0001-8708(82)90029-9.
http://dx.doi.org/10.1016/0001-8708(82)90029-9
A. Björner and M. Wachs. On lexicographically shellable posets. Trans. Amer. Math. Soc., 277(1):323-341, 1983. URL: http://dx.doi.org/10.2307/1999359.
http://dx.doi.org/10.2307/1999359
A. Björner and M. L. Wachs. Shellable nonpure complexes and posets. II. Trans. Amer. Math. Soc., 349(10):3945-3975, 1997. URL: http://dx.doi.org/10.1090/S0002-9947-97-01838-2.
http://dx.doi.org/10.1090/S0002-9947-97-01838-2
H. Bruggesser and P. Mani. Shellable decompositions of cells and spheres. Mathematica Scandinavica, 29(2):197-205, 1972.
G. Danaraj and V. Klee. Shellings of spheres and polytopes. Duke Mathematical Journal, 41(2):443-451, 1974.
G. Danaraj and V. Klee. A representation of 2-dimensional pseudomanifolds and its use in the design of a linear-time shelling algorithm. Ann. Discrete Math., 2:53-63, 1978. Algorithmic aspects of combinatorics (Conf., Vancouver Island, B.C., 1976).
G. Danaraj and V. Klee. Which spheres are shellable? Ann. Discrete Math., 2:33-52, 1978. Algorithmic aspects of combinatorics (Conf., Vancouver Island, B.C., 1976).
Ö. Eğecioğlu and T. F. Gonzalez. A computationally intractable problem on simplicial complexes. Comput. Geom., 6(2):85-98, 1996. URL: http://dx.doi.org/10.1016/0925-7721(95)00015-1.
http://dx.doi.org/10.1016/0925-7721(95)00015-1
X. Goaoc, P. Paták, Z. Patáková, M. Tancer, and U. Wagner. Shellability is NP-complete, 2018. Preprint, URL: https://arxiv.org/abs/1711.08436.
https://arxiv.org/abs/1711.08436
B. Grünbaum. Convex polytopes, volume 221 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. URL: http://dx.doi.org/10.1007/978-1-4613-0019-9.
http://dx.doi.org/10.1007/978-1-4613-0019-9
M. Hachimori. Decompositions of two-dimensional simplicial complexes. Discrete Math., 308(11):2307-2312, 2008.
M. Joswig and M. E. Pfetsch. Computing optimal Morse matchings. SIAM Journal on Discrete Mathematics, 20(1):11-25, 2006.
V. Kaibel and M. E. Pfetsch. Some algorithmic problems in polytope theory. In Algebra, geometry, and software systems, pages 23-47. Springer, Berlin, 2003.
T. Lewiner, H. Lopes, and G. Tavares. Optimal discrete Morse functions for 2-manifolds. Comput. Geom., 26(3):221-233, 2003. URL: http://dx.doi.org/10.1016/S0925-7721(03)00014-2.
http://dx.doi.org/10.1016/S0925-7721(03)00014-2
R. Malgouyres and A. R. Francés. Determining whether a simplicial 3-complex collapses to a 1-complex is NP-complete. DGCI, pages 177-188, 2008.
J. Matoušek. Using the Borsuk-Ulam theorem. Universitext. Springer-Verlag, Berlin, 2007.
P. McMullen. The maximum numbers of faces of a convex polytope. Mathematika, 17(2):179-184, 1970.
A. Nabutovsky. Einstein structures: Existence versus uniqueness. Geom. Funct. Anal., 5(1):76-91, 1995.
I. Peeva, V. Reiner, and B. Sturmfels. How to shell a monoid. Math. Ann., 310(2):379-393, 1998.
J. S. Provan and L. J. Billera. Decompositions of simplicial complexes related to diameters of convex polyhedra. Math. Oper. Res., 5(4):576-594, 1980. URL: http://dx.doi.org/10.1287/moor.5.4.576.
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J. Shareshian. On the shellability of the order complex of the subgroup lattice of a finite group. Trans. Amer. Math. Soc., 353(7):2689-2703, 2001. URL: http://dx.doi.org/10.1090/S0002-9947-01-02730-1.
http://dx.doi.org/10.1090/S0002-9947-01-02730-1
R. P. Stanley. Combinatorics and commutative algebra, volume 41 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, second edition, 1996.
M. Tancer. Recognition of collapsible complexes is NP-complete. Discrete Comput. Geom., 55(1):21-38, 2016.
I.A. Volodin, V.E. Kuznetsov, and A.T. Fomenko. The problem of discriminating algorithmically the standard three-dimensional sphere. Usp. Mat. Nauk, 29(5):71-168, 1974. In Russian. English translation: Russ. Math. Surv. 29,5:71-172 (1974).
M. L. Wachs. Poset topology: Tools and applications. In Geometric combinatorics, volume 13 of IAS/Park City Math. Ser., pages 497-615. American Mathematical Soc., 2007.
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http://dx.doi.org/10.1112/plms/s2-45.1.243
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https://creativecommons.org/licenses/by/3.0/legalcode
Optimal Morphs of Planar Orthogonal Drawings
We describe an algorithm that morphs between two planar orthogonal drawings Gamma_I and Gamma_O of a connected graph G, while preserving planarity and orthogonality. Necessarily Gamma_I and Gamma_O share the same combinatorial embedding. Our morph uses a linear number of linear morphs (linear interpolations between two drawings) and preserves linear complexity throughout the process, thereby answering an open question from Biedl et al. [Biedl et al., 2013].
Our algorithm first unifies the two drawings to ensure an equal number of (virtual) bends on each edge. We then interpret bends as vertices which form obstacles for so-called wires: horizontal and vertical lines separating the vertices of Gamma_O. We can find corresponding wires in Gamma_I that share topological properties with the wires in Gamma_O. The structural difference between the two drawings can be captured by the spirality of the wires in Gamma_I, which guides our morph from Gamma_I to Gamma_O.
Homotopy
Morphing
Orthogonal drawing
Spirality
42:1-42:14
Regular Paper
Arthur
van Goethem
Arthur van Goethem
Kevin
Verbeek
Kevin Verbeek
10.4230/LIPIcs.SoCG.2018.42
Soroush Alamdari, Patrizio Angelini, Fidel Barrera-Cruz, Timothy Chan, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Penny Haxell, Anna Lubiw, Maurizio Patrignani, Vincenzo Roselli, Sahil Singla, and Bryan Wilkinson. How to morph planar graph drawings. SIAM Journal on Computing, 46(2):824-852, 2017.
Greg Aloupis, Luis Barba, Paz Carmi, Vida Dujmović, Fabrizio Frati, and Pat Morin. Compatible connectivity augmentation of planar disconnected graphs. Discrete & Computational Geometry, 54(2):459-480, 2015.
Patrizio Angelini, Fabrizio Frati, Maurizio Patrignani, and Vincenzo Roselli. Morphing planar graph drawings efficiently. In Proc. 21st International Symposium on Graph Drawing, pages 49-60, 2013.
Therese Biedl, Anna Lubiw, Mark Petrick, and Michael Spriggs. Morphing orthogonal planar graph drawings. ACM Transactions on Algorithms, 9(4):29:1-29:24, 2013.
Therese Biedl, Anna Lubiw, and Michael Spriggs. Morphing planar graphs while preserving edge directions. In Proc. 13th International Symposium on Graph Drawing, pages 13-24, 2006.
Thomas Bläsius, Sebastian Lehmann, and Ignaz Rutter. Orthogonal graph drawing with inflexible edges. Computational Geometry, 55:26-40, 2016.
Steward Cairns. Deformations of plane rectilinear complexes. The American Mathematical Monthly, 51(5):247-252, 1944.
Giuseppe Di Battista, Giuseppe Liotta, and Francesco Vargiu. Spirality and optimal orthogonal drawings. SIAM Journal on Computing, 27(6):1764-1811, 1998.
Walter Didimo, Francesco Giordano, and Giuseppe Liotta. Upward spirality and upward planarity testing. SIAM Journal on Discrete Mathematics, 23(4):1842-1899, 2009.
Michael Freedman, Joel Hass, and Peter Scott. Closed geodesics on surfaces. Bulletin of the London Mathematical Society, 14(5):385-391, 1982.
Bettina Speckmann and Kevin Verbeek. Homotopic c-oriented routing with few links and thick edges. Computational Geometry, 67:11-28, 2018.
Carsten Thomassen. Deformations of plane graphs. Journal of Combinatorial Theory, Series B, 34(3):244-257, 1983.
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Computational Topology and the Unique Games Conjecture
Covering spaces of graphs have long been useful for studying expanders (as "graph lifts") and unique games (as the "label-extended graph"). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial's 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-O'Donnell, FOCS '04; SICOMP '07) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new "Unique Games-complete" problem in over a decade.
Our results partially settle an open question of Chen and Freedman (SODA, 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over Z_2, and we show Unique Games-completeness over Z_k for large k.) This equivalence comes from the fact that when the structure group G of the covering space is Abelian - or more generally for principal G-bundles - Maximum Section of a G-Covering Space is the same as the well-studied problem of 1-Homology Localization.
Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future.
Unique Games Conjecture
homology localization
inapproximability
computational topology
graph lift
covering graph
permutation voltage graph
cell
43:1-43:16
Regular Paper
Joshua A.
Grochow
Joshua A. Grochow
Jamie
Tucker-Foltz
Jamie Tucker-Foltz
10.4230/LIPIcs.SoCG.2018.43
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Solving Large-Scale Minimum-Weight Triangulation Instances to Provable Optimality
We consider practical methods for the problem of finding a minimum-weight triangulation (MWT) of a planar point set, a classic problem of computational geometry with many applications. While Mulzer and Rote proved in 2006 that computing an MWT is NP-hard, Beirouti and Snoeyink showed in 1998 that computing provably optimal solutions for MWT instances of up to 80,000 uniformly distributed points is possible, making use of clever heuristics that are based on geometric insights. We show that these techniques can be refined and extended to instances of much bigger size and different type, based on an array of modifications and parallelizations in combination with more efficient geometric encodings and data structures. As a result, we are able to solve MWT instances with up to 30,000,000 uniformly distributed points in less than 4 minutes to provable optimality. Moreover, we can compute optimal solutions for a vast array of other benchmark instances that are not uniformly distributed, including normally distributed instances (up to 30,000,000 points), all point sets in the TSPLIB (up to 85,900 points), and VLSI instances with up to 744,710 points. This demonstrates that from a practical point of view, MWT instances can be handled quite well, despite their theoretical difficulty.
computational geometry
minimum-weight triangulation
44:1-44:14
Regular Paper
Andreas
Haas
Andreas Haas
10.4230/LIPIcs.SoCG.2018.44
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https://creativecommons.org/licenses/by/3.0/legalcode
Dynamic Smooth Compressed Quadtrees
We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in R^d can be made smooth and dynamic subject to insertions and deletions of points. The second version uses the first but must additionally deal with compression and alignment of quadtree components. In both cases our updates take 2^{O(d log d)} time, except for the point location part in the second version which has a lower bound of Omega(log n); but if a pointer (finger) to the correct quadtree cell is given, the rest of the updates take worst-case constant time. Our result implies that several classic and recent results (ranging from ray tracing to planar point location) in computational geometry which use quadtrees can deal with arbitrary point sets on a real RAM pointer machine.
smooth
dynamic
data structure
quadtree
compression
alignment
Real RAM
45:1-45:15
Regular Paper
Ivor
van der Hoog
Ivor van der Hoog
Elena
Khramtcova
Elena Khramtcova
Maarten
Löffler
Maarten Löffler
10.4230/LIPIcs.SoCG.2018.45
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On the Treewidth of Triangulated 3-Manifolds
In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth.
In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs).
We derive these results from work of Agol and of Scharlemann and Thompson, by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 48(k+1) (resp. 4(3k+1)).
computational topology
triangulations of 3-manifolds
thin position
fixed-parameter tractability
congestion
treewidth
46:1-46:15
Regular Paper
Kristóf
Huszár
Kristóf Huszár
Jonathan
Spreer
Jonathan Spreer
Uli
Wagner
Uli Wagner
10.4230/LIPIcs.SoCG.2018.46
I. Agol. Small 3-manifolds of large genus. Geom. Dedicata, 102:53-64, 2003. URL: http://dx.doi.org/10.1023/B:GEOM.0000006584.85248.c5.
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http://dx.doi.org/10.4171/082
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On Partial Covering For Geometric Set Systems
We study a generalization of the Set Cover problem called the Partial Set Cover in the context of geometric set systems. The input to this problem is a set system (X, R), where X is a set of elements and R is a collection of subsets of X, and an integer k <= |X|. Each set in R has a non-negative weight associated with it. The goal is to cover at least k elements of X by using a minimum-weight collection of sets from R. The main result of this article is an LP rounding scheme which shows that the integrality gap of the Partial Set Cover LP is at most a constant times that of the Set Cover LP for a certain projection of the set system (X, R). As a corollary of this result, we get improved approximation guarantees for the Partial Set Cover problem for a large class of geometric set systems.
Partial Set Cover
Geometric Set Cover
47:1-47:14
Regular Paper
Tanmay
Inamdar
Tanmay Inamdar
Kasturi
Varadarajan
Kasturi Varadarajan
10.4230/LIPIcs.SoCG.2018.47
Boris Aronov, Esther Ezra, and Micha Sharir. Small-size ε-nets for axis-parallel rectangles and boxes. SIAM J. Comput., 39(7):3248-3282, 2010.
Reuven Bar-Yehuda. Using homogeneous weights for approximating the partial cover problem. Journal of Algorithms, 39(2):137–144, 2001.
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Timothy M. Chan and Nan Hu. Geometric red-blue set cover for unit squares and related problems. Comput. Geom., 48(5):380-385, 2015.
Kenneth L. Clarkson. New applications of random sampling in computational geometry. Discrete Comput. Geom., 2(1):195-222, 1987.
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Thomas Erlebach and Erik Jan Van Leeuwen. Ptas for weighted set cover on unit squares. In Proceedings of the 13th International Conference on Approximation, and 14 the International Conference on Randomization, and Combinatorial Optimization: Algorithms and Techniques, APPROX/RANDOM'10, pages 166-177, Berlin, Heidelberg, 2010. Springer-Verlag.
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Toshihiro Fujito. On combinatorial approximation of covering 0-1 integer programs and partial set cover. Journal of Combinatorial Optimization, 8(4):439-452, 2001.
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Christian Glaßer, Christian Reitwießner, and Heinz Schmitz. Multiobjective disk cover admits a PTAS. In Algorithms and Computation, 19th International Symposium, ISAAC 2008, Gold Coast, Australia, December 15-17, 2008. Proceedings, pages 40-51, 2008.
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Optimality of Geometric Local Search
Up until a decade ago, the algorithmic status of several basic çlass{NP}-complete problems in geometric combinatorial optimisation was unresolved. This included the existence of polynomial-time approximation schemes (PTASs) for hitting set, set cover, dominating set, independent set, and other problems for some basic geometric objects. These past nine years have seen the resolution of all these problems--interestingly, with the same algorithm: local search. In fact, it was shown that for many of these problems, local search with radius lambda gives a (1+O(lambda^{-1/2}))-approximation with running time n^{O(lambda)}. Setting lambda = Theta(epsilon^{-2}) yields a PTAS with a running time of n^{O(epsilon^{-2})}.
On the other hand, hardness results suggest that there do not exist PTASs for these problems with running time poly(n) * f(epsilon) for any arbitrary f. Thus the main question left open in previous work is in improving the exponent of n to o(epsilon^{-2}).
We show that in fact the approximation guarantee of local search cannot be improved for any of these problems. The key ingredient, of independent interest, is a new lower bound on locally expanding planar graphs, which is then used to show the impossibility results. Our construction extends to other graph families with small separators.
local search
expansion
matchings
Hall's marriage theorem
48:1-48:15
Regular Paper
Bruno
Jartoux
Bruno Jartoux
Nabil H.
Mustafa
Nabil H. Mustafa
10.4230/LIPIcs.SoCG.2018.48
Pankaj K. Agarwal and Nabil H. Mustafa. Independent set of intersection graphs of convex objects in 2D. Computational Geometry, 34(2):83-95, 2006. URL: http://dx.doi.org/10.1016/j.comgeo.2005.12.001.
http://dx.doi.org/10.1016/j.comgeo.2005.12.001
Noga Alon, Paul Seymour, and Robin Thomas. A separator theorem for nonplanar graphs. Journal of the American Mathematical Society, 3(4), 10 1990. URL: http://dx.doi.org/10.1090/S0894-0347-1990-1065053-0.
http://dx.doi.org/10.1090/S0894-0347-1990-1065053-0
Daniel Antunes, Claire Mathieu, and Nabil H. Mustafa. Combinatorics of local search: An optimal 4-local hall’s theorem for planar graphs. In Kirk Pruhs and Christian Sohler, editors, 25th Annual European Symposium on Algorithms (ESA 2017), volume 87 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:13, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2017.8.
http://dx.doi.org/10.4230/LIPIcs.ESA.2017.8
Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing, 33(3):544-562, 2004. URL: http://dx.doi.org/10.1137/S0097539702416402.
http://dx.doi.org/10.1137/S0097539702416402
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http://dx.doi.org/10.1007/978-3-642-36065-7_10
Norbert Bus, Shashwat Garg, Nabil H. Mustafa, and Saurabh Ray. Limits of local search: Quality and efficiency. Discrete & Computational Geometry, 57(3):607-624, 4 2017. URL: http://dx.doi.org/10.1007/s00454-016-9819-x.
http://dx.doi.org/10.1007/s00454-016-9819-x
Sergio Cabello and David Gajser. Simple PTAS’s for families of graphs excluding a minor. Discrete Applied Mathematics, 189:41-48, 2015. URL: http://dx.doi.org/10.1016/j.dam.2015.03.004.
http://dx.doi.org/10.1016/j.dam.2015.03.004
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Matt Gibson, Gaurav Kanade, Erik Krohn, and Kasturi Varadarajan. An approximation scheme for terrain guarding. In Irit Dinur, Klaus Jansen, Joseph Naor, and José Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 12th International Workshop, APPROX 2009, and 13th International Workshop, RANDOM 2009, Berkeley, CA, USA, August 21-23, 2009. Proceedings, pages 140-148, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. URL: http://dx.doi.org/10.1007/978-3-642-03685-9_11.
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http://dx.doi.org/10.1007/978-3-642-15775-2_21
Sathish Govindarajan, Rajiv Raman, Saurabh Ray, and Aniket Basu Roy. Packing and covering with non-piercing regions. In Piotr Sankowski and Christos Zaroliagis, editors, Proceedings of the 22nd Annual European Symposium on Algorithms (ESA), volume 57 of Leibniz International Proceedings in Informatics (LIPIcs), pages 47:1-47:17, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2016.47.
http://dx.doi.org/10.4230/LIPIcs.ESA.2016.47
Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. In Proceedings of the 23rd Annual European Symposium on Algorithms (ESA), pages 717-728, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_60.
http://dx.doi.org/10.1007/978-3-662-48350-3_60
Sariel Har-Peled and Kent Quanrud. Notes on approximation algorithms for polynomial-expansion and low-density graphs, 2016. URL: http://arxiv.org/abs/1603.03098.
http://arxiv.org/abs/1603.03098
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http://dx.doi.org/10.1007/978-3-642-27875-4
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Odd Yao-Yao Graphs are Not Spanners
It is a long standing open problem whether Yao-Yao graphs YY_{k} are all spanners [Li et al. 2002]. Bauer and Damian [Bauer and Damian, 2012] showed that all YY_{6k} for k >= 6 are spanners. Li and Zhan [Li and Zhan, 2016] generalized their result and proved that all even Yao-Yao graphs YY_{2k} are spanners (for k >= 42). However, their technique cannot be extended to odd Yao-Yao graphs, and whether they are spanners are still elusive. In this paper, we show that, surprisingly, for any integer k >= 1, there exist odd Yao-Yao graph YY_{2k+1} instances, which are not spanners.
Odd Yao-Yao Graph
Spanner
Counterexample
49:1-49:15
Regular Paper
Yifei
Jin
Yifei Jin
Jian
Li
Jian Li
Wei
Zhan
Wei Zhan
10.4230/LIPIcs.SoCG.2018.49
Franz Aurenhammer. Voronoi diagrams - survey of a fundamental geometric data structure. ACM Computing Surveys (CSUR), 23(3):345-405, 1991.
Luis Barba, Prosenjit Bose, Mirela Damian, Rolf Fagerberg, Wah Loon Keng, Joseph O'Rourke, André van Renssen, Perouz Taslakian, Sander Verdonschot, and Ge Xia. New and improved spanning ratios for Yao graphs. JoCG, 6(2):19-53, 2015.
Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, André van Renssen, and Sander Verdonschot. On the stretch factor of the Θ₄-graph. In Workshop on Algorithms and Data Structures, pages 109-120. Springer, 2013.
Matthew Bauer and Mirela Damian. An infinite class of sparse-Yao spanners. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 184-196. SIAM, 2013.
Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, and David Ilcinkas. Connections between Θ-graphs, Delaunay triangulations, and orthogonal surfaces. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 266-278. Springer, 2010.
Prosenjit Bose, Mirela Damian, Karim Douïeb, Joseph O'rourke, Ben Seamone, Michiel Smid, and Stefanie Wuhrer. π/2-angle Yao graphs are spanners. International Journal of Computational Geometry &Applications, 22(01):61-82, 2012.
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Prosenjit Bose, Pat Morin, André van Renssen, and Sander Verdonschot. The Θ₅-graph is a spanner. Computational Geometry, 48(2):108-119, 2015.
Prosenjit Bose, André van Renssen, and Sander Verdonschot. On the spanning ratio of theta-graphs. In Workshop on Algorithms and Data Structures, pages 182-194. Springer, 2013.
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Mirela Damian and Kristin Raudonis. Yao graphs span Θ-graphs. In Combinatorial Optimization and Applications, pages 181-194. Springer, 2010.
Nawar M El Molla. Yao spanners for wireless ad hoc networks. Master’s thesis, Villanova University, 2009.
David Eppstein. Spanning trees and spanners. Handbook of computational geometry, pages 425-461, 1999.
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Giri Narasimhan and Michiel Smid. Geometric spanner networks. Cambridge University Press, 2007.
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Christian Schindelhauer, Klaus Volbert, and Martin Ziegler. Geometric spanners with applications in wireless networks. Computational Geometry, 36(3):197-214, 2007.
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Deletion in Abstract Voronoi Diagrams in Expected Linear Time
Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem for a long time. Similarly for various concrete Voronoi diagrams of generalized sites, other than points. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion. We introduce the concept of a Voronoi-like diagram, a relaxed version of a Voronoi construct that has a structure similar to an abstract Voronoi diagram, without however being one. Voronoi-like diagrams serve as intermediate structures, which are considerably simpler to compute, thus, making an expected linear-time construction possible. We formalize the concept and prove that it is robust under an insertion operation, thus, enabling its use in incremental constructions.
Abstract Voronoi diagram
linear-time algorithm
update after deletion
randomized incremental algorithm
50:1-50:14
Regular Paper
Kolja
Junginger
Kolja Junginger
Evanthia
Papadopoulou
Evanthia Papadopoulou
10.4230/LIPIcs.SoCG.2018.50
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From a (p,2)-Theorem to a Tight (p,q)-Theorem
A family F of sets is said to satisfy the (p,q)-property if among any p sets of F some q have a non-empty intersection. The celebrated (p,q)-theorem of Alon and Kleitman asserts that any family of compact convex sets in R^d that satisfies the (p,q)-property for some q >= d+1, can be pierced by a fixed number (independent on the size of the family) f_d(p,q) of points. The minimum such piercing number is denoted by {HD}_d(p,q). Already in 1957, Hadwiger and Debrunner showed that whenever q > (d-1)/d p+1 the piercing number is {HD}_d(p,q)=p-q+1; no exact values of {HD}_d(p,q) were found ever since.
While for an arbitrary family of compact convex sets in R^d, d >= 2, a (p,2)-property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The best-studied among them is axis-parallel boxes in R^d, and specifically, axis-parallel rectangles in the plane. Wegner (1965) and (independently) Dol'nikov (1972) used a (p,2)-theorem for axis-parallel rectangles to show that {HD}_{rect}(p,q)=p-q+1 holds for all q>sqrt{2p}. These are the only values of q for which {HD}_{rect}(p,q) is known exactly.
In this paper we present a general method which allows using a (p,2)-theorem as a bootstrapping to obtain a tight (p,q)-theorem, for families with Helly number 2, even without assuming that the sets in the family are convex or compact. To demonstrate the strength of this method, we show that {HD}_{d-box}(p,q)=p-q+1 holds for all q > c' log^{d-1} p, and in particular, {HD}_{rect}(p,q)=p-q+1 holds for all q >= 7 log_2 p (compared to q >= sqrt{2p}, obtained by Wegner and Dol'nikov more than 40 years ago).
In addition, for several classes of families, we present improved (p,2)-theorems, some of which can be used as a bootstrapping to obtain tight (p,q)-theorems. In particular, we show that any family F of compact convex sets in R^d with Helly number 2 admits a (p,2)-theorem with piercing number O(p^{2d-1}), and thus, satisfies {HD}_{F}(p,q)=p-q+1 for all q>cp^{1-1/(2d-1)}, for a universal constant c.
(p,q)-Theorem
convexity
transversals
(p,2)-theorem
axis-parallel rectangles
51:1-51:14
Regular Paper
Chaya
Keller
Chaya Keller
Shakhar
Smorodinsky
Shakhar Smorodinsky
10.4230/LIPIcs.SoCG.2018.51
N. Alon, G. Kalai, R. Meshulam, and J. Matoušek. Transversal numbers for hypergraphs arising in geometry. Adv. Appl. Math, 29:79-101, 2002.
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http://dx.doi.org/10.1007/s00454-012-9417-5
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https://doi-org.proxy1.athensams.net/10.1007/BF03008396
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Coloring Intersection Hypergraphs of Pseudo-Disks
We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a family of pseudo-disks admits a proper coloring with constantly many colors and a conflict-free coloring with O(log n) colors. Our results serve as a common generalization and strengthening of many earlier results, including ones about proper and conflict-free coloring points with respect to pseudo-disks, coloring regions of linear union complexity with respect to points and coloring disks with respect to disks.
combinatorial geometry
conflict-free coloring
geometric hypergraph coloring
52:1-52:15
Regular Paper
Balázs
Keszegh
Balázs Keszegh
10.4230/LIPIcs.SoCG.2018.52
Eyal Ackerman, Balázs Keszegh, and Mate Vizer. Coloring points with respect to squares. Discrete & Computational Geometry, jun 2017.
B. Aronov, A. Donakonda, E. Ezra, and R. Pinchasi. On Pseudo-disk Hypergraphs. ArXiv e-prints, 2018. URL: http://arxiv.org/abs/1802.08799.
http://arxiv.org/abs/1802.08799
Sarit Buzaglo, Rom Pinchasi, and Günter Rote. Topological hypergraphs. In Thirty Essays on Geometric Graph Theory, pages 71-81. Springer New York, oct 2012.
Jean Cardinal and Matias Korman. Coloring planar homothets and three-dimensional hypergraphs. Computational geometry, 46(9):1027-1035, 2013.
Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48(2):373-392, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9417-5.
http://dx.doi.org/10.1007/s00454-012-9417-5
Chaim Chojnacki. Über wesentlich unplättbare kurven im dreidimensionalen raume. Fundamenta Mathematicae, 23(1):135-142, 1934.
Guy Even, Zvi Lotker, Dana Ron, and Shakhar Smorodinsky. Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM Journal on Computing, 33(1):94-136, jan 2003.
S. P. Fekete and P. Keldenich. Conflict-Free Coloring of Intersection Graphs. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1709.03876.
http://arxiv.org/abs/1709.03876
Andreas F. Holmsen, Hossein Nassajian Mojarrad, János Pach, and Gábor Tardos. Two extensions of the Erdős-Szekeres problem. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1710.11415.
http://arxiv.org/abs/1710.11415
Klara Kedem, Ron Livne, János Pach, and Micha Sharir. On the union of jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete & Computational Geometry, 1(1):59-71, Mar 1986.
C. Keller and S. Smorodinsky. Conflict-Free Coloring of Intersection Graphs of Geometric Objects. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1704.02018.
http://arxiv.org/abs/1704.02018
Balázs Keszegh. Weak conflict-free colorings of point sets and simple regions. In Prosenjit Bose, editor, Proceedings of the 19th Annual Canadian Conference on Computational Geometry, CCCG 2007, August 20-22, 2007, Carleton University, Ottawa, Canada, pages 97-100. Carleton University, Ottawa, Canada, 2007. URL: http://cccg.ca/proceedings/2007/04b2.pdf.
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Balázs Keszegh. Coloring half-planes and bottomless rectangles. Computational geometry, 45(9):495-507, 2012. URL: http://dx.doi.org/10.1016/j.comgeo.2011.09.004.
http://dx.doi.org/10.1016/j.comgeo.2011.09.004
Balázs Keszegh and Dömötör Pálvölgyi. Octants are cover-decomposable. Discrete & Computational Geometry, 47(3):598-609, 2012.
Balázs Keszegh and Dömötör Pálvölgyi. An abstract approach to polychromatic coloring: Shallow hitting sets in aba-free hypergraphs and pseudohalfplanes. In Ernst W. Mayr, editor, Graph-Theoretic Concepts in Computer Science - 41st International Workshop, WG 2015, Garching, Germany, June 17-19, 2015, Revised Papers, volume 9224 of Lecture Notes in Computer Science, pages 266-280. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-53174-7_19.
http://dx.doi.org/10.1007/978-3-662-53174-7_19
Balázs Keszegh and Dömötör Pálvölgyi. More on decomposing coverings by octants. Journal of Computational Geometry, 6(1):300-315, 2015.
Balázs Keszegh and Dömötör Pálvölgyi. Proper coloring of geometric hypergraphs. In Symposium on Computational Geometry, volume 77 of LIPIcs, pages 47:1-47:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017.
István Kovács. Indecomposable coverings with homothetic polygons. Discrete &Computational Geometry, 53(4):817-824, 2015.
János Pach and Dömötör Pálvölgyi. Unsplittable coverings in the plane. In Ernst W. Mayr, editor, Graph-Theoretic Concepts in Computer Science - 41st International Workshop, WG 2015, Garching, Germany, June 17-19, 2015, Revised Papers, volume 9224 of Lecture Notes in Computer Science, pages 281-296. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-53174-7_20.
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Minimizing Crossings in Constrained Two-Sided Circular Graph Layouts
Circular layouts are a popular graph drawing style, where vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is NP-hard. One way to allow for fewer crossings in practice are two-sided layouts that draw some edges as curves in the exterior of the circle. In fact, one- and two-sided circular layouts are equivalent to one-page and two-page book drawings, i.e., graph layouts with all vertices placed on a line (the spine) and edges drawn in one or two distinct half-planes (the pages) bounded by the spine. In this paper we study the problem of minimizing the crossings for a fixed cyclic vertex order by computing an optimal k-plane set of exteriorly drawn edges for k >= 1, extending the previously studied case k=0. We show that this relates to finding bounded-degree maximum-weight induced subgraphs of circle graphs, which is a graph-theoretic problem of independent interest. We show NP-hardness for arbitrary k, present an efficient algorithm for k=1, and generalize it to an explicit XP-time algorithm for any fixed k. For the practically interesting case k=1 we implemented our algorithm and present experimental results that confirm the applicability of our algorithm.
Graph Drawing
Circular Layouts
Crossing Minimization
Circle Graphs
Bounded-Degree Maximum-Weight Induced Subgraphs
53:1-53:14
Regular Paper
Fabian
Klute
Fabian Klute
Martin
Nöllenburg
Martin Nöllenburg
10.4230/LIPIcs.SoCG.2018.53
Md Jawaherul Alam, Martin Fink, and Sergey Pupyrev. The bundled crossing number. In Y. Hu and M. Nöllenburg, editors, Graph Drawing and Network Visualization (GD'16), volume 9801 of LNCS, pages 399-412. Springer, 2016.
Michael Baur and Ulrik Brandes. Crossing reduction in circular layouts. In Graph-Theoretic Concepts in Computer Science (WG'04), volume 3353 of LNCS, pages 332-343. Springer, 2004.
Nadja Betzler, Robert Bredereck, Rolf Niedermeier, and Johannes Uhlmann. On bounded-degree vertex deletion parameterized by treewidth. Discrete Applied Mathematics, 160(1-2):53-60, 2012.
Markus Chimani, Carsten Gutwenger, Michael Jünger, Gunnar W Klau, Karsten Klein, and Petra Mutzel. The open graph drawing framework (OGDF). In R. Tamassia, editor, Handbook of Graph Drawing and Visualization, chapter 17, pages 543-569. CRC Press, 2013.
Anders Dessmark, Klaus Jansen, and Andrzej Lingas. The maximum k-dependent and f-dependent set problem. In K. W. Ng, P. Raghavan, N. V. Balasubramanian, and F. Y. L. Chin, editors, Algorithms and Computation (ISAAC'93), volume 762 of LNCS, pages 88-97. Springer, 1993.
Fedor V Fomin, Ioan Todinca, and Yngve Villanger. Large induced subgraphs via triangulations and CMSO. SIAM J. Computing, 44(1):54-87, 2015.
Robert Ganian, Fabian Klute, and Sebastian Ordyniak. On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem. In Theoretical Aspects of Computer Science (STACS'18), volume 96 of Leibniz International Proceedings in Informatics (LIPIcs), pages 33:1-33:14, 2018.
Emden R Gansner and Yehuda Koren. Improved circular layouts. In Graph Drawing (GD'06), volume 4372 of LNCS, pages 386-398. Springer, 2007.
Serge Gaspers, Dieter Kratsch, Mathieu Liedloff, and Ioan Todinca. Exponential time algorithms for the minimum dominating set problem on some graph classes. ACM Trans. Algorithms, 6(1):9:1-9:21, 2009.
Fanika Gavril. Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks, 3(3):261-273, 1973.
Michael Jünger and Petra Mutzel, editors. Graph Drawing Software. Mathematics and Visualization. Springer, 2004.
J Mark Keil. The complexity of domination problems in circle graphs. Discrete Applied Mathematics, 42(1):51-63, 1993.
Jonathan Klawitter, Tamara Mchedlidze, and Martin Nöllenburg. Experimental evaluation of book drawing algorithms. In F. Frati and K.-L. Ma, editors, Graph Drawing and Network Visualization (GD'17), volume 10692 of LNCS, pages 224-238. Springer, 2018. URL: https://arxiv.org/abs/1708.09221.
https://arxiv.org/abs/1708.09221
Ton Kloks. Treewidth of circle graphs. International J. Foundations of Computer Science, 7(02):111-120, 1996.
Stephen G. Kobourov, Giuseppe Liotta, and Fabrizio Montecchiani. An annotated bibliography on 1-planarity. CoRR, abs/1703.02261, 2017. URL: http://arxiv.org/abs/1703.02261.
http://arxiv.org/abs/1703.02261
Martin I. Krzywinski, Jacqueline E. Schein, Inanc Birol, Joseph Connors, Randy Gascoyne, Doug Horsman, Steven J. Jones, and Marco A. Marra. Circos: An information aesthetic for comparative genomics. Genome Research, 19:1639-1645, 2009.
Giuseppe Liotta. Graph drawing beyond planarity: some results and open problems. In Italian Conference on Theoretical Computer Science (ICTCS'14), pages 3-8, 2014.
Sumio Masuda, Toshinobu Kashiwabara, Kazuo Nakajima, and Toshio Fujisawa. On the NP-completeness of a computer network layout problem. In Circuits and Systems (ISCAS'87), pages 292-295. IEEE, 1987.
Sumio Masuda, Kazuo Nakajima, Toshinobu Kashiwabara, and Toshio Fujisawa. Crossing minimization in linear embeddings of graphs. IEEE Trans. Computers, 39(1):124-127, 1990.
Farhad Shahrokhi, László A. Székely, Ondrej Sýkora, and Imrich Vrt'o. The book crossing number of a graph. J. Graph Theory, 21(4):413-424, 1996.
Janet M. Six and Ioannis G. Tollis. Circular drawing algorithms. In R. Tamassia, editor, Handbook of Graph Drawing and Visualization, chapter 9, pages 285-315. CRC Press, 2013.
Wolfgang Thomas. Languages, automata, and logic. Handbook of formal languages, 3:389-455, 1996.
Gabriel Valiente. A new simple algorithm for the maximum-weight independent set problem on circle graphs. In Algorithms and Computation (ISAAC'03), volume 2906 of LNCS, pages 129-137. Springer, 2003.
Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Theory of Computing (STOC'78), pages 253-264. ACM, 1978.
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Discrete Stratified Morse Theory: A User's Guide
Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and MacPherson. We describe the basics of this theory and prove fundamental theorems relating the topology of a general simplicial complex with the critical simplices of a discrete stratified Morse function on the complex. We also provide an algorithm that constructs a discrete stratified Morse function out of an arbitrary function defined on a finite simplicial complex; this is different from simply constructing a discrete Morse function on such a complex. We borrow Forman's idea of a "user's guide," where we give simple examples to convey the utility of our theory.
Discrete Morse theory
stratified Morse theory
topological data analysis
54:1-54:14
Regular Paper
Kevin
Knudson
Kevin Knudson
Bei
Wang
Bei Wang
10.4230/LIPIcs.SoCG.2018.54
Madjid Allili, Tomasz Kaczynski, and Claudia Landi. Reducing complexes in multidimensional persistent homology theory. Journal of Symbolic Computation, 78:61-75, 2017.
Paul Bendich and John Harer. Persistent intersection homology. Foundations of Computational Mathematics, 11(3):305-336, 2011.
Bruno Benedetti. Smoothing discrete Morse theory. Annali della Scuola Normale Superiore di Pisa, 16(2):335-368, 2016.
Ronald Brown. Topology and Groupoids. www.groupoids.org, 2006.
Olaf Delgado-Friedrichs, Vanessa Robins, and Adrian Sheppard. Skeletonization and partitioning of digital images using discrete Morse theory. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(3):654-666, 2015.
Pawel Dlotko and Hubert Wagner. Computing homology and persistent homology using iterated Morse decomposition, 2012. URL: https://arxiv.org/abs/1210.1429.
https://arxiv.org/abs/1210.1429
Herbert Edelsbrunner, John Harer, Vijay Natarajan, and Valerio Pascucci. Morse-Smale complexes for piece-wise linear 3-manifolds. ACM Symposium on Computational Geometry, pages 361-370, 2003.
Herbert Edelsbrunner, John Harer, and Afra J. Zomorodian. Hierarchical Morse-Smale complexes for piecewise linear 2-manifolds. Discrete &Computational Geometry, 30(87-107), 2003.
Herbert Edelsbrunner and Ernst Peter Mücke. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics, 9(1):66-104, 1990.
Robin Forman. Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift, 228(4):629-681, 1998.
Robin Forman. Morse theory for cell complexes. Advances in Mathematics, 134:90-145, 1998.
Robin Forman. Combinatorial differential topology and geometry. New Perspectives in Geometric Combinatorics, 38:177-206, 1999.
Robin Forman. A user’s guide to discrete Morse theory. Séminaire Lotharingien de Combinatoire, 48, 2002.
Greg Friedman. Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata. Topology and its Applications, 134(2):69-109, 2003.
Mark Goresky and Robert MacPherson. Stratified Morse Theory. Springer-Verlag, 1988.
David Günther, Jan Reininghaus, Hans-Peter Seidel, and Tino Weinkauf. Notes on the simplification of the Morse-Smale complex. In Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, editors, Topological Methods in Data Analysis and Visualization III, pages 135-150. Springer International Publishing, 2014.
Attila Gyulassy, Peer-Timo Bremer, Bernd Hamann, and Valerio Pascucci. A practical approach to Morse-Smale complex computation: Scalability and generality. IEEE Transactions on Visualization and Computer Graphics, 14(6):1619-1626, 2008.
Henry King, Kevin Knudson, and Neža Mramor. Generating discrete Morse functions from point data. Experimental Mathematics, 14:435-444, 2005.
Manfred Knebusch. Semialgebraic topology in the last ten years. In Michel Coste, Louis Mahe, and Marie-Francoise Roy, editors, Real Algebraic Geometry, 1991.
Kevin Knudson and Bei Wang. Discrete stratified Morse theory: A user’s guide. 2018. URL: https://arxiv.org/abs/1801.03183.
https://arxiv.org/abs/1801.03183
Konstantin Mischaikow and Vidit Nanda. Morse theory for filtrations and efficient computation of persistent homology. Discrete &Computational Geometry, 50(2):330-353, 2013.
James R. Munkres. Elements of algebraic topology. Addison-Wesley, Redwood City, California, 1984.
Jan Reininghaus. Computational Discrete Morse Theory. PhD thesis, Zuse Institut Berlin (ZIB), 2012.
Jan Reininghaus, Jens Kasten, Tino Weinkauf, and Ingrid Hotz. Efficient computation of combinatorial feature flow fields. IEEE Transactions on Visualization and Computer Graphics, 18(9):1563-1573, 2011.
Vanessa Robins, Peter John Wood, and Adrian P. Sheppard. Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(8):1646-1658, 2011.
Tammo tom Dieck. Algebraic Topology. European Mathematical Society, 2008.
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An Optimal Algorithm to Compute the Inverse Beacon Attraction Region
The beacon model is a recent paradigm for guiding the trajectory of messages or small robotic agents in complex environments. A beacon is a fixed point with an attraction pull that can move points within a given polygon. Points move greedily towards a beacon: if unobstructed, they move along a straight line to the beacon, and otherwise they slide on the edges of the polygon. The Euclidean distance from a moving point to a beacon is monotonically decreasing. A given beacon attracts a point if the point eventually reaches the beacon.
The problem of attracting all points within a polygon with a set of beacons can be viewed as a variation of the art gallery problem. Unlike most variations, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. It is connected and can be computed in linear time for simple polygons. By contrast, it is known that the inverse attraction region of a point - the set of beacon positions that attract it - could have Omega(n) disjoint connected components.
In this paper, we prove that, in spite of this, the total complexity of the inverse attraction region of a point in a simple polygon is linear, and present a O(n log n) time algorithm to construct it. This improves upon the best previous algorithm which required O(n^3) time and O(n^2) space. Furthermore we prove a matching Omega(n log n) lower bound for this task in the algebraic computation tree model of computation, even if the polygon is monotone.
beacon attraction
inverse attraction region
algorithm
optimal
55:1-55:14
Regular Paper
Irina
Kostitsyna
Irina Kostitsyna
Bahram
Kouhestani
Bahram Kouhestani
Stefan
Langerman
Stefan Langerman
David
Rappaport
David Rappaport
10.4230/LIPIcs.SoCG.2018.55
Sang Won Bae, Chan-Su Shin, and Antoine Vigneron. Tight bounds for beacon-based coverage in simple rectilinear polygons. In Proc. 12th Latin American Symposium on Theoretical Informatics, 2016.
Michael Biro. Beacon-based routing and guarding. PhD thesis, Stony Brook University, 2013.
Michael Biro, Jie Gao, Justin Iwerks, Irina Kostitsyna, and Joseph S. B. Mitchell. Beacon-based routing and coverage. In Abstr. 21st Fall Workshop on Computational Geometry, 2011.
Michael Biro, Jie Gao, Justin Iwerks, Irina Kostitsyna, and Joseph S. B. Mitchell. Combinatorics of beacon-based routing and coverage. In Proc. 25th Canadian Conference on Computational Geometry, 2013.
Michael Biro, Justin Iwerks, Irina Kostitsyna, and Joseph S. B. Mitchell. Beacon-based algorithms for geometric routing. In Proc. 13th Algorithms and Data Structures Symposium, 2013.
Prosenjit Bose, Pat Morin, Ivan Stojmenović, and Jorge Urrutia. Routing with guaranteed delivery in ad hoc wireless networks. Wirel. Netw., 7(6):609-616, 2001. URL: http://dx.doi.org/10.1023/A:1012319418150.
http://dx.doi.org/10.1023/A:1012319418150
Gerth S. Brodal and Riko Jacob. Dynamic planar convex hull. In Proc. 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002.
Bernard Chazelle and Leonidas J. Guibas. Visibility and intersection problems in plane geometry. Discrete & Computational Geometry, 4(6):551-581, 1989.
Leonidas J. Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987.
Brad Karp and H. T. Kung. GPSR: Greedy perimeter stateless routing for wireless networks. In Proceedings of the 6th Annual International Conference on Mobile Computing and Networking, MobiCom '00, pages 243-254, New York, NY, USA, 2000. ACM. URL: http://dx.doi.org/10.1145/345910.345953.
http://dx.doi.org/10.1145/345910.345953
Anne-Marie Kermarrec and Guang Tan. Greedy geographic routing in large-scale sensor networks: A minimum network decomposition approach. IEEE/ACM Transactions on Networking, 20:864-877, 2010.
Irina Kostitsyna, Bahram Kouhestani, Stefan Langerman, and David Rappaport. An optimal algorithm to compute the inverse beacon attraction region. ArXiv e-prints, http://arxiv.org/abs/1803.05946, 2018.
http://arxiv.org/abs/1803.05946
Bahram Kouhestani, David Rappaport, and Kai Salomaa. The length of the beacon attraction trajectory. In Proc. 27th Canadian Conference on Computational Geometry, 2015.
Bahram Kouhestani, David Rappaport, and Kai Salomaa. On the inverse beacon attraction region of a point. In Proc. 27th Canadian Conference on Computational Geometry, 2015.
Bahram Kouhestani, David Rappaport, and Kai Salomaa. Routing in a polygonal terrain with the shortest beacon watchtower. International Journal of Computational Geometry &Applications, 68:34-47, 2018.
Der-Tsai Lee and Franco P. Preparata. Euclidean shortest paths in the presence of rectilinear barriers. Networks, 14(3):393-410, 1984.
Thomas Shermer. A combinatorial bound for beacon-based routing in orthogonal polygons. In Proc. 27th Canadian Conference on Computational Geometry, 2015.
Andrew C. Yao. A lower bound to finding convex hulls. Journal of the ACM, 28(4):780-787, 1981.
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On Optimal Polyline Simplification Using the Hausdorff and Fréchet Distance
We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given epsilon>0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most epsilon.
We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than epsilon. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the Fréchet distance, we give an O(kn^5) time algorithm that requires O(kn^2) space, where k is the output complexity of the simplification.
polygonal line simplification
Hausdorff distance
Fréchet distance
Imai-Iri
Douglas-Peucker
56:1-56:14
Regular Paper
Marc
van Kreveld
Marc van Kreveld
Maarten
Löffler
Maarten Löffler
Lionov
Wiratma
Lionov Wiratma
10.4230/LIPIcs.SoCG.2018.56
Mohammad Ali Abam, Mark de Berg, Peter Hachenberger, and Alireza Zarei:. Streaming algorithms for line simplification. Discrete & Computational Geometry, 43(3):497-515, 2010.
Pankaj K. Agarwal, Sariel Har-Peled, Nabil H. Mustafa, and Yusu Wang. Near-linear time approximation algorithms for curve simplification. Algorithmica, 42(3):203-219, 2005.
Helmut Alt, Bernd Behrends, and Johannes Blömer. Approximate matching of polygonal shapes. Annals of Mathematics and Artificial Intelligence, 13(3):251-265, Sep 1995.
Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications, 5(1-2):75-91, 1995.
Gill Barequet, Danny Z. Chen, Ovidiu Daescu, Michael T. Goodrich, and Jack Snoeyink. Efficiently approximating polygonal paths in three and higher dimensions. Algorithmica, 33(2):150-167, 2002.
Lilian Buzer. Optimal simplification of polygonal chain for rendering. In Proceedings 23rd Annual ACM Symposium on Computational Geometry, SCG '07, pages 168-174, 2007.
Jérémie Chalopin and Daniel Gonçalves. Every planar graph is the intersection graph of segments in the plane: Extended abstract. In Proceedings 41st Annual ACM Symposium on Theory of Computing, STOC '09, pages 631-638, 2009.
W.S. Chan and F. Chin. Approximation of polygonal curves with minimum number of line segments or minimum error. International Journal of Computational Geometry & Applications, 06(01):59-77, 1996.
Danny Z. Chen, Ovidiu Daescu, John Hershberger, Peter M. Kogge, Ningfang Mi, and Jack Snoeyink. Polygonal path simplification with angle constraints. Computational Geometry, 32(3):173-187, 2005.
Mark de Berg, Marc van Kreveld, and Stefan Schirra. Topologically correct subdivision simplification using the bandwidth criterion. Cartography and Geographic Information Systems, 25(4):243-257, 1998.
David H. Douglas and Thomas K. Peucker. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartographica, 10(2):112-122, 1973.
Regina Estkowski and Joseph S. B. Mitchell. Simplifying a polygonal subdivision while keeping it simple. In Proceedings 17th Annual ACM Symposium on Computational Geometry, SCG '01, pages 40-49, 2001.
Stefan Funke, Thomas Mendel, Alexander Miller, Sabine Storandt, and Maria Wiebe. Map simplification with topology constraints: Exactly and in practice. In Proc. 19th Workshop on Algorithm Engineering and Experiments (ALENEX), pages 185-196, 2017.
M.R. Garey, D.S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976.
Michael Godau. A natural metric for curves - computing the distance for polygonal chains and approximation algorithms. In Proceedings 8th Annual Symposium on Theoretical Aspects of Computer Science, STACS 91, pages 127-136. Springer-Verlag, 1991.
Leonidas J. Guibas, John E. Hershberger, Joseph S.B. Mitchell, and Jack Scott Snoeyink. Approximating polygons and subdivisions with minimum-link paths. International Journal of Computational Geometry & Applications, 03(04):383-415, 1993.
John Hershberger and Jack Snoeyink. An O(n log n) implementation of the Douglas-Peucker algorithm for line simplification. In Proceedings 10th Annual ACM Symposium on Computational Geometry, SCG '94, pages 383-384, 1994.
Hiroshi Imai and Masao Iri. Polygonal approximations of a curve - formulations and algorithms. In Godfried T. Toussaint, editor, Computational Morphology: A Computational Geometric Approach to the Analysis of Form. North-Holland, Amsterdam, 1988.
V. S. Anil Kumar, Sunil Arya, and H. Ramesh. Hardness of set cover with intersection 1. In Automata, Languages and Programming: 27th International Colloquium, ICALP 2000, pages 624-635. Springer, Berlin, Heidelberg, 2000.
Nimrod Megiddo and Arie Tamir. On the complexity of locating linear facilities in the plane. Operations Research Letters, 1(5):194-197, 1982.
Avraham Melkman and Joseph O'Rourke. On polygonal chain approximation. In Godfried T. Toussaint, editor, Computational Morphology: A Computational Geometric Approach to the Analysis of Form, pages 87-95. North-Holland, Amsterdam, 1988.
E. R. Scheinerman. Intersection Classes and Multiple Intersection Parameters of Graphs. PhD thesis, Princeton University, 1984.
Marc van Kreveld, Maarten Löffler, and Lionov Wiratma. On optimal polyline simplification using the hausdorff and fréchet distance. URL: https://arxiv.org/abs/1803.03550.
https://arxiv.org/abs/1803.03550
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Graph-Based Time-Space Trade-Offs for Approximate Near Neighbors
We take a first step towards a rigorous asymptotic analysis of graph-based methods for finding (approximate) nearest neighbors in high-dimensional spaces, by analyzing the complexity of randomized greedy walks on the approximate nearest neighbor graph. For random data sets of size n = 2^{o(d)} on the d-dimensional Euclidean unit sphere, using near neighbor graphs we can provably solve the approximate nearest neighbor problem with approximation factor c > 1 in query time n^{rho_{q} + o(1)} and space n^{1 + rho_{s} + o(1)}, for arbitrary rho_{q}, rho_{s} >= 0 satisfying (2c^2 - 1) rho_{q} + 2 c^2 (c^2 - 1) sqrt{rho_{s} (1 - rho_{s})} >= c^4. Graph-based near neighbor searching is especially competitive with hash-based methods for small c and near-linear memory, and in this regime the asymptotic scaling of a greedy graph-based search matches optimal hash-based trade-offs of Andoni-Laarhoven-Razenshteyn-Waingarten [Andoni et al., 2017]. We further study how the trade-offs scale when the data set is of size n = 2^{Theta(d)}, and analyze asymptotic complexities when applying these results to lattice sieving.
approximate nearest neighbor problem
near neighbor graphs
locality-sensitive hashing
locality-sensitive filters
similarity search
57:1-57:14
Regular Paper
Thijs
Laarhoven
Thijs Laarhoven
10.4230/LIPIcs.SoCG.2018.57
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A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon
The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is Omega(n+m log m), and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve O((n+m)log (n+m)) and O(n+m log m log^2n) time, which are optimal for m=Omega(n) and m=O(n/(log^3n)), respectively. In this paper, we give a construction algorithm with O(n+m(log m+log^2 n)) time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in O(log n) time, then the construction time will become the optimal O(n+m log m). In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in O(log n) time.
Simple polygons
Voronoi diagrams
Geodesic distance
58:1-58:14
Regular Paper
Chih-Hung
Liu
Chih-Hung Liu
10.4230/LIPIcs.SoCG.2018.58
Boris Aronov. On the geodesic Voronoi diagram of point sites in a simple polygon. Algorithmica, 4(1-4):109-140, 1989.
Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete & Computational Geometry, 6:485-524, 1991.
Herbert Edelsbrunner, Leonidas J. Guibas, and Jorge Stolfi. Optimal point location in a monotone subdivision. SIAM Journal on Computing, 15(2):317-340, 1986.
Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989.
Leonidas J. Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987.
John Hershberger. A new data structure for shortest path queries in a simple polygon. Information Processing Letters, 38(5):231-235, 1991.
Kurt Mehlhorn and Peter Sanders. Sorted sequences. In Algorithms and Data Structures: The Basic Toolbox. Springer-Verlag Berlin Heidelberg, 2008.
Joseph S. B. Mitchell. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, pages 633-701. Elsevier, 2000.
Eunjin Oh and Hee-Kap Ahn. Voronoi diagrams for a moderate-sized point-set in a simple polygon. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, pages 52:1-52:15, 2017.
Evanthia Papadopoulou and D. T. Lee. A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Algorithmica, 20(4):319-352, 1998.
Robert Endre Tarjan. Updating a balanced search tree in O(1) rotations. Information Processing Letters, 16(5):253-257, 1983.
Robert Endre Tarjan. Efficient Top-Down Updating of Red-Black Trees. Technical report, Technical Report TR-006-85. Dapartment of Computer Science, Princeton University, 1985.
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Further Consequences of the Colorful Helly Hypothesis
Let F be a family of convex sets in R^d, which are colored with d+1 colors. We say that F satisfies the Colorful Helly Property if every rainbow selection of d+1 sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family F there is a color class F_i subset F, for 1 <= i <= d+1, whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension d >= 2 there exist numbers f(d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f(d) points, or all the sets in F can be crossed by g(d) lines.
geometric transversals
convex sets
colorful Helly-type theorems
line transversals
weak epsilon-nets
transversal numbers
59:1-59:14
Regular Paper
Leonardo
Martínez-Sandoval
Leonardo Martínez-Sandoval
Edgardo
Roldán-Pensado
Edgardo Roldán-Pensado
Natan
Rubin
Natan Rubin
10.4230/LIPIcs.SoCG.2018.59
N. Alon, I. Bárány, Z. Füredi, and D. J. Kleitman. Point selections and weak ε-nets for convex hulls. Combinatorics Probability and Computing, 1(3):189-200, 1992.
N. Alon and G. Kalai. Bounding the piercing number. Discrete & Computational Geometry, 13(3):245-256, 1995.
N. Alon, G. Kalai, J. Matoušek, and R. Meshulam. Transversal numbers for hypergraphs arising in geometry. Advances in Applied Mathematics, 29(1):79-101, 2002.
N. Alon and D. J. Kleitman. Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem. Advances in Mathematics, 96(1):103-112, 1992.
N. Amenta, J. A. De Loera, and P. Soberón. Helly’s Theorem: New Variations and Applications. Contemporary Mathematics, 685:55-95, 2017. URL: https://arxiv.org/abs/1508.07606.
https://arxiv.org/abs/1508.07606
J. L. Arocha, I. Bárány, J. Bracho, R. Fabila, and L. Montejano. Very colorful theorems. Discrete & Computational Geometry, 42(2):142-154, 2009.
I. Bárány. A generalization of Carathéodory’s theorem. Discrete Mathematics, 40(2-3):141-152, 1982.
I. Bárány. Tensors, colours, octahedra. In Geometry, Structure and Randomness in Combinatorics, pages 1-17. Springer, 2014.
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L. Danzer, B. Grünbaum, and V. Klee. Helly’s theorem and its relatives. Proceedings of symposia in pure mathematics: Convexity. American Mathematical Society, 1963.
J. Eckhoff. Handbook of Convex Geometry, chapter Helly, Radon, and Carathéodory type theorems, pages 389-448. Elsevier, 1990.
J. E. Goodman, R. Pollack, and R. Wenger. New Trends in Discrete and Computational Geometry, chapter Geometric Transversal Theory, pages 163-198. Springer Berlin Heidelberg, Berlin, Heidelberg, 1993.
H. Hadwiger and H. Debrunner. Über eine Variante zum Helly’schen Satz. Arch. Math., 8:309-313, 1957.
E. Helly. Über Mengen konvexer Körper mit gemeinschaftlichen Punkte. Jahresbericht der Deutschen Mathematiker-Vereinigung, 32:175-176, 1923.
A. F. Holmsen, J. Pach, and H. Tverberg. Points surrounding the origin. Combinatorica, 28(6):633-644, 2008.
M. Katchalski and A. Liu. A problem of geometry in Rⁿ. Proceedings of the American Mathematical Society, 75(2):284-288, 1979.
Leonardo Martínez-Sandoval, Edgardo Roldán-Pensado, and Natan Rubin. Further consequences of the colorful helly hypothesis. arXiv preprint, 2018. URL: https://arxiv.org/abs/1803.06229.
https://arxiv.org/abs/1803.06229
L. Santaló. Un teorema sobre conjuntos de paralelepípedos de aristas paralelas. Publicaciones del Instituto de Matemáticas, Universidad Nacional del Litoral, II(4):49-60, 1940.
R. Wenger and A. Holmsen. The Handbook of Discrete and Computational Geometry, chapter Helly-type Theorems and Geometric Transversals, pages 91-124. Chapman and Hall/CRC, third edition edition, 2017.
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Random Walks on Polytopes of Constant Corank
We show that the pivoting process associated with one line and n points in r-dimensional space may need Omega(log^r n) steps in expectation as n -> infty. The only cases for which the bound was known previously were for r <= 3. Our lower bound is also valid for the expected number of pivoting steps in the following applications: (1) The Random-Edge simplex algorithm on linear programs with n constraints in d = n-r variables; and (2) the directed random walk on a grid polytope of corank r with n facets.
polytope
unique sink orientation
grid
random walk
60:1-60:14
Regular Paper
Malte
Milatz
Malte Milatz
10.4230/LIPIcs.SoCG.2018.60
Andrei Z. Broder, Martin E. Dyer, Alan M. Frieze, Prabhakar Raghavan, and Eli Upfal. The worst-case running time of the random simplex algorithm is exponential in the height. Inf. Process. Lett., 56(2):79-81, 1995.
Stefan Felsner, Bernd Gärtner, and Falk Tschirschnitz. Grid orientations, (d,d+2)-polytopes, and arrangements of pseudolines. Discrete & Computational Geometry, 34(3):411-437, 2005.
Oliver Friedmann, Thomas Dueholm Hansen, and Uri Zwick. Subexponential lower bounds for randomized pivoting rules for the simplex algorithm. In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC), pages 283-292, 2011.
Bernd Gärtner, Martin Henk, and Günter M. Ziegler. Randomized simplex algorithms on klee-minty cubes. Combinatorica, 18(3):349-372, 1998.
Bernd Gärtner, Walter D. Morris, Jr., and Leo Rüst. Unique sink orientations of grids. Algorithmica, 51:200-235, 2008.
Bernd Gärtner, József Solymosi, Falk Tschirschnitz, Pavel Valtr, and Emo Welzl. One line and n points. Random Structures &Algorithms, 23(4):453-471, 2003 (preliminary version at STOC 2001).
Branko Grünbaum. Convex polytopes. Springer, New York, 1967/2003.
Kathy Williamson Hoke. Completely unimodal numberings of a simple polytope. Discrete Applied Mathematics, 20(1):69-81, 1988.
Volker Kaibel, Rafael Mechtel, Micha Sharir, and Günter M. Ziegler. The simplex algorithm in dimension three. SIAM Journal on Computing, 34(2):475-497, 2005.
Gil Kalai. A subexponential randomized simplex algorithm. In Proc. 24th ACM Symposium on Theory of Computing, pages 475-482, 1992.
Jiří Matoušek and Tibor Szabó. Random edge can be exponential on abstract cubes. Advances in Mathematics, 204(1):262-277, 2006.
Peter McMullen. On the upper-bound conjecture for convex polytopes. Journal of Combinatorial Theory, Series B, 10(3):187-200, 1971.
Malte Milatz. Directed random walks on polytopes with few facets. Electronic Notes in Discrete Mathematics, 61:869-875, 2017. The European Conference on Combinatorics, Graph Theory and Applications (Eurocomb '17).
Julian Pfeifle and Günter M. Ziegler. On the monotone upper bound problem. Experimental Mathematics, 13(1):1-12, 2004.
Francisco Santos. A counterexample to the Hirsch Conjecture. Annals of Mathematics, 176:383-412, 2012.
Falk Tschirschnitz. LP-related properties of polytopes with few facets. PhD thesis, ETH Zürich, 2003.
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Table Based Detection of Degenerate Predicates in Free Space Construction
The key to a robust and efficient implementation of a computational geometry algorithm is an efficient algorithm for detecting degenerate predicates. We study degeneracy detection in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to another polyhedron. The structure of the free space is determined by the signs of univariate polynomials, called angle polynomials, whose coefficients are polynomials in the coordinates of the vertices of the polyhedra. Every predicate is expressible as the sign of an angle polynomial f evaluated at a zero t of an angle polynomial g. A predicate is degenerate (the sign is zero) when t is a zero of a common factor of f and g. We present an efficient degeneracy detection algorithm based on a one-time factoring of every possible angle polynomial. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.
free space construction
degenerate predicates
robustness
61:1-61:14
Regular Paper
Victor
Milenkovic
Victor Milenkovic
Elisha
Sacks
Elisha Sacks
Nabeel
Butt
Nabeel Butt
10.4230/LIPIcs.SoCG.2018.61
Cgal, Computational Geometry Algorithms Library. http://www.cgal.org.
Peter Hachenberger. Exact Minkowski sums of polyhedra and exact and efficient decomposition of polyhedra into convex pieces. Algorithmica, 55:329-345, 2009.
Dan Halperin. Controlled perturbation for certified geometric computing with fixed-precision arithmetic. In ICMS, pages 92-95, 2010.
Min-Ho Kyung, Elisha Sacks, and Victor Milenkovic. Robust polyhedral Minkowski sums with GPU implementation. Computer-Aided Design, 67–68:48-57, 2015.
Naama Mayer, Efi Fogel, and Dan Halperin. Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space. Computer-Aided Design, 43(10):1258-1269, 2011.
Victor Milenkovic, Elisha Sacks, and Steven Trac. Robust free space computation for curved planar bodies. IEEE Transactions on Automation Science and Engineering, 10(4):875-883, 2013.
Elisha Sacks, Nabeel Butt, and Victor Milenkovic. Robust free space construction for a polyhedron with planar motion. Computer-Aided Design, 90C:18-26, 2017.
Elisha Sacks and Victor Milenkovic. Robust cascading of operations on polyhedra. Computer-Aided Design, 46:216-220, 2014.
J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM, 27(4):701-717, 1980. URL: http://dx.doi.org/10.1145/322217.322225.
http://dx.doi.org/10.1145/322217.322225
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Approximate Range Queries for Clustering
We study the approximate range searching for three variants of the clustering problem with a set P of n points in d-dimensional Euclidean space and axis-parallel rectangular range queries: the k-median, k-means, and k-center range-clustering query problems. We present data structures and query algorithms that compute (1+epsilon)-approximations to the optimal clusterings of P cap Q efficiently for a query consisting of an orthogonal range Q, an integer k, and a value epsilon>0.
Approximate clustering
orthogonal range queries
62:1-62:14
Regular Paper
Eunjin
Oh
Eunjin Oh
Hee-Kap
Ahn
Hee-Kap Ahn
10.4230/LIPIcs.SoCG.2018.62
Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, and Ali D. Mehrabi. Range-Clustering Queries. In Proceedings of the 33rd International Symposium on Computational Geometry (SoCG 2017), volume 77, pages 5:1-5:16, 2017.
Pankaj K. Agarwal and Jeff Erickson. Geometric range searching and its relatives. In Advances in Discrete and Compputational Geometry, volume 223 of Contemporary Mathematics, pages 1-56. American Mathematical Society Press, 1999.
Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Geometric approximation via coresets. In Combinatorial and Computational Geometry, volume 52, pages 1-30. MSRI Publications, 2005.
Pankaj. K. Agarwal and Cecillia M. Procopiuc. Exact and approximation algorithms for clustering. Algorithmica, 33(2):201-226, 2002.
Srinivas Aluru. Quadtrees and octrees. In Dinesh P. Mehta and Sartaj Sahni, editors, Handbook of Data Structures and Applications, chapter 19. Chapman & Hall/CRC, 2005.
Sunil Arya, David M. Mount, and Eunhui Park. Approximate geometric MST range queries. In Proceedings of the 31st International Symposium on Computational Geometry (SoCG 2015), pages 781-795, 2015.
Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing, 33(3):544-562, 2004.
Peter Brass, Christian Knauer, Chan-Su Shin, Michiel Smid, and Ivo Vigan. Range-aggregate queries for geometric extent problems. In Proceedings of the 19th Computing: The Australasian Theory Symposium (CATS 2013), volume 141, pages 3-10, 2013.
Ke Chen. On coresets for k-median and k-means clustering in metric and euclidean spaces and their applications. SIAM Journal on Computing, 39(3):923-947, 2009.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag TELOS, 2008.
James R. Driscoll, Neil Sarnak, Daniel D. Sleator, and Robert E. Tarjan. Making data structures persistent. Journal of Computer and System Sciences, 38(1):86-124, 1989.
Tomás Feder and Daniel Greene. Optimal algorithms for approximate clustering. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC 1988), pages 434-444, 1988.
Dan Feldman and Michael Langberg. A unified framework for approximating and clustering data. In Proceedings of the 43th Annual ACM Symposium on Theory of Computing (STOC 2011), pages 569-578, 2011.
Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38(Supplement C):293-306, 1985.
Prosenjit Gupta, Ravi Janardan, Yokesh Kumar, and Michiel Smid. Data structures for range-aggregate extent queries. Computational Geometry, 47(2, Part C):329-347, 2014.
Sariel Har-Peled. Geometric Approximation Algorithms. Mathematical surveys and monographs. American Mathematical Society, 2011.
Sariel Har-Peled and Akash Kushal. Smaller coresets for k-median and k-means clustering. Discrete & Computational Geometry, 37(1):3-19, Jan 2007.
Sariel Har-Peled and Soham Mazumdar. On coresets for k-means and k-median clustering. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC 2004), pages 291-300, 2004.
Anil Kumar Jain, M. Narasimha Murty, and Patrick J. Flynn. Data clustering: A review. ACM Computing Surveys, 31(3):264-323, 1999.
Jiří Matoušek. Geometric range searching. ACM Computing Surveys, 26(4):422-461, 1994.
Kurt Mehlhorn and Stefan Näher. Dynamic fractional cascading. Algorithmica, 5(1):215-241, 1990.
Yakov Nekrich and Michiel H. M. Smid. Approximating range-aggregate queries using coresets. In Proceedings of the 22nd Annual Canadian Conference on Computational Geometry (CCCG 2010), pages 253-256, 2010.
Dimitris Papadias, Yufei Tao, Kyriakos Mouratidis, and Chun Kit Hui. Aggregate nearest neighbor queries in spatial databases. ACM Transactions on Database System, 30(2):529-576, 2005.
Jing Shan, Donghui Zhang, and Betty Salzberg. On spatial-range closest-pair query. In Proceedings of the 8th International Symposium on Advances in Spatial and Temporal Databases (SSTD 2003), pages 252-269, 2003.
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Point Location in Dynamic Planar Subdivisions
We study the point location problem on dynamic planar subdivisions that allows insertions and deletions of edges. In our problem, the underlying graph of a subdivision is not necessarily connected. We present a data structure of linear size for such a dynamic planar subdivision that supports sublinear-time update and polylogarithmic-time query. Precisely, the amortized update time is O(sqrt{n}log n(log log n)^{3/2}) and the query time is O(log n(log log n)^2), where n is the number of edges in the subdivision. This answers a question posed by Snoeyink in the Handbook of Computational Geometry. When only deletions of edges are allowed, the update time and query time are just O(alpha(n)) and O(log n), respectively.
dynamic point location
general subdivision
63:1-63:14
Regular Paper
Eunjin
Oh
Eunjin Oh
Hee-Kap
Ahn
Hee-Kap Ahn
10.4230/LIPIcs.SoCG.2018.63
Alfred V. Aho and John E. Hopcroft. The Design and Analysis of Computer Algorithms. Addison-Wesley Longman Publishing Co., Inc., 1974.
Lars Arge, Gerth Stølting Brodal, and Loukas Georgiadis. Improved dynamic planar point location. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pages 305-314, 2006.
Hanna Baumgarten, Hermann Jung, and Kurt Mehlhorn. Dynamic point location in general subdivisions. Journal of Algorithms, 17(3):342-380, 1994.
Jon Louis Bentley and James B Saxe. Decomposable searching problems 1: Static-to-dynamic transformations. Journal of Algorithms, 1(4):301-358, 1980.
Timothy M. Chan and Yakov Nekrich. Towards an optimal method for dynamic planar point location. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015), pages 390-409, 2015.
Siu-Wing Cheng and Ravi Janardan. New results on dynamic planar point location. SIAM Journal on Computing, 21(5):972-999, 1992.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag TELOS, 2008.
Jiří Matousěk. Efficient partition trees. Discrete &Computational Geometry, 8(3), 1992.
Mark H. Overmars. Range searching in a set of line segments. Technical report, Rijksuniversiteit Utrecht, 1983.
Mark H. Overmars and Jan van Leeuwen. Worst-case optimal insertion and deletion methods for decomposable searching problem. Information Processing Letters, 12(4):168-173, 1981.
Neil Sarnak and Robert E. Tarjan. Planar point location using persistent search trees. Communications of the ACM, 29(7):669-679, 1986.
Jack Snoeyink. Point location. In Handbook of Discrete and Computational Geometry, Third Edition, pages 1005-1023. Chapman and Hall/CRC, 2017.
Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22(2):215-225, 1975.
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Edge-Unfolding Nearly Flat Convex Caps
The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R^3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron and a halfspace. "Nearly flat" means that every outer face normal forms a sufficiently small angle f < F with the z^-axis orthogonal to the halfspace bounding plane. The size of F depends on the acuteness gap a: if every triangle angle is at most pi/2 {-} a, then F ~~ 0.36 sqrt{a} suffices; e.g., for a=3°, F ~~ 5°. The proof employs the recent concepts of angle-monotone and radially monotone curves. The proof is constructive, leading to a polynomial-time algorithm for finding the edge-cuts, at worst O(n^2); a version has been implemented.
polyhedra
unfolding
64:1-64:14
Regular Paper
Joseph
O'Rourke
Joseph O'Rourke
10.4230/LIPIcs.SoCG.2018.64
Nicholas Barvinok and Mohammad Ghomi. Pseudo-edge unfoldings of convex polyhedra. arXiv:1709.04944: https://arxiv.org/abs/1709.04944, 2017.
https://arxiv.org/abs/1709.04944
Christopher J. Bishop. Nonobtuse triangulations of PSLGs. Discrete &Comput. Geom., 56(1):43-92, 2016.
Nicolas Bonichon, Prosenjit Bose, Paz Carmi, Irina Kostitsyna, Anna Lubiw, and Sander Verdonschot. Gabriel triangulations and angle-monotone graphs: Local routing and recognition. In Internat. Symp. Graph Drawing Network Vis., pages 519-531. Springer, 2016.
Ju. D. Burago and V. A. Zalgaller. Polyhedral embedding of a net. Vestnik Leningrad. Univ, 15(7):66-80, 1960. In Russian.
Hooman Reisi Dehkordi, Fabrizio Frati, and Joachim Gudmundsson. Increasing-chord graphs on point sets. J. Graph Algorithms Applications, 19(2):761-778, 2015.
Erik D. Demaine and Joseph O'Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, 2007. URL: http://www.gfalop.org.
http://www.gfalop.org
Mohammad Ghomi. Affine unfoldings of convex polyhedra. Geometry &Topology, 18(5):3055-3090, 2014.
John M. Lee. Riemannian Manifolds: An Introduction to Curvature, volume 176. Springer Science &Business Media, 2006.
Anna Lubiw and Joseph O'Rourke. Angle-monotone paths in non-obtuse triangulations. In Proc. 29th Canad. Conf. Comput. Geom., 2017. arXiv:1707.00219 [cs.CG]: URL: https://arxiv.org/abs/1707.00219.
https://arxiv.org/abs/1707.00219
Joseph O'Rourke. Dürer’s problem. In Marjorie Senechal, editor, Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, pages 77-86. Springer, 2013.
Joseph O'Rourke. Unfolding convex polyhedra via radially monotone cut trees. arXiv:1607.07421 [cs.CG]: https://arxiv.org/abs/1607.07421, 2016.
https://arxiv.org/abs/1607.07421
Joseph O'Rourke. Addendum to edge-unfolding nearly flat convex caps. arXiv:1709.02433 [cs.CG]: http://arxiv.org/abs/1709.02433, 2017.
http://arxiv.org/abs/1709.02433
Joseph O'Rourke. Edge-unfolding nearly flat convex caps. arXiv:1707.01006v2 [cs.CG]: http://arxiv.org/abs/1707.01006. Version 2, 2017.
http://arxiv.org/abs/1707.01006
Val Pincu. On the fewest nets problem for convex polyhedra. In Proc. 19th Canad. Conf. Comput. Geom., pages 21-24, 2007.
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A Crossing Lemma for Multigraphs
Let G be a drawing of a graph with n vertices and e>4n edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chvátal, Newborn, Szemerédi and Leighton, the number of crossings in G is at least c{e^3 over n^2}, for a suitable constant c>0. In a seminal paper, Székely generalized this result to multigraphs, establishing the lower bound c{e^3 over mn^2}, where m denotes the maximum multiplicity of an edge in G. We get rid of the dependence on m by showing that, as in the original Crossing Lemma, the number of crossings is at least c'{e^3 over n^2} for some c'>0, provided that the "lens" enclosed by every pair of parallel edges in G contains at least one vertex. This settles a conjecture of Bekos, Kaufmann, and Raftopoulou.
crossing number
Crossing Lemma
multigraph
separator theorem
65:1-65:13
Regular Paper
János
Pach
János Pach
Géza
Tóth
Géza Tóth
10.4230/LIPIcs.SoCG.2018.65
Miklós Ajtai, Vašek Chvátal, Monroe M Newborn, and Endre Szemerédi. Crossing-free subgraphs. North-Holland Mathematics Studies, 60(C):9-12, 1982.
Noga Alon, Paul Seymour, and Robin Thomas. Planar separators. SIAM Journal on Discrete Mathematics, 7(2):184-193, 1994.
Tamal K Dey. Improved bounds for planar k-sets and related problems. Discrete &Computational Geometry, 19(3):373-382, 1998.
Michael R Garey and David S Johnson. Crossing number is np-complete. SIAM Journal on Algebraic Discrete Methods, 4(3):312-316, 1983.
M. Kaufmann. Personal communication. Beyond-Planar Graphs: Algorithmics and Combinatorics, Schloss Dagstuhl, Germany, November 6-11, 2016.
Frank Thomson Leighton. Complexity issues in VLSI: optimal layouts for the shuffle-exchange graph and other networks. MIT press, Cambridge, 1983.
J Pach, F Shahrokhi, and M Szegedy. Applications of the crossing number. Algorithmica, 16(1):111-117, 1996.
János Pach and Géza Tóth. Thirteen problems on crossing numbers. Geombinatorics, 9:199-207, 2000.
Marcus Schaefer. Complexity of some geometric and topological problems. In International Symposium on Graph Drawing, Lecture Notes in Computer Science, volume 5849, pages 334-344. Springer, 2010.
Marcus Schaefer. The graph crossing number and its variants: A survey. The electronic journal of combinatorics, 1000:DS21, 2013.
Micha Sharir and Pankaj K Agarwal. Davenport-Schinzel sequences and their geometric applications. Cambridge University Press, 1995.
László A Székely. Crossing numbers and hard erdős problems in discrete geometry. Combinatorics, Probability and Computing, 6(3):353-358, 1997.
László A Székely. A successful concept for measuring non-planarity of graphs: the crossing number. Discrete Mathematics, 276(1-3):331-352, 2004.
Endre Szemerédi and William T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3(3-4):381-392, 1983.
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Near-Optimal Coresets of Kernel Density Estimates
We construct near-optimal coresets for kernel density estimate for points in R^d when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size O(sqrt{d log (1/epsilon)}/epsilon), and we show a near-matching lower bound of size Omega(sqrt{d}/epsilon). The upper bound is a polynomial in 1/epsilon improvement when d in [3,1/epsilon^2) (for all kernels except the Gaussian kernel which had a previous upper bound of O((1/epsilon) log^d (1/epsilon))) and the lower bound is the first known lower bound to depend on d for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.
Coresets
Kernel Density Estimate
Discrepancy
66:1-66:13
Regular Paper
Jeff M.
Phillips
Jeff M. Phillips
Wai Ming
Tai
Wai Ming Tai
10.4230/LIPIcs.SoCG.2018.66
Ery Arias-Castro, David Mason, and Bruno Pelletier. On the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm. Journal of Machine Learning Research, 17(43):1-28, 2016.
Francis Bach, Simon Lacoste-Julien, and Guillaume Obozinski. On the equivalence between herding and conditional gradient algorithms. In ICML 2012 International Conference on Machine Learning, 2012.
Wojciech Banaszczyk. Balancing vectors and gaussian measures of n-dimensional convex bodies. Random Structures &Algorithms, 12(4):351-360, 1998.
Nikhil Bansal, Daniel Dadush, Shashwat Garg, and Shachar Lovett. The Gram-Schmidt walk: A cure for the Banaszczyk blues (to appear). Proceedings of the fiftieth annual ACM symposium on Theory of computing, 2018.
Jon Louis Bentley and James B. Saxe. Decomposable searching problems I: Static-to-dynamic transformations. Journal of Algorithms, 1(4), 1980.
Omer Bobrowski, Sayan Mukherjee, and Jonathan E. Taylor. Topological consistency via kernel estimation. Bernoulli, 23:288-328, 2017.
Bernard Chazelle. The Discrepancy Method. Cambridge, 2000.
Bernard Chazelle and Jiri Matousek. On linear-time deterministic algorithms for optimization problems in fixed dimensions. J. Algorithms, 21:579-597, 1996.
Luc Devroye and László Györfi. Nonparametric Density Estimation: The L₁ View. Wiley, 1984.
Petros Drineas and Michael W. Mahoney. On the Nyström method for approximating a Gram matrix for improved kernel-based learning. JLMR, 6:2153-2175, 2005.
Jianqing Fan and Irene Gijbels. Local polynomial modelling and its applications: monographs on statistics and applied probability 66, volume 66. CRC Press, 1996.
Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh. Confidence sets for persistence diagrams. The Annals of Statistics, 42:2301-2339, 2014.
Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Scholkopf, and Alexander Smola. A kernel two-sample test. Journal of Machine Learning Research, 13:723-773, 2012.
Matthias Hein and Olivier Bousquet. Hilbertian metrics and positive definite kernels on probability measures. In AISTATS, pages 136-143, 2005.
Thomas Hofmann, Bernhard Schölkopf, and Alexander J. Smola. A review of kernel methods in machine learning. Technical Report 156, Max Planck Institute for Biological Cybernetics, 2006.
Sarang Joshi, Raj Varma Kommaraji, Jeff M Phillips, and Suresh Venkatasubramanian. Comparing distributions and shapes using the kernel distance. In Proceedings of the twenty-seventh annual symposium on Computational geometry, pages 47-56. ACM, 2011.
Jiri Matousek. Geometric Discrepancy; An Illustrated Guide, 2nd printing. Springer-Verlag, 2010.
Jiri Matousek, Aleksandar Nikolov, and Kunal Talwar. Factorization norms and hereditary discrepancy. arXiv preprint arXiv:1408.1376, 2014.
Jeff M. Phillips. Algorithms for ε-approximations of terrains. In ICALP, 2008.
Jeff M Phillips. ε-samples for kernels. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 1622-1632. SIAM, 2013.
Jeff M Phillips and Wai Ming Tai. Improved coresets for kernel density estimates. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2718-2727. SIAM, 2018.
Jeff M. Phillips, Bei Wang, and Yan Zheng. Geometric inference on kernel density estimates. In SOCG, 2015.
Alessandro Rinaldo and Larry Wasserman. Generalized density clustering. The Annals of Statistics, pages 2678-2722, 2010.
Isaac J Schoenberg. Metric spaces and completely monotone functions. Annals of Mathematics, pages 811-841, 1938.
Bernhard Scholkopf and Alexander J. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, 2002.
Erich Schubert, Arthur Zimek, and Hans-Peter Kriegel. Generalized outlier detection with flexible kernel density estimates. In Proceedings of the 2014 SIAM International Conference on Data Mining, pages 542-550. SIAM, 2014.
David W. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, 1992.
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Bharath K. Sriperumbudur, Arthur Gretton, Kenji Fukumizu, Bernhard Schölkopf, and Gert R. G. Lanckriet. Hilbert space embeddings and metrics on probability measures. JMLR, 11:1517-1561, 2010.
Yan Zheng and Jeff M. Phillips. l_∞ error and bandwidth selection for kernel density estimates of large data. In KDD, 2015.
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Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line
In the online metric bipartite matching problem, we are given a set S of server locations in a metric space. Requests arrive one at a time, and on its arrival, we need to immediately and irrevocably match it to a server at a cost which is equal to the distance between these locations. A alpha-competitive algorithm will assign requests to servers so that the total cost is at most alpha times the cost of M_{Opt} where M_{Opt} is the minimum cost matching between S and R.
We consider this problem in the adversarial model for the case where S and R are points on a line and |S|=|R|=n. We improve the analysis of the deterministic Robust Matching Algorithm (RM-Algorithm, Nayyar and Raghvendra FOCS'17) from O(log^2 n) to an optimal Theta(log n). Previously, only a randomized algorithm under a weaker oblivious adversary achieved a competitive ratio of O(log n) (Gupta and Lewi, ICALP'12). The well-known Work Function Algorithm (WFA) has a competitive ratio of O(n) and Omega(log n) for this problem. Therefore, WFA cannot achieve an asymptotically better competitive ratio than the RM-Algorithm.
Bipartite Matching
Online Algorithms
Adversarial Model
Line Metric
67:1-67:14
Regular Paper
Sharath
Raghvendra
Sharath Raghvendra
10.4230/LIPIcs.SoCG.2018.67
Antonios Antoniadis, Neal Barcelo, Michael Nugent, Kirk Pruhs, and Michele Scquizzato. A o(n)-competitive deterministic algorithm for online matching on a line. In Approximation and Online Algorithms: 12th International Workshop, WAOA 2014, pages 11-22, 2015.
Koutsoupias Elias and Nanavati Akash. The online matching problem on a line. In Approximation and Online Algorithms: First International Workshop, WAOA 2003, Budapest, Hungary, September 16-18, 2003. Revised Papers, pages 179-191, 2004.
A. Gupta and K. Lewi. The online metric matching problem for doubling metrics. In Proceedings of International Conference on Automata, Languages and Programming, volume 7391, pages 424-435, 2012.
Bala Kalyanasundaram and Kirk Pruhs. Online weighted matching. J. Algorithms, 14(3):478-488, 1993.
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Krati Nayyar and Sharath Raghvendra. An input sensitive online algorithm for the metric bipartite matching problem. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 505-515, 2017.
Sharath Raghvendra. A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016), volume 60, pages 18:1-18:16, 2016.
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Almost All String Graphs are Intersection Graphs of Plane Convex Sets
A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n --> infty). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.
String graph
intersection graph
plane convex set
68:1-68:14
Regular Paper
János
Pach
János Pach
Bruce
Reed
Bruce Reed
Yelena
Yuditsky
Yelena Yuditsky
10.4230/LIPIcs.SoCG.2018.68
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An Improved Bound for the Size of the Set A/A+A
It is established that for any finite set of positive real numbers A, we have |A/A+A| >> |A|^{3/2+1/26} / log^{5/6}|A|.
sum-product estimates
expanders
incidence theorems
discrete geometry
69:1-69:12
Regular Paper
Oliver
Roche-Newton
Oliver Roche-Newton
10.4230/LIPIcs.SoCG.2018.69
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Fractal Dimension and Lower Bounds for Geometric Problems
We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension.
More specifically, we show that for any set of n points in d-dimensional Euclidean space, of fractal dimension delta in (1,d), for any epsilon>0 and c >= 1, any c-spanner must have treewidth at least Omega(n^{1-1/(delta - epsilon)} / c^{d-1}), matching the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type:
- For any delta in (1,d) and any epsilon >0, d-dimensional Euclidean TSP on n points with fractal dimension at most delta cannot be solved in time 2^{O(n^{1-1/(delta - epsilon)})}. The best-known upper bound is 2^{O(n^{1-1/delta} log n)}.
- For any delta in (1,d) and any epsilon >0, the problem of finding k-pairwise non-intersecting d-dimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most delta cannot be solved in time f(k)n^{O (k^{1-1/(delta - epsilon)})} for any computable function f. The best-known upper bound is n^{O(k^{1-1/delta} log n)}. The above results nearly match previously known upper bounds from [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014].
fractal dimension
treewidth
spanners
lower bounds
exponential time hypothesis
70:1-70:14
Regular Paper
Anastasios
Sidiropoulos
Anastasios Sidiropoulos
Kritika
Singhal
Kritika Singhal
Vijay
Sridhar
Vijay Sridhar
10.4230/LIPIcs.SoCG.2018.70
Jochen Alber and Jiří Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. In Foundations of Information Technology in the Era of Network and Mobile Computing, pages 26-37. Springer, 2002.
Yair Bartal, Lee-Ad Gottlieb, and Robert Krauthgamer. The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 663-672. ACM, 2012.
Hubert TH Chan, Anupam Gupta, Bruce M Maggs, and Shuheng Zhou. On hierarchical routing in doubling metrics. In Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pages 762-771. Society for Industrial and Applied Mathematics, 2005.
T-H Hubert Chan and Anupam Gupta. Small hop-diameter sparse spanners for doubling metrics. Discrete &Computational Geometry, 41(1):28-44, 2009.
Chandra Chekuri and Julia Chuzhoy. Polynomial bounds for the grid-minor theorem. Journal of the ACM (JACM), 63(5):40, 2016.
Richard Cole and Lee-Ad Gottlieb. Searching dynamic point sets in spaces with bounded doubling dimension. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 574-583. ACM, 2006.
Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
Lee-Ad Gottlieb and Liam Roditty. An optimal dynamic spanner for doubling metric spaces. In European Symposium on Algorithms, pages 478-489. Springer, 2008.
Anupam Gupta and Kevin Lewi. The online metric matching problem for doubling metrics. In International Colloquium on Automata, Languages, and Programming, pages 424-435. Springer, 2012.
Sariel Har-Peled and Manor Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM Journal on Computing, 35(5):1148-1184, 2006.
T-H Hubert Chan and Anupam Gupta. Approximating tsp on metrics with bounded global growth. SIAM Journal on Computing, 41(3):587-617, 2012.
David R Karger and Matthias Ruhl. Finding nearest neighbors in growth-restricted metrics. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 741-750. ACM, 2002.
Kyohei Kozawa, Yota Otachi, and Koichi Yamazaki. The carving-width of generalized hypercubes. Discrete Math., 310(21):2867-2876, 2010. URL: http://dx.doi.org/10.1016/j.disc.2010.06.039.
http://dx.doi.org/10.1016/j.disc.2010.06.039
Robert Krauthgamer and James R Lee. The black-box complexity of nearest-neighbor search. Theoretical Computer Science, 348(2):262-276, 2005.
Robert Krauthgamer and James R Lee. Algorithms on negatively curved spaces. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 119-132. IEEE, 2006.
Robert Krauthgamer, James R Lee, Manor Mendel, and Assaf Naor. Measured descent: A new embedding method for finite metrics. Geometric &Functional Analysis GAFA, 15(4):839-858, 2005.
Dániel Marx. Efficient approximation schemes for geometric problems. In European Symposium on Algorithms, pages 448-459. Springer, 2005.
Dániel Marx. Can you beat treewidth? Theory OF Computing, 6:85-112, 2010.
Dániel Marx and Anastasios Sidiropoulos. The limited blessing of low dimensionality: When 1-1/d is the best possible exponent for d-dimensional geometric problems. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 67:67-67:76, New York, NY, USA, 2014. ACM. URL: http://dx.doi.org/10.1145/2582112.2582124.
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Neil Robertson, Paul Seymour, and Robin Thomas. Quickly excluding a planar graph. Journal of Combinatorial Theory, Series B, 62(2):323-348, 1994.
Neil Robertson and Paul D Seymour. Graph minors. v. excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92-114, 1986.
Jeffrey S Salowe. Construction of multidimensional spanner graphs, with applications to minimum spanning trees. In Proceedings of the seventh annual symposium on Computational geometry, pages 256-261. ACM, 1991.
Anastasios Sidiropoulos and Vijay Sridhar. Algorithmic interpretations of fractal dimension. In 33rd International Symposium on Computational Geometry, SoCG 2017, July 4-7, 2017, Brisbane, Australia, volume 77 of LIPIcs, pages 58:1-58:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://www.dagstuhl.de/dagpub/978-3-95977-038-5, URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.58.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2017.58
Warren D Smith and Nicholas C Wormald. Geometric separator theorems and applications. In Foundations of Computer Science, 1998. Proceedings. 39th Annual Symposium on, pages 232-243. IEEE, 1998.
Hideki Takayasu. Fractals in the physical sciences. Manchester University Press, 1990.
Kunal Talwar. Bypassing the embedding: algorithms for low dimensional metrics. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pages 281-290. ACM, 2004.
Pravin M Vaidya. A sparse graph almost as good as the complete graph on points ink dimensions. Discrete &Computational Geometry, 6(3):369-381, 1991.
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The Trisection Genus of Standard Simply Connected PL 4-Manifolds
Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. In this note we show that the K3 surface has trisection genus 22. This implies that the trisection genus of all standard simply connected PL 4-manifolds is known. We show that the trisection genus of each of these manifolds is realised by a trisection that is supported by a singular triangulation. Moreover, we explicitly give the building blocks to construct these triangulations.
combinatorial topology
triangulated manifolds
simply connected 4-manifolds
K3 surface
trisections of 4-manifolds
71:1-71:13
Regular Paper
Jonathan
Spreer
Jonathan Spreer
Stephan
Tillmann
Stephan Tillmann
10.4230/LIPIcs.SoCG.2018.71
Biplab Basak and Jonathan Spreer. Simple crystallizations of 4-manifolds. Adv. in Geom., 16(1):111-130, 2016. arXiv:1407.0752 [math.GT]. URL: http://dx.doi.org/10.1515/advgeom-2015-0043.
http://dx.doi.org/10.1515/advgeom-2015-0043
M. Bell, J. Hass, J. Hyam Rubinstein, and S. Tillmann. Computing trisections of 4-manifolds. ArXiv e-prints, nov 2017. URL: http://arxiv.org/abs/1711.02763.
http://arxiv.org/abs/1711.02763
Bruno Benedetti and Frank H. Lutz. Random discrete Morse theory and a new library of triangulations. Exp. Math., 23(1):66-94, 2014.
Benjamin A. Burton and Ryan Budney. A census of small triangulated 4-manifolds. in preparation, ≥2017.
Benjamin A. Burton, Ryan Budney, William Pettersson, et al. Regina: Software for low-dimensional topology. http://regina-normal.github.io/, 1999-2017.
Stewart S. Cairns. Introduction of a Riemannian geometry on a triangulable 4-manifold. Ann. of Math. (2), 45:218-219, 1944. URL: http://dx.doi.org/10.2307/1969264.
http://dx.doi.org/10.2307/1969264
Stewart S. Cairns. The manifold smoothing problem. Bull. Amer. Math. Soc., 67:237-238, 1961. URL: http://dx.doi.org/10.1090/S0002-9904-1961-10588-0.
http://dx.doi.org/10.1090/S0002-9904-1961-10588-0
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P. Feller, M. Klug, T. Schirmer, and D. Zemke. Calculating the homology and intersection form of a 4-manifold from a trisection diagram. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1711.04762.
http://arxiv.org/abs/1711.04762
M. Ferri, C. Gagliardi, and L. Grasselli. A graph-theoretical representation of PL-manifolds - a survey on crystallizations. Aequationes Math., 31(2-3):121-141, 1986. URL: http://dx.doi.org/10.1007/BF02188181.
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http://dx.doi.org/10.4310/MRL.2001.v8.n3.a5
David Gay and Robion Kirby. Trisecting 4-manifolds. Geom. Topol., 20(6):3097-3132, 2016. URL: http://dx.doi.org/10.2140/gt.2016.20.3097.
http://dx.doi.org/10.2140/gt.2016.20.3097
David T. Gay. Trisections of Lefschetz pencils. Algebr. Geom. Topol., 16(6):3523-3531, 2016. URL: http://dx.doi.org/10.2140/agt.2016.16.3523.
http://dx.doi.org/10.2140/agt.2016.16.3523
A. Hatcher. Algebraic Topology. Cambridge University Press, 2002. URL: http://books.google.de/books?id=BjKs86kosqgC.
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Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11):4983-4997, 2016. URL: http://dx.doi.org/10.1090/proc/13105.
http://dx.doi.org/10.1090/proc/13105
Jeffrey Meier, Trent Schirmer, and Alexander Zupan. Classification of trisections and the generalized property R conjecture. Proc. Amer. Math. Soc., 144(11):4983-4997, 2016. URL: http://dx.doi.org/10.1090/proc/13105.
http://dx.doi.org/10.1090/proc/13105
Jeffrey Meier and Alexander Zupan. Bridge trisections of knotted surfaces in S⁴. Trans. Amer. Math. Soc., 369(10):7343-7386, 2017. URL: http://dx.doi.org/10.1090/tran/6934.
http://dx.doi.org/10.1090/tran/6934
Jeffrey Meier and Alexander Zupan. Genus-two trisections are standard. Geom. Topol., 21(3):1583-1630, 2017. URL: http://dx.doi.org/10.2140/gt.2017.21.1583.
http://dx.doi.org/10.2140/gt.2017.21.1583
John Milnor and Dale Husemoller. Symmetric bilinear forms, volume 73 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York, 1973.
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J. Hyam Rubinstein and S. Tillmann. Multisections of piecewise linear manifolds. ArXiv e-prints, 2016. URL: http://arxiv.org/abs/1602.03279.
http://arxiv.org/abs/1602.03279
Nikolai Saveliev. Lectures on the topology of 3-manifolds. De Gruyter Textbook. Walter de Gruyter &Co., Berlin, revised edition, 2012. An introduction to the Casson invariant.
Martin Tancer. Recognition of collapsible complexes is np-complete. Discrete & Computational Geometry, 55(1):21-38, Jan 2016. URL: http://dx.doi.org/10.1007/s00454-015-9747-1.
http://dx.doi.org/10.1007/s00454-015-9747-1
J. H. C. Whitehead. On C¹-complexes. Ann. of Math. (2), 41:809-824, 1940. URL: http://dx.doi.org/10.2307/1968861.
http://dx.doi.org/10.2307/1968861
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An O(n log n)-Time Algorithm for the k-Center Problem in Trees
We consider a classical k-center problem in trees. Let T be a tree of n vertices and every vertex has a nonnegative weight. The problem is to find k centers on the edges of T such that the maximum weighted distance from all vertices to their closest centers is minimized. Megiddo and Tamir (SIAM J. Comput., 1983) gave an algorithm that can solve the problem in O(n log^2 n) time by using Cole's parametric search. Since then it has been open for over three decades whether the problem can be solved in O(n log n) time. In this paper, we present an O(n log n) time algorithm for the problem and thus settle the open problem affirmatively.
k-center
trees
facility locations
72:1-72:15
Regular Paper
Haitao
Wang
Haitao Wang
Jingru
Zhang
Jingru Zhang
10.4230/LIPIcs.SoCG.2018.72
P.K. Agarwal and J.M. Phillips. An efficient algorithm for 2D Euclidean 2-center with outliers. In Proceedings of the 16th Annual European Conference on Algorithms(ESA), pages 64-75, 2008.
M. Ajtai, J. Komlós, and E. Szemerédi. An O(nlog n) sorting network. In Proc. of the 15th Annual ACM Symposium on Theory of Computing (STOC), pages 1-9, 1983.
A. Banik, B. Bhattacharya, S. Das, T. Kameda, and Z. Song. The p-center problem in tree networks revisited. In Proc. of the 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 6:1-6:15, 2016.
B. Bhattacharya and Q. Shi. Optimal algorithms for the weighted p-center problems on the real line for small p. In Proc. of the 10th International Workshop on Algorithms and Data Structures, pages 529-540, 2007.
P. Brass, C. Knauer, H.-S. Na, C.-S. Shin, and A. Vigneron. The aligned k-center problem. International Journal of Computational Geometry and Applications, 21:157-178, 2011.
H Brönnimann and B. Chazelle. Optimal slope selection via cuttings. Computational Geometry: Theory and Applications, 10(1):23-29, 1998.
T.M. Chan. More planar two-center algorithms. Computational Geometry: Theory and Applications, 13:189-198, 1999.
R. Chandrasekaran and A. Tamir. Polynomially bounded algorithms for locating p-centers on a tree. Mathematical Programming, 22(1):304-315, 1982.
D.Z. Chen, J. Li, and H. Wang. Efficient algorithms for the one-dimensional k-center problem. Theoretical Computer Science, 592:135-142, 2015.
D.Z. Chen and H. Wang. Approximating points by a piecewise linear function. Algorithmica, 88:682-713, 2013.
D.Z. Chen and H. Wang. A note on searching line arrangements and applications. Information Processing Letters, 113:518-521, 2013.
R. Cole. Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM, 34(1):200-208, 1987.
G. Cormode and A. McGregor. Approximation algorithms for clustering uncertain data. In Proc. of the 27t Symposium on Principles of Database Systems (PODS), pages 191-200, 2008.
G. Frederickson and D. Johnson. Generalized selection and ranking: Sorted matrices. SIAM Journal on Computing, 13(1):14-30, 1984.
G.N. Frederickson. Optimal algorithms for tree partitioning. In Proc. of the 2nd Annual ACM-SIAM Symposium of Discrete Algorithms (SODA), pages 168-177, 1991.
G.N. Frederickson. Parametric search and locating supply centers in trees. In Proc. of the 2nd International Workshop on Algorithms and Data Structures (WADS), pages 299-319, 1991.
G.N. Frederickson and D.B. Johnson. Finding kth paths and p-centers by generating and searching good data structures. Journal of Algorithms, 4(1):61-80, 1983.
L. Huang and J. Li. Stochastic k-center and j-flat-center problems. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 110-129, 2017.
M. Jeger and O. Kariv. Algorithms for finding P-centers on a weighted tree (for relatively small P). Networks, 15(3):381-389, 1985.
O. Kariv and S.L. Hakimi. An algorithmic approach to network location problems. I: The p-centers. SIAM Journal on Applied Mathematics, 37(3):513-538, 1979.
A. Karmakar, S. Das, S.C. Nandy, and B.K. Bhattacharya. Some variations on constrained minimum enclosing circle problem. Journal of Combinatorial Optimization, 25(2):176-190, 2013.
M. Katz and M. Sharir. Optimal slope selection via expanders. Information Processing Letters, 47(3):115-122, 1993.
N. Megiddo. Linear-time algorithms for linear programming in R³ and related problems. SIAM Journal on Computing, 12(4):759-776, 1983.
N. Megiddo and K.J. Supowit. On the complexity of some common geometric location problems. SIAM Journal on Comuting, 13:182-196, 1984.
N. Megiddo and A. Tamir. New results on the complexity of p-centre problems. SIAM Journal on Computing, 12(4):751-758, 1983.
N. Megiddo, A. Tamir, E. Zemel, and R. Chandrasekaran. An O(n log² n) algorithm for the k-th longest path in a tree with applications to location problems. SIAM Journal on Computing, 10:328-337, 1981.
H. Wang and J. Zhang. One-dimensional k-center on uncertain data. Theoretical Computer Science, 602:114-124, 2015.
H. Wang and J. Zhang. Line-constrained k-median, k-means, and k-center problems in the plane. International Journal of Computational Geometry and Applications, 26:185-210, 2016.
H. Wang and J. Zhang. A note on computing the center of uncertain data on the real line. Operations Research Letters, 44:370-373, 2016.
H. Wang and J. Zhang. Computing the center of uncertain points on tree networks. Algorithmica, 609:32-48, 2017.
H. Wang and J. Zhang. Covering uncertain points in a tree. In Proc. of the 15th Algorithms and Data Structures Symposium (WADS), pages 557-568, 2017.
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New Bounds for Range Closest-Pair Problems
Given a dataset S of points in R^2, the range closest-pair (RCP) problem aims to preprocess S into a data structure such that when a query range X is specified, the closest-pair in S cap X can be reported efficiently. The RCP problem can be viewed as a range-search version of the classical closest-pair problem, and finds applications in many areas. Due to its non-decomposability, the RCP problem is much more challenging than many traditional range-search problems. This paper revisits the RCP problem, and proposes new data structures for various query types including quadrants, strips, rectangles, and halfplanes. Both worst-case and average-case analyses (in the sense that the data points are drawn uniformly and independently from the unit square) are applied to these new data structures, which result in new bounds for the RCP problem. Some of the new bounds significantly improve the previous results, while the others are entirely new.
Closest-pair
Range search
Candidate pairs
Average-case analysis
73:1-73:14
Regular Paper
Jie
Xue
Jie Xue
Yuan
Li
Yuan Li
Saladi
Rahul
Saladi Rahul
Ravi
Janardan
Ravi Janardan
10.4230/LIPIcs.SoCG.2018.73
Mohammad Ali Abam, Paz Carmi, Mohammad Farshi, and Michiel Smid. On the power of the semi-separated pair decomposition. In Workshop on Algorithms and Data Structures, pages 1-12. Springer, 2009.
Pankaj K Agarwal, Jeff Erickson, et al. Geometric range searching and its relatives. Contemporary Mathematics, 223:1-56, 1999.
M. de Berg, M. van Kreveld, M. Overmars, and O. C. Schwarzkopf. Computational geometry. In Computational geometry, pages 1-17. Springer, 2000.
Herbert Edelsbrunner, Leonidas J Guibas, and Jorge Stolfi. Optimal point location in a monotone subdivision. SIAM Journal on Computing, 15(2):317-340, 1986.
Prosenjit Gupta. Range-aggregate query problems involving geometric aggregation operations. Nordic journal of Computing, 13(4):294-308, 2006.
Prosenjit Gupta, Ravi Janardan, Yokesh Kumar, and Michiel Smid. Data structures for range-aggregate extent queries. Computational Geometry: Theory and Applications, 2(47):329-347, 2014.
Saladi Rahul and Yufei Tao. On top-k range reporting in 2D space. In Proceedings of the 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 265-275. ACM, 2015.
Neil Sarnak and Robert E Tarjan. Planar point location using persistent search trees. Communications of the ACM, 29(7):669-679, 1986.
Jing Shan, Donghui Zhang, and Betty Salzberg. On spatial-range closest-pair query. In International Symposium on Spatial and Temporal Databases, pages 252-269. Springer, 2003.
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Jie Xue, Yuan Li, Saladi Rahul, and Ravi Janardan. New bounds for range closest-pair problems. arXiv preprint arXiv:1712.09749, 2017.
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Coordinated Motion Planning: The Video (Multimedia Exposition)
We motivate, visualize and demonstrate recent work for minimizing the total execution time of a coordinated, parallel motion plan for a swarm of N robots in the absence of obstacles. Under relatively mild assumptions on the separability of robots, the algorithm achieves constant stretch: If all robots want to move at most d units from their respective starting positions, then the total duration of the overall schedule (and hence the distance traveled by each robot) is O(d) steps; this implies constant-factor approximation for the optimization problem. Also mentioned is an NP-hardness result for finding an optimal schedule, even in the case in which robot positions are restricted to a regular grid. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Omega(N^{1/4}) is required in the worst case; we establish an achievable stretch factor of O(N^{1/2}) even in this case. We also sketch geometric difficulties of computing optimal trajectories, even for just two unit disks.
Motion planning
robot swarms
complexity
stretch
approximation
74:1-74:6
Multimedia Exposition
Aaron T.
Becker
Aaron T. Becker
Sándor P.
Fekete
Sándor P. Fekete
Phillip
Keldenich
Phillip Keldenich
Matthias
Konitzny
Matthias Konitzny
Lillian
Lin
Lillian Lin
Christian
Scheffer
Christian Scheffer
10.4230/LIPIcs.SoCG.2018.74
Erik D. Demaine, Sándor P. Fekete, Phillip Keldenich, Henk Meijer, and Christian Scheffer. Coordinated motion planning: Reconfiguring a swarm of labeled robots with bounded stretch. In Csaba Tóth and Bettina Speckmann, editors, 34th International Symposium on Computational Geometry (SoCG 2018, these proceedings), volume 99 of Leibniz International Proceedings in Informatics (LIPIcs), pages 29:1-29:17, 2018. Full version at https://arxiv.org/abs/1801.01689. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.29.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.29
Jacob T. Schwartz and Micha Sharir. On the piano movers' problem: III. Coordinating the motion of several independent bodies: the special case of circular bodies moving amidst polygonal barriers. Int. J. Robotics Research, 2(3):46-75, 1983.
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Geometric Realizations of the 3D Associahedron (Multimedia Exposition)
The associahedron is a convex polytope whose 1-skeleton is isomorphic to the flip graph of a convex polygon. There exists an elegant geometric realization of the associahedron, using the remarkable theory of secondary polytopes, based on the geometry of the underlying polygon. We present an interactive application that visualizes this correspondence in the 3D case.
associahedron
secondary polytope
realization
75:1-75:4
Multimedia Exposition
Satyan L.
Devadoss
Satyan L. Devadoss
Daniel D.
Johnson
Daniel D. Johnson
Justin
Lee
Justin Lee
Jackson
Warley
Jackson Warley
10.4230/LIPIcs.SoCG.2018.75
Louis J Billera, Susan P Holmes, and Karen Vogtmann. Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4):733-767, 2001.
Michael Bostock, Vadim Ogievetsky, and Jeffrey Heer. D³ data-driven documents. IEEE transactions on visualization and computer graphics, 17(12):2301-2309, 2011.
Ricardo Cabello et al. Three.js, 2010. URL: https://github.com/mrdoob/three.js.
https://github.com/mrdoob/three.js
Cesar Ceballos, Francisco Santos, and Günter M Ziegler. Many non-equivalent realizations of the associahedron. Combinatorica, 35:513-551, 2015.
Frederic Chapoton, Sergey Fomin, and Andrei Zelevinsky. Polytopal realizations of generalized associahedra. Bulletin Canadien de Mathématiques, 45:537-566, 2002.
Satyan L Devadoss. Tessellations of moduli spaces and the mosaic operad. Contemporary Mathematics, 239:91-114, 1999.
Satyan L Devadoss and Joseph O'Rourke. Discrete and Computational Geometry. Princeton University Press, 2011.
Satyan L Devadoss, Rahul Shah, Xuancheng Shao, and Ezra Winston. Deformations of associahedra and visibility graphs. Contributions to Discrete Mathematics, 7:68-81, 2012.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Lagrangian intersection Floer theory: Anomaly and obstruction. American Mathematical Society, 2010.
Israel M Gelfand, Mikhail Kapranov, and Andrei Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Springer, 2008.
Christophe Hohlweg and Carsten Lange. Realizations of the associahedron and cyclohedron. Discrete &Computational Geometry, 37(4):517-543, 2007.
Ivan Kuckir. PolyK.js. URL: http://polyk.ivank.net/.
http://polyk.ivank.net/
Carl W Lee. The associahedron and triangulations of the n-gon. European Journal of Combinatorics, 10:551-560, 1989.
Chiu-Chu Melissa Liu. Moduli of J-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for an S¹-equivariant pair. arXiv math/0210257, 2002.
Sébastien Loisel. Numeric javascript. URL: http://www.numericjs.com/.
http://www.numericjs.com/
Alexander Postnikov. Permutohedra, associahedra, and beyond. International Mathematics Research Notices, 2009:1026-1106, 2009.
Jim Stasheff. From operads to ‘physically’ inspired theories. In Operads: Proceedings of Renaissance Conferences, volume 202, page 53. American Mathematical Society, 1997.
M Yoshida. Hypergeometric functions, my love. Vieweg, 1997.
Creative Commons Attribution 3.0 Unported license
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Star Unfolding of Boxes (Multimedia Exposition)
Given a convex polyhedron, the star unfolding of its surface is obtained by cutting along the shortest paths from a fixed source point to each of its vertices. We present an interactive application that visualizes the star unfolding of a box, such that its dimensions and source point locations can be continuously toggled by the user.
star unfolding
source unfolding
Voronoi diagram
76:1-76:4
Multimedia Exposition
Dani
Demas
Dani Demas
Satyan L.
Devadoss
Satyan L. Devadoss
Yu Xuan
Hong
Yu Xuan Hong
10.4230/LIPIcs.SoCG.2018.76
A.D. Alexandrov. Convex Polyhedra. Springer Monographs in Mathematics. Springer Berlin Heidelberg, 2006. URL: https://books.google.com/books?id=aoMreDT_DwcC.
https://books.google.com/books?id=aoMreDT_DwcC
Boris Aronov and Joseph O'Rourke. Nonoverlap of the star unfolding. Discrete & Computational Geometry, 8(3):219-250, Sep 1992. URL: http://dx.doi.org/10.1007/BF02293047.
http://dx.doi.org/10.1007/BF02293047
Satyan L. Devadoss and Joseph O'Rourke. Discrete and Computational Geometry. Princeton University Press, 2011.
Raymond Hill. Javascript-voronoi. https://github.com/gorhill/Javascript-Voronoi, 2013.
https://github.com/gorhill/Javascript-Voronoi
Creative Commons Attribution 3.0 Unported license
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VoroCrust Illustrated: Theory and Challenges (Multimedia Exposition)
Over the past decade, polyhedral meshing has been gaining popularity as a better alternative to tetrahedral meshing in certain applications. Within the class of polyhedral elements, Voronoi cells are particularly attractive thanks to their special geometric structure. What has been missing so far is a Voronoi mesher that is sufficiently robust to run automatically on complex models. In this video, we illustrate the main ideas behind the VoroCrust algorithm, highlighting both the theoretical guarantees and the practical challenges imposed by realistic inputs.
sampling
surface reconstruction
polyhedral meshing
Voronoi
77:1-77:4
Multimedia Exposition
Ahmed
Abdelkader
Ahmed Abdelkader
Chandrajit L.
Bajaj
Chandrajit L. Bajaj
Mohamed S.
Ebeida
Mohamed S. Ebeida
Ahmed H.
Mahmoud
Ahmed H. Mahmoud
Scott A.
Mitchell
Scott A. Mitchell
John D.
Owens
John D. Owens
Ahmad A.
Rushdi
Ahmad A. Rushdi
10.4230/LIPIcs.SoCG.2018.77
A. Abdelkader, C. Bajaj, M. Ebeida, A. Mahmoud, S. Mitchell, J. Owens, and A. Rushdi. VoroCrust illustrated: theory and challenges (video content). URL: http://computational-geometry.org/SoCG-videos/socg18video/videos/VoroCrust/.
http://computational-geometry.org/SoCG-videos/socg18video/videos/VoroCrust/
A. Abdelkader, C. Bajaj, M. Ebeida, A. Mahmoud, S. Mitchell, J. Owens, and A. Rushdi. Sampling conditions for conforming Voronoi meshing by the VoroCrust algorithm. In 34th International Symposium on Computational Geometry (SoCG 2018), pages 1:1-1:16, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.1.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.1
A. Abdelkader, C. Bajaj, M. Ebeida, A. Mahmoud, S. Mitchell, J. Owens, and A. Rushdi. VoroCrust: Voronoi meshing without clipping. Manuscript, In preparation.
A. Abdelkader, C. Bajaj, M. Ebeida, and S. Mitchell. A Seed Placement Strategy for Conforming Voronoi Meshing. In Canadian Conference on Computational Geometry, 2017.
J. Ahrens, B. Geveci, and C. Law. Paraview: An end-user tool for large-data visualization. In C. Hansen and C. Johnson, editors, Visualization Handbook, pages 717-731. Butterworth-Heinemann, 2005.
N. Amenta and R.-K. Kolluri. The medial axis of a union of balls. Computational Geometry, 20(1):25-37, 2001. Selected papers from the 12th Annual Canadian Conference.
N. Bellomo, F. Brezzi, and G. Manzini. Recent techniques for PDE discretizations on polyhedral meshes. Mathematical Models and Methods in Applied Sciences, 24(08):1453-1455, 2014.
T. Brochu, C. Batty, and R. Bridson. Matching fluid simulation elements to surface geometry and topology. ACM Trans. Graph., 29(4):47:1-47:9, 2010.
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M. Ebeida and S. Mitchell. Uniform random Voronoi meshes. In International Meshing Roundtable (IMR), pages 258-275, 2011.
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M. Peric and S. Ferguson. The advantage of polyhedral meshes. Dynamics - Issue 24, page 4–5, Spring 2005. The customer magazine of the CD-adapco Group, currently maintained by Siemens at http://siemens.com/mdx. The issue is available at http://mdx2.plm.automation.siemens.com/magazine/dynamics-24 (accessed March 29, 2018).
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M. Yip, J. Mohle, and J. Bolander. Automated modeling of three‐dimensional structural components using irregular lattices. Computer-Aided Civil and Infrastructure Engineering, 20(6):393-407, 2005.
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