33rd International Symposium on Computational Geometry (SoCG 2017), SoCG 2017, July 4-7, 2017, Brisbane, Australia
SoCG 2017
July 4-7, 2017
Brisbane, Australia
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
Boris
Aronov
Boris Aronov
Matthew J.
Katz
Matthew J. Katz
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
77
2017
978-3-95977-038-5
https://www.dagstuhl.de/dagpub/978-3-95977-038-5
Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers, Sponsors
Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers, Sponsors
Front Matter
Table of Contents
Foreword
Conference Organization
External Reviewers
Sponsors
0:i-0:xviii
Front Matter
Boris
Aronov
Boris Aronov
Matthew J.
Katz
Matthew J. Katz
10.4230/LIPIcs.SoCG.2017.0
Creative Commons Attribution 3.0 Unported license
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The Geometry and Topology of Crystals: From Sphere-Packing to Tiling, Nets, and Knots (Invited Talk)
Crystal structures have inspired developments in geometry since the Ancient Greeks conceived of Platonic solids after observing tetrahedral, cubical and octahedral mineral forms in their local environment. The internal structure of crystals became accessible with the development of x-ray diffraction techniques just over 100 years ago, and a key step in developing this method was understanding the arrangement of atoms in the simplest crystals as close-packings of spheres. Determining a crystal structure via x-ray diffraction unavoidably requires prior models, and this has led to the intense study of sphere packing, atom-bond networks, and arrangements of polyhedra by crystallographers investigating ever more complex compounds. In the 21st century, chemists are exploring the possibilities of coordination polymers, a wide class of crystalline materials that self-assemble from metal cations and organic ligands into periodic framework materials. Longer organic ligands mean these compounds can form multi-component interwoven network structures where the "edges" are no longer constrained to join nearest-neighbour "nodes" as in simpler atom-bond networks. The challenge for geometers is to devise algorithms for enumerating relevant structures and to devise invariants that will distinguish between different modes of interweaving. This talk will survey various methods from computational geometry and topology that are currently used to describe crystalline structures and outline research directions to address some of the open questions suggested above.
Mathematical crystallography
Combinatorial tiling theory
Graphs and surfaces in the 3-torus
1:1-1:1
Invited Talk
Vanessa
Robins
Vanessa Robins
10.4230/LIPIcs.SoCG.2017.1
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The Algebraic Revolution in Combinatorial and Computational Geometry: State of the Art (Invited Talk)
For the past 10 years, combinatorial geometry (and to some extent, computational geometry too) has gone through a dramatic revolution, due to the infusion of techniques from algebraic geometry and algebra that have proven effective in solving a variety of hard problems that were thought to be unreachable with more traditional techniques. The new era has begun with two groundbreaking papers of Guth and Katz, the second of which has (almost completely) solved the distinct distances problem of Erdos, open since 1946.
In this talk I will survey some of the progress that has been made since then, including a variety of problems on distinct and repeated distances and other configurations, on incidences between points and lines, curves, and surfaces in two, three, and higher dimensions, on polynomials vanishing on Cartesian products with applications, on cycle elimination for lines and triangles in three dimensions, on range searching with semialgebraic sets, and I will most certainly run out of time while doing so.
Combinatorial Geometry
Incidences
Polynomial method
Algebraic Geometry
Distances
2:1-2:1
Invited Talk
Micha
Sharir
Micha Sharir
10.4230/LIPIcs.SoCG.2017.2
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Irrational Guards are Sometimes Needed
In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon so that the guards together can see the whole polygon. We say that a guard at position x sees a point y if the line segment xy is contained in the polygon.
Despite an extensive study of the art gallery problem, it remained an open question whether there are polygons given by integer coordinates that require guard positions with irrational coordinates in any optimal solution. We give a positive answer to this question by constructing a monotone polygon with integer coordinates that can be guarded by three guards only when we allow to place the guards at points with irrational coordinates. Otherwise, four guards are needed. By extending this example, we show that for every n, there is a polygon which can be guarded by 3n guards with irrational coordinates but needs 4n guards if the coordinates have to be rational. Subsequently, we show that there are rectilinear polygons given by integer coordinates that require guards with irrational coordinates in any optimal solution.
art gallery problem
computational geometry
irrational numbers
3:1-3:15
Regular Paper
Mikkel
Abrahamsen
Mikkel Abrahamsen
Anna
Adamaszek
Anna Adamaszek
Tillmann
Miltzow
Tillmann Miltzow
10.4230/LIPIcs.SoCG.2017.3
Pankaj Kumar Agarwal, Kurt Mehlhorn, and Monique Teillaud. Dagstuhl Seminar 11111, Computational Geometry, March 13-18 , 2011.
Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in real algebraic geometry. Springer-Verlag Berlin Heidelberg, 2006.
Patrice Belleville. Computing two-covers of simple polygons. Master’s thesis, McGill University, 1991.
Édouard Bonnet and Tillmann Miltzow. An approximation algorithm for the art gallery problem. CoRR, abs/1607.05527, 2016.
Édouard Bonnet and Tillmann Miltzow. Parameterized hardness of art gallery problems. In Proceedings of the 24th Annual European Symposium on Algorithms (ESA), pages 19:1-19:17, 2016.
Björn Brodén, Mikael Hammar, and Bengt J. Nilsson. Guarding lines and 2-link polygons is APX-hard. In Proceedings of the 13th Canadian Conference on Computational Geometry (CCCG), pages 45-48, 2001.
John Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the twentieth annual ACM symposium on Theory of computing (STOC), pages 460-467. ACM, 1988.
Jean Cardinal. Computational geometry column 62. SIGACT News, 46(4):69-78, December 2015. URL: http://dx.doi.org/10.1145/2852040.2852053.
http://dx.doi.org/10.1145/2852040.2852053
Vasek Chvátal. A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B, 18(1):39-41, 1975.
Pedro Jussieu de Rezende, Cid C. de Souza, Stephan Friedrichs, Michael Hemmer, Alexander Kröller, and Davi C. Tozoni. Engineering art galleries. In Algorithm Engineering: Selected Results and Surveys, LNCS, pages 379-417. Springer, 2016.
The Sage Developers. SageMath, the Sage Mathematics Software System (Version 7.4), 2016. http://www.sagemath.org.
Alon Efrat and Sariel Har-Peled. Guarding galleries and terrains. Inf. Process. Lett., 100(6):238-245, 2006.
Stephan Eidenbenz, Christoph Stamm, and Peter Widmayer. Inapproximability results for guarding polygons and terrains. Algorithmica, 31(1):79-113, 2001.
Sándor Fekete. Private communication.
Steve Fisk. A short proof of Chvátal’s watchman theorem. J. Comb. Theory, Ser. B, 24(3):374, 1978.
Stephan Friedrichs, Michael Hemmer, James King, and Christiane Schmidt. The continuous 1.5D terrain guarding problem: Discretization, optimal solutions, and PTAS. Journal of Computational Geometry, 7(1):256-284, 2016.
Erik Krohn and Bengt J. Nilsson. Approximate guarding of monotone and rectilinear polygons. Algorithmica, 66(3):564-594, 2013.
Der-Tsai Lee and Arthur K. Lin. Computational complexity of art gallery problems. IEEE Transactions on Information Theory, 32(2):276-282, 1986.
Jiří Matoušek. Intersection graphs of segments and ∃ ℝ. CoRR, abs/1406.2636, 2014.
Joseph O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, 1987.
Joseph O'Rourke and Kenneth Supowit. Some NP-hard polygon decomposition problems. IEEE Transactions on Information Theory, 29(2):181-190, 1983.
Joseph O’Rourke. Visibility. In Jacob E. Goodman and Joseph O’Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 28. Chapman &Hall/CRC, second edition, 2004.
Günter Rote. EuroCG open problem session, 2011. See the personal webpage of Günter Rote: URL: http://page.mi.fu-berlin.de/rote/Papers/slides/Open-Problem_artgallery-Morschach-EuroCG-2011.pdf.
http://page.mi.fu-berlin.de/rote/Papers/slides/Open-Problem_artgallery-Morschach-EuroCG-2011.pdf
Marcus Schaefer. Complexity of some geometric and topological problems. In International Symposium on Graph Drawing, pages 334-344. Springer, 2009.
Dietmar Schuchardt and Hans-Dietrich Hecker. Two NP-hard art-gallery problems for ortho-polygons. Math. Log. Q., 41:261-267, 1995.
Thomas C. Shermer. Recent results in art galleries. Proceedings of the IEEE, 80(9):1384-1399, 1992.
Ana Paula Tomás. Guarding thin orthogonal polygons is hard. In Fundamentals of Computation Theory, pages 305-316. Springer, 2013.
Jorge Urrutia. Art gallery and illumination problems. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 973-1027. Elsevier, 2000.
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Minimum Perimeter-Sum Partitions in the Plane
Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time.
Computational geometry
clustering
minimum-perimeter partition
convex hull
4:1-4:15
Regular Paper
Mikkel
Abrahamsen
Mikkel Abrahamsen
Mark
de Berg
Mark de Berg
Kevin
Buchin
Kevin Buchin
Mehran
Mehr
Mehran Mehr
Ali D.
Mehrabi
Ali D. Mehrabi
10.4230/LIPIcs.SoCG.2017.4
Creative Commons Attribution 3.0 Unported license
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Range-Clustering Queries
In a geometric k-clustering problem the goal is to partition a set of points in R^d into k subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering queries on a point set S: given a query box Q and an integer k > 2, compute an optimal k-clustering for the subset of S inside Q. We obtain the following results.
* We present a general method to compute a (1+epsilon)-approximation to a range-clustering query, where epsilon>0 is a parameter that can be specified as part of the query. Our method applies to a large class of clustering problems, including k-center clustering in any Lp-metric and a variant of k-center clustering where the goal is to minimize the sum (instead of maximum) of the cluster sizes.
* We extend our method to deal with capacitated k-clustering problems, where each of the clusters should not contain more than a given number of points.
* For the special cases of rectilinear k-center clustering in R^1, and in R^2 for k = 2 or 3, we present data structures that answer range-clustering queries exactly.
Geometric data structures
clustering
k-center problem
5:1-5:16
Regular Paper
Mikkel
Abrahamsen
Mikkel Abrahamsen
Mark
de Berg
Mark de Berg
Kevin
Buchin
Kevin Buchin
Mehran
Mehr
Mehran Mehr
Ali D.
Mehrabi
Ali D. Mehrabi
10.4230/LIPIcs.SoCG.2017.5
M. Abam, P. Carmi, M. Farshi, and M. Smid. On the power of the semi-separated pair decomposition. Compututational Geometry: Theory and Applications, 46:631-639, 2013.
P. K. Agarwal, R. Ben Avraham, and M. Sharir. The 2-center problem in three dimensions. Compututational Geometry: Theory and Applications, 46:734-746, 2013.
P. K. Agarwal and Cecilia M. Procopiuc. Exact and approximation algorithms for clustering. Algorithmica, 33:201-226, 2002.
Sunil Arya, David M. Mount, and Eunhui Park. Approximate geometric MST range queries. In Proc. 36th International Symposium on Computational Geometry (SoCG), pages 781-795, 2015.
Peter Brass, Christian Knauer, Chan-Su Shin, Michiel H. M. Smid, and Ivo Vigan. Range-aggregate queries for geometric extent problems. In Computing: The Australasian Theory Symposium 2013, CATS'13, pages 3-10, 2013.
V. Capoyleas, G. Rote, and G. Woeginger. Geometric clusterings. Journal of Algorithms, 12:341-356, 1991.
T. M. Chan. Geometric applications of a randomized optimization technique. Discrete &Compututational Geometry, 22:547-567, 1999.
T. M. Chan. More planar two-center algorithms. Compututational Geometry: Theory and Applications, 13:189-198, 1999.
Bernard Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17:427-462, 1988.
A. W. Das, P. Gupta, K. Kothapalli, and K. Srinathan. On reporting the L₁-metric closest pair in a query rectangle. Information Processing Letters, 114:256-263, 2014.
Mark de Berg, Marc van Kreveld, and Jack Snoeyink. Two- and three-dimensional point location in rectangular subdivisions. Journal of Algorithms, 18:256-277, 1995.
D. Eppstein. Faster construction of planar two-centers. In Proc. 8th Annual ACM-SIAM Symposiun on Discrete Algorithms (SODA), pages 131-138, 1997.
P. Gupta, R. Janardan, Y. Kumar, and M. Smid. Data structures for range-aggregate extent queries. Compututational Geometry: Theory and Applications, 47:329-347, 2014.
Sariel Har-Peled. Geometric Approximation Algorithms, volume 173 of Mathematical surveys and monographs. American Mathematical Society, 2011.
Sariel Har-Peled and Soham Mazumdar. On coresets for k-means and k-median clustering. In Proc. 36th Annual ACM Symposium on Theory of Computing (STOC), pages 291-300, 2004.
M. Hoffmann. A simple linear algorithm for computing rectilinear 3-centers. Compututational Geometry: Theory and Applications, 31:150-165, 2005.
R. Z. Hwang, R. Lee, and R. C. Chang. The generalized searching over separators strategy to solve some NP-hard problems in subexponential time. Algorithmica, 9:398-423, 1993.
S. Khare, J. Agarwal, N. Moidu, and K. Srinathan. Improved bounds for smallest enclosing disk range queries. In Proc. 26th Canadian Conference on Computational Geometry (CCCG), 2014.
H.-P. Lenhof and M. H. M. Smid. Using persistent data structures for adding range restrictions to searching problems. Theoretical Informatics and Applications, 28:25-49, 1994.
Yakov Nekrich and Michiel H. M. Smid. Approximating range-aggregate queries using coresets. In Proc. 22nd Canadian Conference on Computational Geometry (CCCG), pages 253-256, 2010.
Jeff M. Phillips. Algorithms for ε-approximations of terrains. In Proc. 35th International Colloquium on Automata, Languages, and Programming (ICALP), pages 447-458, 2008.
M. Sharir. A near-linear time algorithm for the planar 2-center problem. Discrete &Compututational Geometry, 18:125-134, 1997.
M. Sharir and E. Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th International Symposium on Computational Geometry (SoCG), pages 122-132, 1996.
Gelin Zhou. Two-dimensional range successor in optimal time and almost linear space. Information Processing Letters, 116:171-174, 2016.
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Best Laid Plans of Lions and Men
We answer the following question dating back to J.E. Littlewood (1885-1977): Can two lions catch a man in a bounded area with rectifiable lakes? The lions and the man are all assumed to be points moving with at most unit speed. That the lakes are rectifiable means that their boundaries are finitely long. This requirement is to avoid pathological examples where the man survives forever because any path to the lions is infinitely long. We show that the answer to the question is not always "yes", by giving an example of a region R in the plane where the man has a strategy to survive forever. R is a polygonal region with holes and the exterior and interior boundaries are pairwise disjoint, simple polygons. Our construction is the first truly two-dimensional example where the man can survive.
Next, we consider the following game played on the entire plane instead of a bounded area: There is any finite number of unit speed lions and one fast man who can run with speed 1+epsilon for some value epsilon>0. Can the man always survive? We answer the question in the affirmative for any constant epsilon>0.
Lion and man game
Pursuit evasion game
Winning strategy
6:1-6:16
Regular Paper
Mikkel
Abrahamsen
Mikkel Abrahamsen
Jacob
Holm
Jacob Holm
Eva
Rotenberg
Eva Rotenberg
Christian
Wulff-Nilsen
Christian Wulff-Nilsen
10.4230/LIPIcs.SoCG.2017.6
M. Abrahamsen, J. Holm, E. Rotenberg, and C. Wulff-Nilsen. Best laid plans of lions and men. CoRR, abs/1703.03687, 2017. URL: https://arxiv.org/abs/1703.03687.
https://arxiv.org/abs/1703.03687
Martin Aigner and Michael Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8(1):1-12, 1984.
Noga Alon and Abbas Mehrabian. Chasing a fast robber on planar graphs and random graphs. Journal of Graph Theory, 78(2):81-96, 2015.
Alessandro Berarducci and Benedetto Intrigila. On the cop number of a graph. Advances in Applied Mathematics, 14(4):389-403, 1993.
B. Bollobás, I. Leader, and M. Walters. Lion and man - can both win? Israel Journal of Mathematics, 189(1):267-286, 2012.
Béla Bollobás. The Art of Mathematics: Coffee Time in Memphis. Cambridge University Press, 2006.
Béla Bollobás. The lion and the christian, and other pursuit and evasion games. In Dierk Schleicher and Malte Lackmann, editors, An Invitation to Mathematics: From Competitions to Research, pages 181-193. Springer-Verlag Berlin Heidelberg, 2011.
Hallard T. Croft. "Lion and man": A postscript. Journal of the London Mathematical Society, 39:385-390, 1964.
James Flynn. Lion and man: The boundary constraint. SIAM Journal on Control, 11:397-411, 1973.
James Flynn. Lion and man: The general case. SIAM Journal on Control, 12:581-597, 1974.
Robbert Fokkink, Leonhard Geupel, and Kensaku Kikuta. Open problems on search games. In Steve Alpern, Robbert Fokkink, Leszek Antoni Gąsieniec, Roy Lindelauf, and V. S. Subrahmanian, editors, Search Theory: A Game Theoretic Perspective, chapter 5, pages 181-193. Springer-Verlag New York, 2013.
Fedor V. Fomin, Petr A. Golovach, Jan Kratochvíl, Nicolas Nisse, and Karol Suchan. Pursuing a fast robber on a graph. Theoretical Computer Science, 411:1167-1181, 2010.
Vladimir Janković. About a man and lions. Matematički Vesnik, 2:359-361, 1978.
J. Lewin. The lion and man problem revisited. Journal of Optimization Theory and Applications, 49(3):411-430, 1986.
John Edensor Littlewood. Littlewood’s miscellany: edited by Béla Bollobás. Cambridge University Press, 1986.
Peter A. Rado and Richard Rado. More about lions and other animals. Mathematical Sprectrum, 7(3):89-93, 1974/75.
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Faster Algorithms for the Geometric Transportation Problem
Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric:
* For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost.
* For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point.
* An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2.
transportation map
earth mover's distance
shape matching
approximation algorithms
7:1-7:16
Regular Paper
Pankaj K.
Agarwal
Pankaj K. Agarwal
Kyle
Fox
Kyle Fox
Debmalya
Panigrahi
Debmalya Panigrahi
Kasturi R.
Varadarajan
Kasturi R. Varadarajan
Allen
Xiao
Allen Xiao
10.4230/LIPIcs.SoCG.2017.7
Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput., 29(3):912-953, 1999. URL: http://dx.doi.org/10.1137/S0097539795295936.
http://dx.doi.org/10.1137/S0097539795295936
Pankaj K. Agarwal and Jeff Erickson. Geometric range searching and its relatives. Contemporary Mathematics, 223:1-56, 1999.
Pankaj K. Agarwal and Kasturi R. Varadarajan. A near-linear constant-factor approximation for Euclidean bipartite matching? In Proc. of the 20superscriptth ACM Symp. on Comp. Geometry, pages 247-252, 2004. URL: http://dx.doi.org/10.1145/997817.997856.
http://dx.doi.org/10.1145/997817.997856
Alexandr Andoni, Aleksandar Nikolov, Krzysztof Onak, and Grigory Yaroslavtsev. Parallel algorithms for geometric graph problems. In Proc. of the 46superscriptth Ann. ACM Symp. on Theory of Comp., pages 574-583, 2014. URL: http://dx.doi.org/10.1145/2591796.2591805.
http://dx.doi.org/10.1145/2591796.2591805
Sanjeev Arora. Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In Proc. of the 37superscriptth Ann. IEEE Symp. on Found. of Comp. Sci., pages 2-11, 1996. URL: http://dx.doi.org/10.1109/SFCS.1996.548458.
http://dx.doi.org/10.1109/SFCS.1996.548458
Sunil Arya, Gautam Das, David M. Mount, Jeffrey S. Salowe, and Michiel H. M. Smid. Euclidean spanners: short, thin, and lanky. In Proc. of the 27superscriptth Ann. ACM Symp. on Theory of Comp., pages 489-498, 1995. URL: http://dx.doi.org/10.1145/225058.225191.
http://dx.doi.org/10.1145/225058.225191
Sunil Arya, David M. Mount, and Michiel H. M. Smid. Randomized and deterministic algorithms for geometric spanners of small diameter. In Proc. of the 35superscriptth Ann. IEEE Symp. on Found. of Comp. Sci., pages 703-712, 1994. URL: http://dx.doi.org/10.1109/SFCS.1994.365722.
http://dx.doi.org/10.1109/SFCS.1994.365722
David S. Atkinson and Pravin M. Vaidya. Using geometry to solve the transportation problem in the plane. Algorithmica, 13(5):442-461, 1995.
Sergio Cabello, Panos Giannopoulos, Christian Knauer, and Günter Rote. Matching point sets with respect to the Earth Mover’s Distance. Comput. Geom., 39(2):118-133, 2008. URL: http://dx.doi.org/10.1016/j.comgeo.2006.10.001.
http://dx.doi.org/10.1016/j.comgeo.2006.10.001
Paul B. Callahan and S. Rao Kosaraju. Faster algorithms for some geometric graph problems in higher dimensions. In Proc. of the 4superscriptth Ann. ACM/SIGACT-SIAM Symp. on Discrete Algo., pages 291-300, 1993. URL: http://dl.acm.org/citation.cfm?id=313559.313777.
http://dl.acm.org/citation.cfm?id=313559.313777
Paul B. Callahan and S. Rao Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM, 42(1):67-90, 1995. URL: http://dx.doi.org/10.1145/200836.200853.
http://dx.doi.org/10.1145/200836.200853
Marco Cuturi and Arnaud Doucet. Fast computation of Wasserstein barycenters. In Proc. of the 31superscriptth Internat. Conf. on Machine Learning, pages 685-693, 2014. URL: http://jmlr.org/proceedings/papers/v32/cuturi14.html.
http://jmlr.org/proceedings/papers/v32/cuturi14.html
Alexandre Gramfort, Gabriel Peyré, and Marco Cuturi. Fast optimal transport averaging of neuroimaging data. In Proc. of the 24superscriptth Internat. Conf. on Infor. Processing in Medical Imaging, pages 261-272. Springer, 2015.
Kristen Grauman and Trevor Darrell. Fast contour matching using approximate earth mover’s distance. In Proc. of the 24superscriptth Ann. IEEE Conf. on Comp. Vision and Pattern Recog., volume 1, pages I-220. IEEE, 2004.
Sariel Har-Peled. Geometric Approximation Algorithms, volume 173. American Mathematical Society Providence, 2011.
Piotr Indyk. A near linear time constant factor approximation for Euclidean bichromatic matching (cost). In Proc. of the 18superscriptth Ann. ACM-SIAM Symp. on Discrete Algo., pages 39-42, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283388.
http://dl.acm.org/citation.cfm?id=1283383.1283388
Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic planar Voronoi diagrams for general distance functions and their algorithmic applications. CoRR, abs/1604.03654, 2016. URL: http://arxiv.org/abs/1604.03654.
http://arxiv.org/abs/1604.03654
Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in Õ(vrank) iterations and faster algorithms for maximum flow. In Proc. of the 55superscriptth Ann. IEEE Symp. on Found. of Comp. Sci., pages 424-433, 2014.
James B. Orlin. A faster strongly polynominal minimum cost flow algorithm. In Proc. of the 20superscriptth Annual ACM Symp. on Theory of Comp., May 2-4, 1988, Chicago, Illinois, USA, pages 377-387, 1988. URL: http://dx.doi.org/10.1145/62212.62249.
http://dx.doi.org/10.1145/62212.62249
Yossi Rubner, Carlo Tomasi, and Leonidas J. Guibas. A metric for distributions with applications to image databases. In 6superscriptth Internat. Conf. on Comp. Vision, pages 59-66, 1998. URL: http://dx.doi.org/10.1109/ICCV.1998.710701.
http://dx.doi.org/10.1109/ICCV.1998.710701
R. Sharathkumar and Pankaj K. Agarwal. Algorithms for the transportation problem in geometric settings. In Proc. of the 23superscriptrd Ann. ACM-SIAM Symp. on Discrete Algo., pages 306-317, 2012. URL: http://portal.acm.org/citation.cfm?id=2095145&CFID=63838676&CFTOKEN=79617016.
http://portal.acm.org/citation.cfm?id=2095145&CFID=63838676&CFTOKEN=79617016
R. Sharathkumar and Pankaj K. Agarwal. A near-linear time ε-approximation algorithm for geometric bipartite matching. In Proc. of the 44superscriptth Ann. ACM Symp. on Theory of Comp., pages 385-394, 2012. URL: http://dx.doi.org/10.1145/2213977.2214014.
http://dx.doi.org/10.1145/2213977.2214014
Justin Solomon, Raif M. Rustamov, Leonidas J. Guibas, and Adrian Butscher. Earth mover’s distances on discrete surfaces. ACM Transactions on Graphics, 33(4):67:1-67:12, 2014. URL: http://dx.doi.org/10.1145/2601097.2601175.
http://dx.doi.org/10.1145/2601097.2601175
Kunal Talwar. Bypassing the embedding: Algorithms for low dimensional metrics. In Proc. of the 36superscriptth Ann. ACM Symp. on Theory of Comp., pages 281-290, 2004. URL: http://dx.doi.org/10.1145/1007352.1007399.
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Kasturi R. Varadarajan and Pankaj K. Agarwal. Approximation algorithms for bipartite and non-bipartite matching in the plane. In Proc. of the 10superscriptth Ann. ACM-SIAM Symp. on Discrete Algo., pages 805-814, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.314918.
http://dl.acm.org/citation.cfm?id=314500.314918
Cédric Villani. Optimal Transport: Old and New, volume 338. Springer Science &Business Media, 2008.
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A Superlinear Lower Bound on the Number of 5-Holes
Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h_5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position.
Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h_5(n) have been of order Omega(n) and O(n^2), respectively. We show that h_5(n) = Omega(n(log n)^(4/5)), obtaining the first superlinear lower bound on h_5(n).
The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted.
Erdös-Szekeres type problem
k-hole
empty k-gon
empty pentagon
planar point set
8:1-8:16
Regular Paper
Oswin
Aichholzer
Oswin Aichholzer
Martin
Balko
Martin Balko
Thomas
Hackl
Thomas Hackl
Jan
Kyncl
Jan Kyncl
Irene
Parada
Irene Parada
Manfred
Scheucher
Manfred Scheucher
Pavel
Valtr
Pavel Valtr
Birgit
Vogtenhuber
Birgit Vogtenhuber
10.4230/LIPIcs.SoCG.2017.8
O. Aichholzer. Enumerating order types for small point sets with applications. URL: http://www.ist.tugraz.at/aichholzer/research/rp/triangulations/ordertypes/.
http://www.ist.tugraz.at/aichholzer/research/rp/triangulations/ordertypes/
O. Aichholzer. [Empty] [colored] k-gons. Recent results on some Erdős-Szekeres type problems. In Proceedings of XIII Encuentros de Geometría Computacional, pages 43-52, Zaragoza, Spain, 2009.
O. Aichholzer, F. Aurenhammer, and H. Krasser. Enumerating order types for small point sets with applications. Order, 19(3):265-281, 2002.
O. Aichholzer, M. Balko, T. Hackl, J. Kynčl, I. Parada, M. Scheucher, P. Valtr, and B. Vogtenhuber. A superlinear lower bound on the number of 5-holes. http://arXiv.org/abs/1703.05253, 2017.
http://arXiv.org/abs/1703.05253
O. Aichholzer, R. Fabila-Monroy, T. Hackl, C. Huemer, A. Pilz, and B. Vogtenhuber. Lower bounds for the number of small convex k-holes. Computational Geometry: Theory and Applications, 47(5):605-613, 2014.
O. Aichholzer, T. Hackl, and B. Vogtenhuber. On 5-gons and 5-holes. Lecture Notes in Computer Science, 7579:1-13, 2012.
O. Aichholzer and H. Krasser. Abstract order type extension and new results on the rectilinear crossing number. Computational Geometry: Theory and Applications, 36(1):2-15, 2007.
M. Balko. URL: http://kam.mff.cuni.cz/~balko/superlinear5Holes.
http://kam.mff.cuni.cz/~balko/superlinear5Holes
M. Balko, R. Fulek, and J. Kynčl. Crossing numbers and combinatorial characterization of monotone drawings of K_n. Discrete &Computational Geometry, 53(1):107-143, 2015.
I. Bárány and Z. Füredi. Empty simplices in Euclidean space. Canadian Mathematical Bulletin, 30(4):436-445, 1987.
I. Bárány and Gy. Károlyi. Problems and results around the Erdős-Szekeres convex polygon theorem. In Akiyama, Kano, and Urabe, editors, Discrete and Computational Geometry, volume 2098 of Lecture Notes in Computer Science, pages 91-105. Springer, 2001.
I. Bárány and P. Valtr. Planar point sets with a small number of empty convex polygons. Studia Scientiarum Mathematicarum Hungarica, 41(2):243-266, 2004.
P. Brass, W. Moser, and J. Pach. Research Problems in Discrete Geometry. Springer, 2005.
K. Dehnhardt. Leere konvexe Vielecke in ebenen Punktmengen. PhD thesis, TU Braunschweig, Germany, 1987. In German.
P. Erdős. Some more problems on elementary geometry. Australian Mathematical Society Gazette, 5(2):52-54, 1978.
P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935.
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A. García. A note on the number of empty triangles. Lecture Notes in Computer Science, 7579:249-257, 2012.
T. Gerken. Empty convex hexagons in planar point sets. Discrete & Computational Geometry, 39(1-3):239-272, 2008.
J. E. Goodman and R. Pollack. Multidimensional sorting. SIAM Journal on Computing, 12(3):484-507, 1983.
H. Harborth. Konvexe Fünfecke in ebenen Punktmengen. Elemente der Mathematik, 33:116-118, 1978. In German.
J. D. Horton. Sets with no empty convex 7-gons. Canadian Mathematical Bulletin, 26(4):482-484, 1983.
C. M. Nicolás. The empty hexagon theorem. Discrete & Computational Geometry, 38(2):389-397, 2007.
R. Pinchasi, R. Radoičić, and M. Sharir. On empty convex polygons in a planar point set. Journal of Combinatorial Theory, Series A, 113(3):385-419, 2006.
M. Scheucher. URL: http://www.ist.tugraz.at/scheucher/5holes.
http://www.ist.tugraz.at/scheucher/5holes
M. Scheucher. Counting convex 5-holes, Bachelor’s thesis, 2013. In German.
M. Scheucher. On order types, projective classes, and realizations, Bachelor’s thesis, 2014.
W. Steiger and J. Zhao. Generalized ham-sandwich cuts. Discrete &Computational Geometry, 44(3):535-545, 2010.
P. Valtr. Convex independent sets and 7-holes in restricted planar point sets. Discrete & Computational Geometry, 7(2):135-152, 1992.
P. Valtr. Sets in ℝ^d with no large empty convex subsets. Discrete Mathematics, 108(1):115-124, 1992.
P. Valtr. On empty pentagons and hexagons in planar point sets. In Proceedings of Computing: The Eighteenth Australasian Theory Symposium (CATS 2012), pages 47-48, Melbourne, Australia, 2012.
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A Universal Slope Set for 1-Bend Planar Drawings
We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1).
Slope number
1-bend drawings
planar graphs
angular resolution
9:1-9:16
Regular Paper
Patrizio
Angelini
Patrizio Angelini
Michael A.
Bekos
Michael A. Bekos
Giuseppe
Liotta
Giuseppe Liotta
Fabrizio
Montecchiani
Fabrizio Montecchiani
10.4230/LIPIcs.SoCG.2017.9
Patrizio Angelini, Michael A. Bekos, Giuseppe Liotta, and Fabrizio Montecchiani. Universal slope sets for 1-bend planar drawings. CoRR, 1703.04283, 2017.
János Barát, Jirí Matoušek, and David R. Wood. Bounded-degree graphs have arbitrarily large geometric thickness. Electr. J. Comb., 13(1), 2006.
Michael A. Bekos, Martin Gronemann, Michael Kaufmann, and Robert Krug. Planar octilinear drawings with one bend per edge. J. Graph Algorithms Appl., 19(2):657-680, 2015. URL: http://dx.doi.org/10.7155/jgaa.00369.
http://dx.doi.org/10.7155/jgaa.00369
Michael A. Bekos, Michael Kaufmann, and Robert Krug. On the total number of bends for planar octilinear drawings. In LATIN, volume 9644 of LNCS, pages 152-163. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-662-49529-2_12.
http://dx.doi.org/10.1007/978-3-662-49529-2_12
Therese C. Biedl and Goos Kant. A better heuristic for orthogonal graph drawings. Comput. Geom., 9(3):159-180, 1998. URL: http://dx.doi.org/10.1016/S0925-7721(97)00026-6.
http://dx.doi.org/10.1016/S0925-7721(97)00026-6
Thomas Bläsius, Marcus Krug, Ignaz Rutter, and Dorothea Wagner. Orthogonal graph drawing with flexibility constraints. Algorithmica, 68(4):859-885, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9705-8.
http://dx.doi.org/10.1007/s00453-012-9705-8
Thomas Bläsius, Sebastian Lehmann, and Ignaz Rutter. Orthogonal graph drawing with inflexible edges. Comput. Geom., 55:26-40, 2016.
Hans L. Bodlaender and Gerard Tel. A note on rectilinearity and angular resolution. J. Graph Algorithms Appl., 8:89-94, 2004.
Nicolas Bonichon, Bertrand Le Saëc, and Mohamed Mosbah. Optimal area algorithm for planar polyline drawings. In WG, volume 2573 of LNCS, pages 35-46. Springer, 2002.
Hubert de Fraysseix, Patrice Ossona de Mendez, and Pierre Rosenstiehl. On triangle contact graphs. Combinatorics, Probability & Computing, 3:233-246, 1994. URL: http://dx.doi.org/10.1017/S0963548300001139.
http://dx.doi.org/10.1017/S0963548300001139
Hubert de Fraysseix, János Pach, and Richard Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41-51, 1990. URL: http://dx.doi.org/10.1007/BF02122694.
http://dx.doi.org/10.1007/BF02122694
Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, 1999.
Emilio Di Giacomo, Giuseppe Liotta, and Fabrizio Montecchiani. The planar slope number of subcubic graphs. In LATIN, volume 8392 of LNCS, pages 132-143. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-642-54423-1_12.
http://dx.doi.org/10.1007/978-3-642-54423-1_12
Emilio Di Giacomo, Giuseppe Liotta, and Fabrizio Montecchiani. Drawing outer 1-planar graphs with few slopes. J. Graph Algorithms Appl., 19(2):707-741, 2015. URL: http://dx.doi.org/10.7155/jgaa.00376.
http://dx.doi.org/10.7155/jgaa.00376
Vida Dujmović, David Eppstein, Matthew Suderman, and David R. Wood. Drawings of planar graphs with few slopes and segments. Comput. Geom., 38(3):194-212, 2007.
Christian A. Duncan, David Eppstein, Michael T. Goodrich, Stephen G. Kobourov, and Martin Nöllenburg. Drawing trees with perfect angular resolution and polynomial area. Discrete & Computational Geometry, 49(2):157-182, 2013.
Christian A. Duncan and Stephen G. Kobourov. Polar coordinate drawing of planar graphs with good angular resolution. J. Graph Algorithms Appl., 7(4):311-333, 2003.
Stephane Durocher and Debajyoti Mondal. Trade-offs in planar polyline drawings. In GD, volume 8871 of LNCS, pages 306-318. Springer, 2014.
Michael Formann, Torben Hagerup, James Haralambides, Michael Kaufmann, Frank Thomson Leighton, Antonios Symvonis, Emo Welzl, and Gerhard J. Woeginger. Drawing graphs in the plane with high resolution. SIAM J. Comput., 22(5):1035-1052, 1993. URL: http://dx.doi.org/10.1137/0222063.
http://dx.doi.org/10.1137/0222063
Ashim Garg and Roberto Tamassia. Planar drawings and angular resolution: Algorithms and bounds (extended abstract). In ESA, volume 855 of LNCS, pages 12-23. Springer, 1994.
Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput., 31(2):601-625, 2001. URL: http://dx.doi.org/10.1137/S0097539794277123.
http://dx.doi.org/10.1137/S0097539794277123
Carsten Gutwenger and Petra Mutzel. Planar polyline drawings with good angular resolution. In GD, volume 1547 of LNCS, pages 167-182. Springer, 1998.
Carsten Gutwenger and Petra Mutzel. A linear time implementation of SPQR-trees. In GD, volume 1984 of LNCS, pages 77-90. Springer, 2000. URL: http://dx.doi.org/10.1007/3-540-44541-2_8.
http://dx.doi.org/10.1007/3-540-44541-2_8
Udo Hoffmann. On the complexity of the planar slope number problem. J. Graph Algorithms Appl., 21(2):183-193, 2017.
Vít Jelínek, Eva Jelínková, Jan Kratochvíl, Bernard Lidický, Marek Tesar, and Tomás Vyskocil. The planar slope number of planar partial 3-trees of bounded degree. Graphs and Comb., 29(4):981-1005, 2013. URL: http://dx.doi.org/10.1007/s00373-012-1157-z.
http://dx.doi.org/10.1007/s00373-012-1157-z
Michael Jünger and Petra Mutzel, editors. Graph Drawing Software. Springer, 2004.
Goos Kant. Drawing planar graphs using the lmc-ordering (extended abstract). In FOCS, pages 101-110. IEEE Computer Society, 1992. URL: http://dx.doi.org/10.1109/SFCS.1992.267814.
http://dx.doi.org/10.1109/SFCS.1992.267814
Goos Kant. Hexagonal grid drawings. In WG, volume 657 of LNCS, pages 263-276. Springer, 1992. URL: http://dx.doi.org/10.1007/3-540-56402-0_53.
http://dx.doi.org/10.1007/3-540-56402-0_53
Goos Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16(1):4-32, 1996. URL: http://dx.doi.org/10.1007/BF02086606.
http://dx.doi.org/10.1007/BF02086606
Balázs Keszegh, János Pach, and Dömötör Pálvölgyi. Drawing planar graphs of bounded degree with few slopes. SIAM J. Discrete Math., 27(2):1171-1183, 2013. URL: http://dx.doi.org/10.1137/100815001.
http://dx.doi.org/10.1137/100815001
Kolja Knauer and Bartosz Walczak. Graph drawings with one bend and few slopes. In LATIN, volume 9644 of LNCS, pages 549-561. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-662-49529-2_41.
http://dx.doi.org/10.1007/978-3-662-49529-2_41
Kolja B. Knauer, Piotr Micek, and Bartosz Walczak. Outerplanar graph drawings with few slopes. Comput. Geom., 47(5):614-624, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2014.01.003.
http://dx.doi.org/10.1016/j.comgeo.2014.01.003
William Lenhart, Giuseppe Liotta, Debajyoti Mondal, and Rahnuma Islam Nishat. Planar and plane slope number of partial 2-trees. In GD, volume 8242 of LNCS, pages 412-423. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-319-03841-4_36.
http://dx.doi.org/10.1007/978-3-319-03841-4_36
Yanpei Liu, Aurora Morgana, and Bruno Simeone. A linear algorithm for 2-bend embeddings of planar graphs in the two-dimensional grid. Discrete Applied Mathematics, 81(1-3):69-91, 1998. URL: http://dx.doi.org/10.1016/S0166-218X(97)00076-0.
http://dx.doi.org/10.1016/S0166-218X(97)00076-0
Padmini Mukkamala and Dömötör Pálvölgyi. Drawing cubic graphs with the four basic slopes. In GD, volume 7034 of LNCS, pages 254-265. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-25878-7_25.
http://dx.doi.org/10.1007/978-3-642-25878-7_25
Martin Nöllenburg. Automated drawings of metro maps. Technical Report 2005-25, Fakultät für Informatik, Universität Karlsruhe, 2005.
Martin Nöllenburg and Alexander Wolff. Drawing and labeling high-quality metro maps by mixed-integer programming. IEEE Trans. Vis. Comput. Graph., 17(5):626-641, 2011. URL: http://dx.doi.org/10.1109/TVCG.2010.81.
http://dx.doi.org/10.1109/TVCG.2010.81
Jonathan M. Stott, Peter Rodgers, Juan Carlos Martinez-Ovando, and Stephen G. Walker. Automatic metro map layout using multicriteria optimization. IEEE Trans. Vis. Comput. Graph., 17(1):101-114, 2011. URL: http://dx.doi.org/10.1109/TVCG.2010.24.
http://dx.doi.org/10.1109/TVCG.2010.24
Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput., 16(3):421-444, 1987. URL: http://dx.doi.org/10.1137/0216030.
http://dx.doi.org/10.1137/0216030
Roberto Tamassia, editor. Handbook on Graph Drawing and Visualization. CRC Press, 2013.
Greg A. Wade and Jiang-Hsing Chu. Drawability of complete graphs using a minimal slope set. Comput. J., 37(2):139-142, 1994. URL: http://dx.doi.org/10.1093/comjnl/37.2.139.
http://dx.doi.org/10.1093/comjnl/37.2.139
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Near-Optimal epsilon-Kernel Construction and Related Problems
The computation of (i) eps-kernels, (ii) approximate diameter, and (iii) approximate bichromatic closest pair are fundamental problems in geometric approximation. In each case the input is a set of points in d-dimensional space for a constant d and an approximation parameter eps > 0. In this paper, we describe new algorithms for these problems, achieving significant improvements to the exponent of the eps-dependency in their running times, from roughly d to d/2 for the first two problems and from roughly d/3 to d/4 for problem (iii).
These results are all based on an efficient decomposition of a convex body using a hierarchy of Macbeath regions, and contrast to previous solutions that decomposed the space using quadtrees and grids. By further application of these techniques, we also show that it is possible to obtain near-optimal preprocessing time for the most efficient data structures for (iv) approximate nearest neighbor searching, (v) directional width queries, and (vi) polytope membership queries.
Approximation
diameter
kernel
coreset
nearest neighbor
polytope membership
bichromatic closest pair
Macbeath regions
10:1-10:15
Regular Paper
Sunil
Arya
Sunil Arya
Guilherme D.
da Fonseca
Guilherme D. da Fonseca
David M.
Mount
David M. Mount
10.4230/LIPIcs.SoCG.2017.10
P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan. Approximating extent measures of points. J. Assoc. Comput. Mach., 51:606-635, 2004.
P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan. Geometric approximation via coresets. In J. E. Goodman, J. Pach, and E. Welzl, editors, Combinatorial and Computational Geometry. MSRI Publications, 2005.
P. K. Agarwal, J. Matoušek, and S. Suri. Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Comput. Geom. Theory Appl., 1(4):189-201, 1992.
S. Arya, G. D. da Fonseca, and D. M. Mount. Approximate polytope membership queries. In Proc. 43rd Annu. ACM Sympos. Theory Comput., pages 579-586, 2011. URL: http://dx.doi.org/10.1145/1993636.1993713.
http://dx.doi.org/10.1145/1993636.1993713
S. Arya, G. D. da Fonseca, and D. M. Mount. Optimal area-sensitive bounds for polytope approximation. In Proc. 28th Annu. Sympos. Comput. Geom., pages 363-372, 2012.
S. Arya, G. D. da Fonseca, and D. M. Mount. On the combinatorial complexity of approximating polytopes. In Proc. 32nd Internat. Sympos. Comput. Geom., pages 11:1-11:15, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.11.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.11
S. Arya, G. D. da Fonseca, and D. M. Mount. Optimal approximate polytope membership. In Proc. 28th Annu. ACM-SIAM Sympos. Discrete Algorithms, pages 270-288, 2017.
S. Arya, T. Malamatos, and D. M. Mount. The effect of corners on the complexity of approximate range searching. Discrete Comput. Geom., 41:398-443, 2009.
S. Arya and D. M. Mount. A fast and simple algorithm for computing approximate Euclidean minimum spanning trees. In Proc. 27th Annu. ACM-SIAM Sympos. Discrete Algorithms, pages 1220-1233, 2016.
S. Arya, D. M. Mount, and J. Xia. Tight lower bounds for halfspace range searching. Discrete Comput. Geom., 47:711-730, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9412-x.
http://dx.doi.org/10.1007/s00454-012-9412-x
Sunil Arya and Timothy M. Chan. Better ε-dependencies for offline approximate nearest neighbor search, Euclidean minimum spanning trees, and ε-kernels. In Proc. 30th Annu. Sympos. Comput. Geom., pages 416-425, 2014.
I. Bárány. The technique of M-regions and cap-coverings: A survey. Rend. Circ. Mat. Palermo, 65:21-38, 2000.
G. Barequet and S. Har-Peled. Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms, 38(1):91-109, 2001.
J. L. Bentley, M. G. Faust, and F. P. Preparata. Approximation algorithms for convex hulls. Commun. ACM, 25(1):64-68, 1982. URL: http://dx.doi.org/10.1145/358315.358392.
http://dx.doi.org/10.1145/358315.358392
H. Brönnimann, B. Chazelle, and J. Pach. How hard is halfspace range searching. Discrete Comput. Geom., 10:143-155, 1993.
E. M. Bronshteyn and L. D. Ivanov. The approximation of convex sets by polyhedra. Siberian Math. J., 16:852-853, 1976.
T. M. Chan. Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput. Geom. Theory Appl., 35(1):20-35, 2006. URL: http://dx.doi.org/10.1016/j.comgeo.2005.10.002.
http://dx.doi.org/10.1016/j.comgeo.2005.10.002
T. M. Chan. Applications of Chebyshev polynomials to low-dimensional computational geometry. In Proc. 33rd Internat. Sympos. Comput. Geom., pages 26:1-15, 2017.
R. M. Dudley. Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory, 10(3):227-236, 1974.
K. Dutta, A. Ghosh, B. Jartoux, and N. H. Mustafa. Shallow packings, semialgebraic set systems, Macbeath regions and polynomial partitioning. In Proc. 33rd Internat. Sympos. Comput. Geom., pages 38:1-15, 2017.
G. Ewald, D. G. Larman, and C. A. Rogers. The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space. Mathematika, 17:1-20, 1970.
F. John. Extremum problems with inequalities as subsidiary conditions. In Studies and Essays Presented to R. Courant on his 60th Birthday, pages 187-204. Interscience Publishers, Inc., New York, 1948.
S. Khuller and Y. Matias. A simple randomized sieve algorithm for the closest-pair problem. Information and Computation, 118(1):34-37, 1995.
A. M. Macbeath. A theorem on non-homogeneous lattices. Ann. of Math., 56:269-293, 1952.
N. H. Mustafa and S. Ray. Near-optimal generalisations of a theorem of Macbeath. In Proc. 31st Internat. Sympos. on Theoret. Aspects of Comp. Sci., pages 578-589, 2014.
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Exact Algorithms for Terrain Guarding
Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size on T that guards all of the points on T. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an n^O(sqrt{k})-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable.
Terrain Guarding
Art Gallery
Exponential-Time Algorithms
11:1-11:15
Regular Paper
Pradeesha
Ashok
Pradeesha Ashok
Fedor V.
Fomin
Fedor V. Fomin
Sudeshna
Kolay
Sudeshna Kolay
Saket
Saurabh
Saket Saurabh
Meirav
Zehavi
Meirav Zehavi
10.4230/LIPIcs.SoCG.2017.11
J. Abello, O. Egecioglu, and K. Kumar. Visibility graphs of staircase polygons and the weak bruhat order I: From visibility graphs to maximal chains. Discrete and Computational Geometry, 14(3):331-358, 1995.
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M. Gibson, G. Kanade, E. Krohn, and K. Varadarajan. Guarding terrains via local search. JoCG, 5(1):168-178, 2014.
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Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences
Let d and k be integers with 1 <= k <= d-1. Let Lambda be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in the intersection of Lambda with K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n * ... * n grid is at least Omega(n^(d(d-k)/(d-1)-epsilon)) and at most O(n^(d(d-k)/(d-1))), where epsilon > 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach. We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover the intersection of Lambda with K.
We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer. For d > =3 and epsilon in (0,1), we show that there is an integer r=r(d,epsilon) such that for all positive integers n, m the following statement is true. There is a set of n points in R^d and an arrangement of m hyperplanes in R^d with no K_(r,r) in their incidence graph and with at least Omega((mn)^(1-(2d+3)/((d+2)(d+3)) - epsilon)) incidences if d is odd and Omega((mn)^(1-(2d^2+d-2)/((d+2)(d^2+2d-2)) - epsilon)) incidences if d is even.
lattice point
covering
linear subspace
point-hyperplane incidence
12:1-12:16
Regular Paper
Martin
Balko
Martin Balko
Josef
Cibulka
Josef Cibulka
Pavel
Valtr
Pavel Valtr
10.4230/LIPIcs.SoCG.2017.12
E. Ackerman. On topological graphs with at most four crossings per edge. http://arxiv.org/abs/1509.01932, 2015.
http://arxiv.org/abs/1509.01932
R. Apfelbaum and M. Sharir. Large complete bipartite subgraphs in incidence graphs of points and hyperplanes. SIAM J. Discrete Math., 21(3):707-725, 2007.
M. Balko, J. Cibulka, and P. Valtr. Covering lattice points by subspaces and counting point-hyperplane incidences. http://arxiv.org/abs/1703.04767, 2017.
http://arxiv.org/abs/1703.04767
W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Math. Ann., 296(4):625-635, 1993.
I. Bárány, G. Harcos, J. Pach, and G. Tardos. Covering lattice points by subspaces. Period. Math. Hungar., 43(1-2):93-103, 2001.
P. Brass and C. Knauer. On counting point-hyperplane incidences. Comput. Geom., 25(1-2):13-20, 2003.
P. Brass, W. Moser, and J. Pach. Research problems in discrete geometry. Springer, New York, 2005.
B. Chazelle. Cutting hyperplanes for Divide-and-Conquer. Discrete Comput. Geom., 9(2):145-158, 1993.
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J. Fox, J. Pach, A. Sheffer, and A. Suk. A semi-algebraic version of Zarankiewicz’s problem. http://arxiv.org/abs/1407.5705, 2014.
http://arxiv.org/abs/1407.5705
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H. Lefmann. Extensions of the No-Three-In-Line Problem. https://www.tu-chemnitz.de/informatik/ThIS/downloads/publications/lefmann_no_three_submitted.pdf, 2012.
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János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Combinatorica, 17:427-439, 1997.
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Adam Sheffer. Lower bounds for incidences with hypersurfaces. Discrete Anal., 2016. Paper No. 16, 14.
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E. Szemerédi and W. T. Trotter Jr. Extremal problems in discrete geometry. Combinatorica, 3(3-4):381-392, 1983.
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Subquadratic Algorithms for Algebraic Generalizations of 3SUM
The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n^{11/6}) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. Finally, we shed light on the General Position Testing (GPT) problem: "Given n points in the plane, do three of them lie on a line?", a key problem in computational geometry.
We prove that there exist bounded-degree algebraic decision trees of depth O(n^{12/7+e}) that solve 3POL, and that 3POL can be solved in O(n^2 (log log n)^{3/2} / (log n)^{1/2}) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in subquadratic time when the input points lie on o((log n)^{1/6}/(log log n)^{1/2}) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.
3SUM
subquadratic algorithms
general position testing
range searching
dominance reporting
polynomial curves
13:1-13:15
Regular Paper
Luis
Barba
Luis Barba
Jean
Cardinal
Jean Cardinal
John
Iacono
John Iacono
Stefan
Langerman
Stefan Langerman
Aurélien
Ooms
Aurélien Ooms
Noam
Solomon
Noam Solomon
10.4230/LIPIcs.SoCG.2017.13
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In FOCS, pages 434-443. IEEE Computer Society, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In ICALP (1), volume 8572 of LNCS, pages 39-51, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In STOC, pages 41-50. ACM, 2015.
Nir Ailon and Bernard Chazelle. Lower bounds for linear degeneracy testing. J. ACM, 52(2):157-171, 2005.
Amihood Amir, Timothy M. Chan, Moshe Lewenstein, and Noa Lewenstein. On hardness of jumbled indexing. In ICALP (1), volume 8572 of LNCS, pages 114-125, 2014.
Ilya Baran, Erik D. Demaine, and Mihai Pătrascu. Subquadratic algorithms for 3SUM. Algorithmica, 50(4):584-596, 2008.
Gill Barequet and Sariel Har-Peled. Polygon containment and translational min Hausdorff distance between segment sets are 3SUM-hard. Int. J. Comput. Geometry Appl., 11(4):465-474, 2001.
Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Computing roadmaps of semi-algebraic sets (extended abstract). In STOC, pages 168-173. ACM, 1996.
Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in real algebraic geometry, volume 10 of Algorithms and Computation in Mathematics. Springer, 2006.
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Marco L. Carmosino, Jiawei Gao, Russell Impagliazzo, Ivan Mihajlin, Ramamohan Paturi, and Stefan Schneider. Nondeterministic extensions of the strong exponential time hypothesis and consequences for non-reducibility. In ITCS, pages 261-270. ACM, 2016.
Bob F. Caviness and Jeremy R. Johnson. Quantifier elimination and cylindrical algebraic decomposition. Springer, 2012.
Timothy M. Chan. All-pairs shortest paths with real weights in O(n³/log n) time. Algorithmica, 50(2):236-243, 2008.
George E. Collins. Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In Automata Theory and Formal Languages, volume 33 of LNCS, pages 134-183. Springer, 1975.
James H. Davenport and Joos Heintz. Real quantifier elimination is doubly exponential. J. Symb. Comput., 5(1/2):29-35, 1988.
György Elekes and Lajos Rónyai. A combinatorial problem on polynomials and rational functions. J. Comb. Theory, Ser. A, 89(1):1-20, 2000.
György Elekes and Endre Szabó. How to find groups? (and how to use them in Erdős geometry?). Combinatorica, 32(5):537-571, 2012.
Jeff Erickson. Lower bounds for linear satisfiability problems. Chicago J. Theor. Comput. Sci., 1999.
Michael L. Fredman. How good is the information theory bound in sorting? Theor. Comput. Sci., 1(4):355-361, 1976.
Ari Freund. Improved subquadratic 3SUM. Algorithmica, pages 1-19, 2015.
Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Comput. Geom., 5:165-185, 1995.
Omer Gold and Micha Sharir. Improved bounds for 3SUM, k-SUM, and linear degeneracy. ArXiv e-prints, 2015. URL: https://arxiv.org/abs/1512.05279.
https://arxiv.org/abs/1512.05279
Allan Grønlund and Seth Pettie. Threesomes, degenerates, and love triangles. In Foundations of Computer Science (FOCS 2014), pages 621-630. IEEE, 2014.
Joe Harris. Algebraic geometry: a first course, volume 133. Springer, 2013.
Robin Hartshorne. Algebraic geometry, volume 52. Springer, 1977.
Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In STOC, pages 21-30. ACM, 2015.
Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3SUM conjecture. In SODA, pages 1272-1287. SIAM, 2016.
Jirí Matoušek. Range searching with efficient hierarchical cutting. Discrete & Computational Geometry, 10:157-182, 1993.
Bhubaneswar Mishra. Computational real algebraic geometry. In Handbook of Discrete and Computational Geometry, 2nd Ed., pages 743-764. Chapman and Hall/CRC, 2004.
H. Nassajian Mojarrad, T. Pham, C. Valculescu, and F. de Zeeuw. Schwartz-Zippel bounds for two-dimensional products. ArXiv e-prints, 2016. URL: https://arxiv.org/abs/1507.08181.
https://arxiv.org/abs/1507.08181
János Pach and Micha Sharir. On the number of incidences between points and curves. Combinatorics, Probability & Computing, 7(1):121-127, 1998.
Mihai Pătrascu. Towards polynomial lower bounds for dynamic problems. In STOC, pages 603-610. ACM, 2010.
Franco P. Preparata and Michael Ian Shamos. Computational Geometry - An Introduction. Texts and Monographs in Computer Science. Springer, 1985.
Michael O. Rabin. Proving simultaneous positivity of linear forms. J. Comput. Syst. Sci., 6(6):639-650, 1972.
Orit E. Raz, Micha Sharir, and Frank de Zeeuw. Polynomials vanishing on cartesian products: The Elekes-Szabó theorem revisited. In SoCG, volume 34 of LIPIcs, pages 522-536, 2015.
Orit E. Raz, Micha Sharir, and Frank de Zeeuw. The elekes-szabó theorem in four dimensions. ArXiv e-prints, 2016. URL: https://arxiv.org/abs/1607.03600.
https://arxiv.org/abs/1607.03600
Orit E. Raz, Micha Sharir, and József Solymosi. Polynomials vanishing on grids: The Elekes-Rónyai problem revisited. In SoCG, page 251. ACM, 2014.
Abraham Seidenberg. Constructions in algebra. Transactions of the AMS, 197:273-313, 1974.
J. Michael Steele and Andrew Yao. Lower bounds for algebraic decision trees. J. Algorithms, 3(1):1-8, 1982.
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Andrew Yao. A lower bound to finding convex hulls. J. ACM, 28(4):780-787, 1981.
David Yun. On square-free decomposition algorithms. In SYMSACC, pages 26-35, 1976.
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Towards a Topology-Shape-Metrics Framework for Ortho-Radial Drawings
Ortho-Radial drawings are a generalization of orthogonal drawings to grids that are formed by concentric circles and straight-line spokes emanating from the circles' center. Such drawings have applications in schematic graph layouts, e.g., for metro maps and destination maps.
A plane graph is a planar graph with a fixed planar embedding. We give a combinatorial characterization of the plane graphs that admit a planar ortho-radial drawing without bends. Previously, such a characterization was only known for paths, cycles, and theta graphs, and in the special case of rectangular drawings for cubic graphs, where the contour of each face is required to be a rectangle.
The characterization is expressed in terms of an ortho-radial representation that, similar to Tamassia's orthogonal representations for orthogonal drawings describes such a drawing combinatorially in terms of angles around vertices and bends on the edges. In this sense our characterization can be seen as a first step towards generalizing the Topology-Shape-Metrics framework of Tamassia to ortho-radial drawings.
Graph Drawing
Ortho-Radial Drawings
Combinatorial Characterization
Bend Minimization
Topology-Shape-Metrics
14:1-14:16
Regular Paper
Lukas
Barth
Lukas Barth
Benjamin
Niedermann
Benjamin Niedermann
Ignaz
Rutter
Ignaz Rutter
Matthias
Wolf
Matthias Wolf
10.4230/LIPIcs.SoCG.2017.14
Lukas Barth, Benjamin Niedermann, Ignaz Rutter, and Matthias Wolf. Towards a topology-shape-metrics framework for ortho-radial drawings. CoRR, arXiv:1703.06040, 2017.
Therese Biedl and Goos Kant. A better heuristic for orthogonal graph drawings. Computational Geometry: Theory and Applications, 9:159-180, 1998.
Thomas Bläsius, Marcus Krug, Ignaz Rutter, and Dorothea Wagner. Orthogonal graph drawing with flexibility constraints. Algorithmica, 68(4):859-885, 2014.
Thomas Bläsius, Sebastian Lehmann, and Ignaz Rutter. Orthogonal graph drawing with inflexible edges. Computational Geometry: Theory and Applications, 55:26-40, 2016.
Thomas Bläsius, Ignaz Rutter, and Dorothea Wagner. Optimal orthogonal graph drawing with convex bend costs. ACM Transactions on Algorithms, 12:33:1-33:32, 2016.
Sabine Cornelsen and Andreas Karrenbauer. Accelerated bend minimization. Journal of Graph Algorithms and Applications, 16(3):635-650, 2012.
Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing - Algorithms for the Visualization of Graphs. Prentice Hall, 1999.
Giuseppe Di Battista, Giuseppe Liotta, and Francesco Vargiu. Spirality and optimal orthogonal drawings. SIAM Journal on Computing, 27(6):1764-1811, 1998. URL: http://dx.doi.org/10.1137/S0097539794262847.
http://dx.doi.org/10.1137/S0097539794262847
Christian A. Duncan and Michael T. Goodrich. Handbook of Graph Drawing and Visualization, chapter Planar Orthogonal and Polyline Drawing Algorithms, pages 223-246. CRC Press, 2013.
Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. SIAM Journal on Computing, 31(2):601-625, 2001.
Madieh Hasheminezhad, S. Mehdi Hashemi, Brendan D. McKay, and Maryam Tahmasbi. Rectangular-radial drawings of cubic plane graphs. Computational Geometry: Theory and Applications, 43:767-780, 2010.
Mahdie Hasheminezhad, S. Mehdi Hashemi, and Maryam Tahmabasi. Ortho-radial drawings of graphs. Australasian Journal of Combinatorics, 44:171-182, 2009.
Md. Saidur Rahman, Shin-ichi Nakano, and Takao Nishizeki. Rectangular grid drawings of plane graphs. Computational Geometry: Theory and Applications, 10:203-220, 1998.
Md. Saidur Rahman, Takao Nishizeki, and Mahmuda Naznin. Orthogonal drawings of plane graphs without bends. Journal of Graph Algorithms and Applications, 7(3):335-362, 2003.
Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM Journal on Computing, 16(3):421-444, 1987.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Number of Ordinary Lines Determined by Sets in Complex Space
Kelly's theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.
Incidences
Combinatorial Geometry
Designs
Polynomial Method
15:1-15:15
Regular Paper
Abdul
Basit
Abdul Basit
Zeev
Dvir
Zeev Dvir
Shubhangi
Saraf
Shubhangi Saraf
Charles
Wolf
Charles Wolf
10.4230/LIPIcs.SoCG.2017.15
N. Alon. Perturbed identity matrices have high rank: Proof and applications. Combinatorics, Probability and Computing, 18(1-2):3-15, 2009.
B. Barak, Z. Dvir, A. Wigderson, and A. Yehudayoff. Fractional Sylvester-Gallai theorems. Proceedings of the National Academy of Sciences, 110(48):19213-19219, 2013.
P. Borwein and W. Moser. A survey of Sylvester’s problem and its generalizations. Aequationes Mathematicae, 40(1):111-135, 1990.
R. K. Brayton, D. Coppersmith, and A. J. Hoffman. Self-orthogonal latin squares of all orders n ≠ 2, 3, 6. Bulletin of the American Mathematical Society, 80, 1974.
J. Csima and E. T. Sawyer. There exist 6n/13 ordinary points. Discrete &Computational Geometry, 9(2):187-202, 1993.
Z. Dvir, S. Saraf, and A. Wigderson. Improved rank bounds for design matrices and a new proof of Kelly’s theorem. In Forum of Mathematics, Sigma, volume 2, page e4. Cambridge University Press, 2014.
N. Elkies, L. M. Pretorius, and K. Swanepoel. Sylvester-Gallai theorems for complex numbers and quaternions. Discrete &Computational Geometry, 35(3):361-373, 2006.
T. Gallai. Solution of problem 4065. American Mathematical Monthly, 51:169-171, 1944.
B. Green and T. Tao. On sets defining few ordinary lines. Discrete &Computational Geometry, 50(2):409-468, 2013.
A. J. W. Hilton. On double diagonal and cross latin squares. Journal of the London Mathematical Society, 2(4):679-689, 1973.
F. Hirzebruch. Arrangements of lines and algebraic surfaces. In Arithmetic and geometry, pages 113-140. Springer, 1983.
L. Kelly. A resolution of the Sylvester-Gallai problem of J.-P. Serre. Discrete &Computational Geometry, 1(1):101-104, 1986.
L. Kelly and W. Moser. On the number of ordinary lines determined by n points. Canadian Journal of Mathematics, 10:210-219, 1958.
E. Melchior. Über Vielseite der projektiven Ebene. Deutsche Math, 5:461-475, 1940.
Th. Motzkin. The lines and planes connecting the points of a finite set. Transactions of the American Mathematical Society, pages 451-464, 1951.
U. Rothblum and H. Schneider. Scalings of matrices which have prespecified row sums and column sums via optimization. Linear Algebra and its Applications, 114:737-764, 1989.
J.-P. Serre. Advanced problem 5359. American Mathematical Monthly, 73(1):89, 1966.
J. J. Sylvester. Mathematical question 11851. Educational Times, 59(98):256, 1893.
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On Optimal 2- and 3-Planar Graphs
A graph is k-planar if it can be drawn in the plane such that no edge is crossed more than k times. While for k=1, optimal 1-planar graphs, i.e., those with n vertices and exactly 4n-8 edges, have been completely characterized, this has not been the case for k > 1. For k=2,3 and 4, upper bounds on the edge density have been developed for the case of simple graphs by Pach and Tóth, Pach et al. and Ackerman, which have been used to improve the well-known "Crossing Lemma". Recently, we proved that these bounds also apply to non-simple 2- and 3-planar graphs without homotopic parallel edges and self-loops.
In this paper, we completely characterize optimal 2- and 3-planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes.
topological graphs
optimal k-planar graphs
characterization
16:1-16:16
Regular Paper
Michael A.
Bekos
Michael A. Bekos
Michael
Kaufmann
Michael Kaufmann
Chrysanthi N.
Raftopoulou
Chrysanthi N. Raftopoulou
10.4230/LIPIcs.SoCG.2017.16
E. Ackerman. On topological graphs with at most four crossings per edge. CoRR, 1509.01932, 2015.
M. Ajtai, V. Chvátal, M. Newborn, and E. Szemerédi. Crossing-free subgraphs. In Theory and Practice of Combinatorics, pages 9-12. North-Holland Mathematics Studies, 1982.
N. Alon and P. Erdős. Disjoint edges in geometric graphs. Discrete & Computational Geometry, 4:287-290, 1989. URL: http://dx.doi.org/10.1007/BF02187731.
http://dx.doi.org/10.1007/BF02187731
P. Angelini, M. A. Bekos, F.-J. Brandenburg, G. Da Lozzo, G. Di Battista, W. Didimo, G. Liotta, F. Montecchiani, and I. Rutter. On the relationship between k-planar and k-quasi planar graphs. CoRR, 1702.08716, 2017.
M. A. Bekos, T. Bruckdorfer, M. Kaufmann, and C. N. Raftopoulou. 1-planar graphs have constant book thickness. In ESA, volume 9294 of LNCS, pages 130-141. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_12.
http://dx.doi.org/10.1007/978-3-662-48350-3_12
M. A. Bekos, M. Kaufmann, and C. N. Raftopoulou. On the density of non-simple 3-planar graphs. In Graph Drawing, volume 9801 of LNCS, pages 344-356. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-319-50106-2_27.
http://dx.doi.org/10.1007/978-3-319-50106-2_27
C. Binucci, E. Di Giacomo, W. Didimo, F. Montecchiani, M. Patrignani, A. Symvonis, and I. G. Tollis. Fan-planarity: Properties and complexity. Theor. Comp. Sci., 589:76-86, 2015. URL: http://dx.doi.org/10.1016/j.tcs.2015.04.020.
http://dx.doi.org/10.1016/j.tcs.2015.04.020
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O. Borodin. A new proof of the 6 color theorem. J. of Graph Theory, 19(4):507-521, 1995. URL: http://dx.doi.org/10.1002/jgt.3190190406.
http://dx.doi.org/10.1002/jgt.3190190406
F.-J. Brandenburg. 1-visibility representations of 1-planar graphs. J. Graph Algorithms Appl., 18(3):421-438, 2014. URL: http://dx.doi.org/10.7155/jgaa.00330.
http://dx.doi.org/10.7155/jgaa.00330
F.-J. Brandenburg. Recognizing optimal 1-planar graphs in linear time. CoRR, 1602.08022, 2016.
O. Cheong, S. Har-Peled, H. Kim, and H.-S. Kim. On the number of edges of fan-crossing free graphs. Algorithmica, 73(4):673-695, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9935-z.
http://dx.doi.org/10.1007/s00453-014-9935-z
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E. Di Giacomo, W. Didimo, G. Liotta, H. Meijer, and S. K. Wismath. Planar and quasi-planar simultaneous geometric embedding. Comput. J., 58(11):3126-3140, 2015. URL: http://dx.doi.org/10.1093/comjnl/bxv048.
http://dx.doi.org/10.1093/comjnl/bxv048
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http://dx.doi.org/10.1007/978-3-642-32241-9_29
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http://dx.doi.org/10.1007/PL00009322
A. Bekos M. M. Kaufmann, and N. Raftopoulou C.Ȯn optimal 2- and 3-planar graphs. CoRR, 1703.06526, 2017.
J. Pach, R. Radoičić, G. Tardos, and G. Tóth. Improving the crossing lemma by finding more crossings in sparse graphs. Discrete & Computational Geometry, 36(4):527-552, 2006. URL: http://dx.doi.org/10.1007/s00454-006-1264-9.
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J. Pach and G. Tóth. Graphs drawn with few crossings per edge. Combinatorica, 17(3):427-439, 1997. URL: http://dx.doi.org/10.1007/BF01215922.
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http://dx.doi.org/10.1002/jgt.3190010105
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Reachability in a Planar Subdivision with Direction Constraints
Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. We give an O(n log n)-time algorithm for this problem, as well as an Omega(n log n) lower bound in the algebraic computation tree model. We prove that the generalization where two cones of directions per face are allowed is NP-hard.
Design and analysis of geometric algorithms
Path planning
Reachability
17:1-17:15
Regular Paper
Daniel
Binham
Daniel Binham
Pedro Machado
Manhaes de Castro
Pedro Machado Manhaes de Castro
Antoine
Vigneron
Antoine Vigneron
10.4230/LIPIcs.SoCG.2017.17
Michael Ben-Or. Lower bounds for algebraic computation trees. In Proc. 15th ACM Symposium on Theory of Computing, pages 80-86, 1983. URL: http://dx.doi.org/10.1145/800061.808735.
http://dx.doi.org/10.1145/800061.808735
Siu-Wing Cheng and Jiongxin Jin. Approximate shortest descending paths. In Proc. 24th ACM-SIAM Symposium on Discrete Algorithms, pages 144-155, 2013.
Siu-Wing Cheng, Hyeon-Suk Na, Antoine Vigneron, and Yajun Wang. Approximate shortest paths in anisotropic regions. SIAM Journal on Computing, 38(3):802-824, 2008.
Mark de Berg, Herman J. Haverkort, and Constantinos P. Tsirogiannis. Implicit flow routing on terrains with applications to surface networks and drainage structures. In Proc. 22nd ACM-SIAM Symposium on Discrete Algorithms, pages 285-296, 2011.
Mark de Berg and Marc J. van Kreveld. Trekking in the alps without freezing or getting tired. Algorithmica, 18(3):306-323, 1997. URL: http://dx.doi.org/10.1007/PL00009159.
http://dx.doi.org/10.1007/PL00009159
John H. Reif and Zheng Sun. Movement planning in the presence of flows. Algorithmica, 39(2):127-153, 2004. URL: http://dx.doi.org/10.1007/s00453-003-1079-5.
http://dx.doi.org/10.1007/s00453-003-1079-5
Zheng Sun and John H. Reif. On finding energy-minimizing paths on terrains. IEEE Transactions on Robotics, 21(1):102-114, 2005. URL: http://dx.doi.org/10.1109/TRO.2004.837232.
http://dx.doi.org/10.1109/TRO.2004.837232
Robert Endre Tarjan. Data Structures and Network Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1983.
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Fine-Grained Complexity of Coloring Unit Disks and Balls
On planar graphs, many classic algorithmic problems enjoy a certain "square root phenomenon" and can be solved significantly faster than what is known to be possible on general graphs: for example, Independent Set, 3-Coloring, Hamiltonian Cycle, Dominating Set can be solved in time 2^O(sqrt{n}) on an n-vertex planar graph, while no 2^o(n) algorithms exist for general graphs, assuming the Exponential Time Hypothesis (ETH). The square root in the exponent seems to be best possible for planar graphs: assuming the ETH, the running time for these problems cannot be improved to 2^o(sqrt{n}). In some cases, a similar speedup can be obtained for 2-dimensional geometric problems, for example, there are 2^O(sqrt{n}log n) time algorithms for Independent Set on unit disk graphs or for TSP on 2-dimensional point sets.
In this paper, we explore whether such a speedup is possible for geometric coloring problems. On the one hand, geometric objects can behave similarly to planar graphs: 3-Coloring can be solved in time 2^O(sqrt{n}) on the intersection graph of n unit disks in the plane and, assuming the ETH, there is no such algorithm with running time 2^o(sqrt{n}). On the other hand, if the number L of colors is part of the input, then no such speedup is possible: Coloring the intersection graph of n unit disks with L colors cannot be solved in time 2^o(n), assuming the ETH. More precisely, we exhibit a smooth increase of complexity as the number L of colors increases: If we restrict the number of colors to L=Theta(n^alpha) for some 0<=alpha<=1, then the problem of coloring the intersection graph of n unit disks with L colors
* can be solved in time exp(O(n^{{1+alpha}/2}log n))=exp( O(sqrt{nL}log n)), and
* cannot be solved in time exp(o(n^{{1+alpha}/2}))=exp(o(sqrt{nL})), unless the ETH fails.
More generally, we consider the problem of coloring d-dimensional unit balls in the Euclidean space and obtain analogous results showing that the problem
* can be solved in time exp(O(n^{{d-1+alpha}/d}log n))=exp(O(n^{1-1/d}L^{1/d}log n)), and
* cannot be solved in time exp(n^{{d-1+alpha}/d-epsilon})= exp (O(n^{1-1/d-epsilon}L^{1/d})) for any epsilon>0, unless the ETH fails.
unit disk graphs
unit ball graphs
coloring
exact algorithm
18:1-18:16
Regular Paper
Csaba
Biró
Csaba Biró
Édouard
Bonnet
Édouard Bonnet
Dániel
Marx
Dániel Marx
Tillmann
Miltzow
Tillmann Miltzow
Pawel
Rzazewski
Pawel Rzazewski
10.4230/LIPIcs.SoCG.2017.18
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
Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and Dániel Marx. Tight bounds for Planar Strongly Connected Steiner Subgraph with fixed number of terminals (and extensions). In SODA 2014 Proc., pages 1782-1801, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.129.
http://dx.doi.org/10.1137/1.9781611973402.129
Marek Cygan, Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin, Jakub W. Pachocki, and Arkadiusz Socała. Tight lower bounds on graph embedding problems. CoRR, abs/1602.05016, 2016. URL: http://arxiv.org/abs/1602.05016.
http://arxiv.org/abs/1602.05016
Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Bidimensional parameters and local treewidth. SIAM J. Discrete Math., 18(3):501-511, 2004. URL: http://dx.doi.org/10.1137/S0895480103433410.
http://dx.doi.org/10.1137/S0895480103433410
Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Fixed-parameter algorithms for (k,r)-Center in planar graphs and map graphs. ACM Transactions on Algorithms, 1(1):33-47, 2005. URL: http://dx.doi.org/10.1145/1077464.1077468.
http://dx.doi.org/10.1145/1077464.1077468
Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM, 52(6):866-893, 2005. URL: http://dx.doi.org/10.1145/1101821.1101823.
http://dx.doi.org/10.1145/1101821.1101823
Erik D. Demaine and Mohammad Taghi Hajiaghayi. Fast algorithms for hard graph problems: Bidimensionality, minors, and local treewidth. In GD 2014 Proc., pages 517-533, 2004. URL: http://dx.doi.org/10.1007/978-3-540-31843-9_57.
http://dx.doi.org/10.1007/978-3-540-31843-9_57
Erik D. Demaine and MohammadTaghi Hajiaghayi. The bidimensionality theory and its algorithmic applications. Comput. J., 51(3):292-302, 2008. URL: http://dx.doi.org/10.1093/comjnl/bxm033.
http://dx.doi.org/10.1093/comjnl/bxm033
Erik D. Demaine and MohammadTaghi Hajiaghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica, 28(1):19-36, 2008. URL: http://dx.doi.org/10.1007/s00493-008-2140-4.
http://dx.doi.org/10.1007/s00493-008-2140-4
Frederic Dorn, Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs. In STACS 2010 Proc., pages 251-262, 2010. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2010.2459.
http://dx.doi.org/10.4230/LIPIcs.STACS.2010.2459
Frederic Dorn, Fedor V. Fomin, and Dimitrios M. Thilikos. Subexponential parameterized algorithms. Computer Science Review, 2(1):29-39, 2008. URL: http://dx.doi.org/10.1016/j.cosrev.2008.02.004.
http://dx.doi.org/10.1016/j.cosrev.2008.02.004
Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica, 58(3):790-810, 2010. URL: http://dx.doi.org/10.1007/s00453-009-9296-1.
http://dx.doi.org/10.1007/s00453-009-9296-1
Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Yngve Villanger. Tight bounds for parameterized complexity of cluster editing with a small number of clusters. J. Comput. Syst. Sci., 80(7):1430-1447, 2014. URL: http://dx.doi.org/10.1016/j.jcss.2014.04.015.
http://dx.doi.org/10.1016/j.jcss.2014.04.015
Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Subexponential algorithms for partial cover problems. Inf. Process. Lett., 111(16):814-818, 2011. URL: http://dx.doi.org/10.1016/j.ipl.2011.05.016.
http://dx.doi.org/10.1016/j.ipl.2011.05.016
Fedor V. Fomin and Dimitrios M. Thilikos. Dominating sets in planar graphs: Branch-width and exponential speed-up. SIAM J. Comput., 36(2):281-309, 2006. URL: http://dx.doi.org/10.1137/S0097539702419649.
http://dx.doi.org/10.1137/S0097539702419649
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
Philip N. Klein and Dániel Marx. Solving Planar k-Terminal Cut in O(n^c√k) time. In ICALP 2012 Proc., pages 569-580, 2012. URL: http://dx.doi.org/10.1007/978-3-642-31594-7_48.
http://dx.doi.org/10.1007/978-3-642-31594-7_48
Philip N. Klein and Dániel Marx. A subexponential parameterized algorithm for Subset TSP on planar graphs. In SODA 2014 Proc., pages 1812-1830, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.131.
http://dx.doi.org/10.1137/1.9781611973402.131
J. Kratochvíl and J. Matoušek. Intersection graphs of segments. Journal of Combinatorial Theory, Series B, 62(2):289-315, 1994. URL: http://dx.doi.org/10.1006/jctb.1994.1071.
http://dx.doi.org/10.1006/jctb.1994.1071
Dániel Marx. Efficient approximation schemes for geometric problems? In ESA 2005 Proc., pages 448-459, 2005. URL: http://dx.doi.org/10.1007/11561071_41.
http://dx.doi.org/10.1007/11561071_41
Dániel Marx and Michal Pilipczuk. Optimal parameterized algorithms for planar facility location problems using voronoi diagrams. In Nikhil Bansal and Irene Finocchi, editors, ESA 2015 Proc., volume 9294 of LNCS, pages 865-877. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_72.
http://dx.doi.org/10.1007/978-3-662-48350-3_72
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 2014 Proc., pages 67:67-67:76, New York, NY, USA, 2014. ACM. URL: http://dx.doi.org/10.1145/2582112.2582124.
http://dx.doi.org/10.1145/2582112.2582124
Gary L. Miller, Shang-Hua Teng, William Thurston, and Stephen A. Vavasis. Separators for sphere-packings and nearest neighbor graphs. J. ACM, 44(1):1-29, January 1997. URL: http://dx.doi.org/10.1145/256292.256294.
http://dx.doi.org/10.1145/256292.256294
Marcin Pilipczuk, Michał Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Subexponential-time parameterized algorithm for Steiner Tree on planar graphs. In STACS 2013 Proc., pages 353-364, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.353.
http://dx.doi.org/10.4230/LIPIcs.STACS.2013.353
Marcin Pilipczuk, Michal Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Network sparsification for steiner problems on planar and bounded-genus graphs. In FOCS 2014 Proc., pages 276-285. IEEE Computer Society, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.37.
http://dx.doi.org/10.1109/FOCS.2014.37
W. D. Smith and N. C. Wormald. Geometric separator theorems. available online at URL: https://www.math.uwaterloo.ca/~nwormald/papers/focssep.ps.gz.
https://www.math.uwaterloo.ca/~nwormald/papers/focssep.ps.gz
W. D. Smith and N. C. Wormald. Geometric separator theorems and applications. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science, FOCS 1998 Proc., pages 232-243, Washington, DC, USA, 1998. IEEE Computer Society. URL: http://dl.acm.org/citation.cfm?id=795664.796397.
http://dl.acm.org/citation.cfm?id=795664.796397
Dimitrios M. Thilikos. Fast sub-exponential algorithms and compactness in planar graphs. In ESA 2011 Proc., pages 358-369, 2011. URL: http://dx.doi.org/10.1007/978-3-642-23719-5_31.
http://dx.doi.org/10.1007/978-3-642-23719-5_31
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Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams
The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R^2 and on surfaces embedded in R^3 as detailed in our experimental companion paper.
In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Omega equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Omega under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened.
Riemannian Geometry
Voronoi diagram
Delaunay triangulation
19:1-19:16
Regular Paper
Jean-Daniel
Boissonnat
Jean-Daniel Boissonnat
Mael
Rouxel-Labbé
Mael Rouxel-Labbé
Mathijs
Wintraecken
Mathijs Wintraecken
10.4230/LIPIcs.SoCG.2017.19
F. Aurenhammer and R. Klein. Voronoi diagrams. In J. Sack and G. Urrutia, editors, Handbook of Computational Geometry, pages 201-290. Elsevier Science Publishing, 2000.
J.-D. Boissonnat, R. Dyer, and A. Ghosh. Delaunay triangulation of manifolds. Foundations of Computational Mathematics, pages 1-33, 2017.
J.-D. Boissonnat, R. Dyer, A. Ghosh, and S. Y. Oudot. Only distances are required to reconstruct submanifolds. Comp. Geom. Theory and Appl., 2016. To appear.
J.-D. Boissonnat, M. Rouxel-Labbé, and M. Wintraecken. Anisotropic triangulations via discrete Riemannian Voronoi diagrams, 2017. URL: https://arxiv.org/abs/1703.06487.
https://arxiv.org/abs/1703.06487
M. Campen, M. Heistermann, and L. Kobbelt. Practical anisotropic geodesy. In Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing, SGP'13, pages 63-71. Eurographics Association, 2013.
G. D. Cañas and S. J. Gortler. Orphan-free anisotropic Voronoi diagrams. Discrete and Computational Geometry, 46(3), 2011.
G. D. Cañas and S. J. Gortler. Duals of orphan-free anisotropic Voronoi diagrams are embedded meshes. In SoCG, pages 219-228. ACM, 2012.
T. Cao, H. Edelsbrunner, and T. Tan. Proof of correctness of the digital Delaunay triangulation algorithm. Comp. Geo.: Theory and Applications, 48, 2015.
S.-W. Cheng, T. K. Dey, E. A. Ramos, and R. Wenger. Anisotropic surface meshing. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 202-211. Society for Industrial and Applied Mathematics, 2006.
E. F. D'Azevedo and R. B. Simpson. On optimal interpolation triangle incidences. SIAM J. Sci. Statist. Comput., 10(6):1063-1075, 1989.
T. K. Dey, F. Fan, and Y. Wang. Graph induced complex on point data. Computational Geometry, 48(8):575-588, 2015.
Q. Du and D. Wang. Anisotropic centroidal Voronoi tessellations and their applications. SIAM Journal on Scientific Computing, 26(3):737-761, 2005.
R. Dyer, G. Vegter, and M. Wintraecken. Riemannian simplices and triangulations. Preprint: arXiv:1406.3740, 2014.
R. Dyer, H. Zhang, and T. Möller. Surface sampling and the intrinsic Voronoi diagram. Computer Graphics Forum, 27(5):1393-1402, 2008.
M. Garland and P. S. Heckbert. Surface simplification using quadric error metrics. In ACM SIGGRAPH, pages 209-216, 1997.
H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30:509-541, 1977.
F. Labelle and J. R. Shewchuk. Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. In SCG'03: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, pages 191-200. ACM, 2003.
G. Leibon. Random Delaunay triangulations, the Thurston-Andreev theorem, and metric uniformization. PhD thesis, UCSD, 1999.
J.-M. Mirebeau. Optimal meshes for finite elements of arbitrary order. Constructive approximation, 32(2):339-383, 2010.
P. Niyogi, S. Smale, and S. Weinberger. Finding the homology of submanifolds with high confidence from random samples. Discrete &Comp. Geom., 39(1-3), 2008.
G. Peyré, M. Péchaud, R. Keriven, and L. D. Cohen. Geodesic methods in computer vision and graphics. Found. Trends. Comput. Graph. Vis., 2010.
C. Rourke and B. Sanderson. Introduction to piecewise-linear topology. Springer Science &Business Media, 2012.
M. Rouxel-Labbé, M. Wintraecken, and J.-D. Boissonnat. Discretized Riemannian Delaunay triangulations. In Proc. of the 25th Intern. Mesh. Round. Elsevier, 2016.
J. R. Shewchuk. What is a good linear finite element? Interpolation, conditioning, anisotropy, and quality measures, Manuscript 2002.
E. Sperner. Fifty years of further development of a combinatorial lemma. Numerical solution of highly nonlinear problems, pages 183-197, 1980.
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An Approximation Algorithm for the Art Gallery Problem
Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum-size set S such that every point in P is visible from a point in S. The set S is referred to as guards. Assuming integer coordinates and a specific general position on the vertices of P, we present the first O(log OPT)-approximation algorithm for the point guard problem. This algorithm combines ideas in papers of Efrat and Har-Peled and Deshpande et al. We also point out a mistake in the latter.
computational geometry
art gallery
approximation algorithm
20:1-20:15
Regular Paper
Édouard
Bonnet
Édouard Bonnet
Tillmann
Miltzow
Tillmann Miltzow
10.4230/LIPIcs.SoCG.2017.20
Eyüp Serdar Ayaz and Alper Üngör. Minimal witness sets for art gallery problems. EuroCG, 2016.
János Barát, Vida Dujmovic, Gwenaël Joret, Michael S. Payne, Ludmila Scharf, Daria Schymura, Pavel Valtr, and David R. Wood. Empty pentagons in point sets with collinearities. SIAM J. Discrete Math., 29(1):198-209, 2015.
Saugata Basu, Richard Pollack, and Marie-Francoise Roy. Algorithms in real algebraic geometry. Springer, 2007.
Patrice Belleville. Computing two-covers of simple polygons. Master’s thesis, McGill University, 1991.
Vijay V. S. P. Bhattiprolu and Sariel Har-Peled. Separating a voronoi diagram via local search. In SOCG, pages 18:1-18:16, 2016.
Édouard Bonnet and Tillmann Miltzow. An approximation algorithm for the art gallery problem. CoRR, 1607.05527, 2016. URL: http://arxiv.org/abs/1607.05527.
http://arxiv.org/abs/1607.05527
Édouard Bonnet and Tillmann Miltzow. The parameterized hardness of the art gallery problem. In ESA 2016, pages 19:1-19:17, 2016. Arxiv identifier: 1603.08116.
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Kyung-Yong Chwa, Byung-Cheol Jo, Christian Knauer, Esther Moet, René van Oostrum, and Chan-Su Shin. Guarding art galleries by guarding witnesses. Int. J. Comput. Geometry Appl., 16(2-3):205-226, 2006.
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Pedro Jussieu de Rezende, Cid C. de Souza, Stephan Friedrichs, Michael Hemmer, Alexander Kröller, and Davi C. Tozoni. Engineering art galleries. CoRR, abs/1410.8720, 2014. URL: http://arxiv.org/abs/1410.8720.
http://arxiv.org/abs/1410.8720
Ajay Deshpande. A pseudo-polynomial time O(log² n)-approximation algorithm for art gallery problems. Master’s thesis, Department of Mechanical Engineering, Department of Electrical Engineering and Computer Science, MIT, 2006.
Ajay Deshpande, Taejung Kim, Erik D. Demaine, and Sanjay E. Sarma. A pseudopolynomial time O(log n)-approximation algorithm for art gallery problems. In WADS 2007, 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
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Khaled Elbassioni. Finding small hitting sets in infinite range spaces of bounded VC-dimension. CoRR, abs/1610.03812, 2016. accepted to SoCG 2017.
Steve Fisk. A short proof of Chvátal’s watchman theorem. J. Comb. Theory, Ser. B, 24(3):374, 1978. URL: http://dx.doi.org/10.1016/0095-8956(78)90059-X.
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Self-Approaching Paths in Simple Polygons
We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, |ac| >= |bc|. We analyze the properties, and present a characterization of shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points inside a polygon can be forced to follow a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find a self-approaching path inside a polygon connecting two points under a model of computation which assumes that we can calculate involute curves of high order.
Lastly, we provide an algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.
self-approaching path
simple polygon
shortest path
involute curve
21:1-21:15
Regular Paper
Prosenjit
Bose
Prosenjit Bose
Irina
Kostitsyna
Irina Kostitsyna
Stefan
Langerman
Stefan Langerman
10.4230/LIPIcs.SoCG.2017.21
O. Aichholzer, F. Aurenhammer, C. Icking, R. Klein, E. Langetepe, and G. Rote. Generalized self-approaching curves. Discrete Applied Mathematics, 109(1-2):3-24, 2001. URL: http://dx.doi.org/10.1016/S0166-218X(00)00233-X.
http://dx.doi.org/10.1016/S0166-218X(00)00233-X
S. Alamdari, T. M. Chan, E. Grant, A. Lubiw, and V. Pathak. Self-approaching Graphs. In 20th International Symposium on Graph Drawing (GD), pages 260-271, 2012. URL: http://dx.doi.org/10.1007/978-3-642-36763-2_23.
http://dx.doi.org/10.1007/978-3-642-36763-2_23
E. M. Arkin, R. Connelly, and J. S. B. Mitchell. On monotone paths among obstacles with applications to planning assemblies. In 5th Annual Symposium on Computational Geometry (SCG), pages 334-343. ACM Press, 1989. URL: http://dx.doi.org/10.1145/73833.73870.
http://dx.doi.org/10.1145/73833.73870
M. A. Bender and M. Farach-Colton. The LCA Problem Revisited. In Latin American Symposium on Theoretical Informatics, pages 88-94, 2000. URL: http://dx.doi.org/10.1007/10719839_9.
http://dx.doi.org/10.1007/10719839_9
M. Biro, J. Iwerks, I. Kostitsyna, and J. S. B. Mitchell. Beacon-Based Algorithms for Geometric Routing. In 13th Algorithms and Data Structures Symposium (WADS), pages 158-169. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40104-6_14.
http://dx.doi.org/10.1007/978-3-642-40104-6_14
P. Bose, I. Kostitsyna, and S. Langerman. Self-approaching paths in simple polygons. Preprint, http://arxiv.org/abs/1703.06107, 2017.
http://arxiv.org/abs/1703.06107
B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, J. Hershberger, M. Sharir, and J. Snoeyink. Ray shooting in polygons using geodesic triangulations. Algorithmica, 12(1):54-68, 1994. URL: http://dx.doi.org/10.1007/BF01377183.
http://dx.doi.org/10.1007/BF01377183
H. Dehkordi, F. Frati, and J. Gudmundsson. Increasing-Chord Graphs On Point Sets. In 22nd International Symposium on Graph Drawing, pages 464-475. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-45803-7_39.
http://dx.doi.org/10.1007/978-3-662-45803-7_39
L. E. Dubins. On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents. American Journal of Mathematics, 79(3):497-516, 1957. URL: http://dx.doi.org/10.2307/2372560.
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L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987. URL: http://dx.doi.org/10.1007/BF01840360.
http://dx.doi.org/10.1007/BF01840360
C. Icking and R. Klein. Searching for the kernel of a polygon - a competitive strategy. In 11th Annual Symposium on Computational Geometry (SCG), pages 258-266. ACM Press, 1995. URL: http://dx.doi.org/10.1145/220279.220307.
http://dx.doi.org/10.1145/220279.220307
C. Icking, R. Klein, and E. Langetepe. Self-approaching curves. Mathematical Proceedings of the Cambridge Philosophical Society, 125(3):441-453, 1999. URL: http://dx.doi.org/10.1017/S0305004198003016.
http://dx.doi.org/10.1017/S0305004198003016
M. Laczkovich. The removal of π from some undecidable problems involving elementary functions. Proceedings of the American Mathematical Society, 131(07):2235-2241, 2003. URL: http://dx.doi.org/10.1090/S0002-9939-02-06753-9.
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J. S. B. Mitchell, C. Piatko, and E. M. Arkin. Computing a shortest k-link path in a polygon. In 33rd Annual Symposium on Foundations of Computer Science, pages 573-582. IEEE, 1992. URL: http://dx.doi.org/10.1109/SFCS.1992.267794.
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M. Nöllenburg, R. Prutkin, and I. Rutter. On self-approaching and increasing-chord drawings of 3-connected planar graphs. Journal of Computational Geometry, 7(1):47-69, 2016. URL: http://dx.doi.org/10.20382/jocg.v7i1a3.
http://dx.doi.org/10.20382/jocg.v7i1a3
G. Rote. Curves with increasing chords. Mathematical Proceedings of the Cambridge Philosophical Society, 115(01):1, 1994. URL: http://dx.doi.org/10.1017/S0305004100071875.
http://dx.doi.org/10.1017/S0305004100071875
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Maximum Volume Subset Selection for Anchored Boxes
Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that:
* The problem is NP-hard already in 3 dimensions.
* In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm.
* For any constant dimension d, we give an efficient polynomial-time approximation scheme.
geometric optimization
subset selection
hypervolume indicator
Klee’s 23 measure problem
boxes
NP-hardness
PTAS
22:1-22:15
Regular Paper
Karl
Bringmann
Karl Bringmann
Sergio
Cabello
Sergio Cabello
Michael T. M.
Emmerich
Michael T. M. Emmerich
10.4230/LIPIcs.SoCG.2017.22
A. Adamaszek and A. Wiese. Approximation schemes for maximum weight independent set of rectangles. In Proc. of the 54th IEEE Symp. on Found. of Comp. Science (FOCS), pages 400-409. IEEE, 2013.
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A. Auger, J. Bader, D. Brockhoff, and E. Zitzler. Hypervolume-based multiobjective optimization: Theoretical foundations and practical implications. Theoretical Comp. Science, 425:75-103, 2012.
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N. Beume, B. Naujoks, and M. Emmerich. SMS-EMOA: Multiobjective selection based on dominated hypervolume. European J. of Operational Research, 181(3):1653-1669, 2007.
K. Bringmann. Bringing order to special cases of Klee’s measure problem. In Int. Symp. on Mathematical Foundations of Comp. Science, pages 207-218. Springer, 2013.
K. Bringmann and T. Friedrich. Approximating the volume of unions and intersections of high-dimensional geometric objects. Computational Geometry, 43(6):601-610, 2010.
K. Bringmann and T. Friedrich. An efficient algorithm for computing hypervolume contributions. Evolutionary Computation, 18(3):383-402, 2010.
K. Bringmann and T. Friedrich. Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. Theoretical Comp. Science, 425:104-116, 2012.
K. Bringmann, T. Friedrich, and P. Klitzke. Generic postprocessing via subset selection for hypervolume and epsilon-indicator. In Int. Conf. on Parallel Problem Solving from Nature, pages 518-527. Springer, 2014.
K. Bringmann, T. Friedrich, and P. Klitzke. Two-dimensional subset selection for hypervolume and epsilon-indicator. In Proc. of the 2014 Conf. on Genetic and Evolutionary Comput., pages 589-596. ACM, 2014.
T. M. Chan. A (slightly) faster algorithm for Klee’s measure problem. Computational Geometry, 43(3):243-250, 2010.
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M. Emmerich, A. H. Deutz, and I. Yevseyeva. A Bayesian approach to portfolio selection in multicriteria group decision making. Procedia Comp. Science, 64:993-1000, 2015.
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A. P. Guerreiro, C. M. Fonseca, and L. Paquete. Greedy hypervolume subset selection in low dimensions. Evolutionary Computation, 24(3):521-544, 2016.
D. S. Hochbaum and W. Maass. Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM, 32(1):130-136, 1985.
H. Imai and T. Asano. Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. of Algorithms, 4(4):310-323, 1983.
J. D. Knowles, D. W. Corne, and M. Fleischer. Bounded archiving using the Lebesgue measure. In Proc. of the 2003 Congress on Evolutionary Computation (CEC), volume 4, pages 2490-2497. IEEE, 2003.
T. Kuhn, C. M. Fonseca, L. Paquete, S. Ruzika, M. M. Duarte, and J. R. Figueira. Hypervolume subset selection in two dimensions: Formulations and algorithms. Evolutionary Computation, 2015.
G. L. Miller. Finding small simple cycle separators for 2-connected planar graphs. J. Comput. Syst. Sci., 32(3):265-279, 1986.
J. S. B. Mitchell and M. Sharir. New results on shortest paths in three dimensions. In Proc. of the 20th ACM Symp. on Computational Geometry, pages 124-133, 2004.
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G. Rote, K. Buchin, K. Bringmann, S. Cabello, and M. Emmerich. Selecting k points that maximize the convex hull volume (extended abstract). In JCDCG3 2016; The 19th Japan Conf. on Discrete and Computational Geometry, Graphs, and Games, pages 58-60, 9 2016. URL: http://www.jcdcgg.u-tokai.ac.jp/JCDCG3_abstracts.pdf.
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L. While, P. Hingston, L. Barone, and S. Huband. A faster algorithm for calculating hypervolume. IEEE Trans. on Evolutionary Computation, 10(1):29-38, 2006.
J. Wu and S. Azarm. Metrics for quality assessment of a multiobjective design optimization solution set. J. of Mechanical Design, 123(1):18-25, 2001.
E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. G. Da Fonseca. Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. on Evolutionary Computation, 7(2):117-132, 2003.
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Declutter and Resample: Towards Parameter Free Denoising
In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth K in a metric space, but it got corrupted with noise so that some of the data points lie far away from K creating outliers also termed as ambient noise. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of K. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases.
Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms
are effective in practice.
denoising
parameter free
k-distance,compact sets
23:1-23:16
Regular Paper
Mickael
Buchet
Mickael Buchet
Tamal K.
Dey
Tamal K. Dey
Jiayuan
Wang
Jiayuan Wang
Yusu
Wang
Yusu Wang
10.4230/LIPIcs.SoCG.2017.23
N. Amenta and M. Bern. Surface reconstruction by voronoi filtering. Discr. Comput. Geom., 22:481-504, 1999.
G. Biau et al. A weighted k-nearest neighbor density estimate for geometric inference. Electronic Journal of Statistics, 5:204-237, 2011.
J.-D. Boissonnat, L. J. Guibas, and S. Y. Oudot. Manifold reconstruction in arbitrary dimensions using witness complexes. Discr. Comput. Geom., 42(1):37-70, 2009.
M. Buchet. Topological inference from measures. PhD thesis, Paris 11, 2014.
M. Buchet, F. Chazal, T. K. Dey, F. Fan, S. Y. Oudot, and Y. Wang. Topological analysis of scalar fields with outliers. In Proc. 31st Sympos. Comput. Geom., pages 827-841, 2015.
M. Buchet, T. K. Dey, J. Wang, and Y. Wang. Declutter and resample: Towards parameter free denoising. arXiv version of this paper: arXiv 1511.05479, 2015.
C. Caillerie, F. Chazal, J. Dedecker, and B. Michel. Deconvolution for the wasserstein metric and geometric inference. In Geom. Sci. Info., pages 561-568. 2013.
T.-H. H. Chan, M. Dinitz, and A. Gupta. Spanners with slack. In Euro. Sympos. Algo., pages 196-207, 2006.
F. Chazal, D. Cohen-Steiner, and A. Lieutier. A sampling theory for compact sets in Euclidean space. Discr. Comput. Geom., 41(3):461-479, 2009.
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T. K. Dey, Z. Dong, and Y. Wang. Parameter-free topology inference and sparsification for data on manifolds. In Proc. ACM-SIAM Sympos. Discr. Algo., pages 2733-2747, 2017.
T. K. Dey, J. Giesen, S. Goswami, and W. Zhao. Shape dimension and approximation from samples. Discr. Comput. Geom., 29:419-434, 2003.
D. L. Donoho. De-noising by soft-thresholding. IEEE Trans. Info. Theory., 41(3):613-627, 1995.
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J. Zhang. Advancements of outlier detection: A survey. EAI Endorsed Trans. Scalable Information Systems, 1(1):e2, 2013.
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Ham Sandwich is Equivalent to Borsuk-Ulam
The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in R^n can always be simultaneously bisected by an (n-1)-dimensional hyperplane.
In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and the Ham Sandwich theorem. The main technical result we show towards establishing the equivalence is the following: For every odd polynomial restricted to the hypersphere f:S^n->R, there exists a compact set A in R^{n+1}, such that for every x in S^n we have f(x)=vol(A cap H^+) - vol(A cap H^-), where H is the oriented hyperplane containing the origin with x as the normal. A noteworthy aspect of the proof of the above result is the use of hyperspherical harmonics.
Finally, using the above result we prove that there exist constants n_0, epsilon_0>0 such that for every n>= n_0 and epsilon <= epsilon_0/sqrt{48n}, any query algorithm to find an epsilon-bisecting (n-1)-dimensional hyperplane of n compact set in [-n^4.51,n^4.51]^n, even with success probability 2^-Omega(n), requires 2^Omega(n) queries.
Ham Sandwich theorem
Borsuk-Ulam theorem
Query Complexity
Hyperspherical Harmonics
24:1-24:15
Regular Paper
Karthik
C. S.
Karthik C. S.
Arpan
Saha
Arpan Saha
10.4230/LIPIcs.SoCG.2017.24
James Aisenberg, Maria Luisa Bonet, and Sam Buss. 2-D Tucker is PPA complete. Electronic Colloquium on Computational Complexity (ECCC), 22:163, 2015. URL: http://eccc.hpi-web.de/report/2015/163.
http://eccc.hpi-web.de/report/2015/163
Kendall Atkinson and Weimin Han. Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Springer-Verlag Berlin Heidelberg, 2012. URL: http://dx.doi.org/10.1007/978-3-642-25983-8.
http://dx.doi.org/10.1007/978-3-642-25983-8
Yakov Babichenko. Query complexity of approximate nash equilibria. J. ACM, 63(4):36, 2016. URL: http://dx.doi.org/10.1145/2908734.
http://dx.doi.org/10.1145/2908734
Yakov Babichenko and Aviad Rubinstein. Communication complexity of approximate nash equilibria. CoRR, abs/1608.06580, 2016. URL: http://arxiv.org/abs/1608.06580.
http://arxiv.org/abs/1608.06580
Kim Border. Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, 1989. URL: http://dx.doi.org/10.1137/1028074.
http://dx.doi.org/10.1137/1028074
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L. E. J. Brouwer. Über Abbildung von Mannigfaltigkeiten. Mathematische Annalen, 71:97-115, 1912. URL: http://eudml.org/doc/158520.
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Xi Chen and Xiaotie Deng. Matching algorithmic bounds for finding a brouwer fixed point. J. ACM, 55(3), 2008. URL: http://dx.doi.org/10.1145/1379759.1379761.
http://dx.doi.org/10.1145/1379759.1379761
Xi Chen and Xiaotie Deng. On the complexity of 2D discrete fixed point problem. Theor. Comput. Sci., 410(44):4448-4456, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2009.07.052.
http://dx.doi.org/10.1016/j.tcs.2009.07.052
Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Settling the complexity of computing two-player nash equilibria. J. ACM, 56(3), 2009. URL: http://dx.doi.org/10.1145/1516512.1516516.
http://dx.doi.org/10.1145/1516512.1516516
Xi Chen and Shang-Hua Teng. Paths beyond local search: A tight bound for randomized fixed-point computation. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007, Providence, RI, USA, Proceedings, pages 124-134, 2007. URL: http://dx.doi.org/10.1109/FOCS.2007.53.
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Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a nash equilibrium. SIAM J. Comput., 39(1):195-259, 2009. URL: http://dx.doi.org/10.1137/070699652.
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http://dx.doi.org/10.4230/LIPIcs.STACS.2011.649
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Local Equivalence and Intrinsic Metrics between Reeb Graphs
As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as functional distortion or interleaving that are provably more discriminative than the so-called bottleneck distance, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough locally, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the intrinsic metrics induced by these distances, which turn out to be all globally equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them.
Reeb Graphs
Extended Persistence
Induced Metrics
Topological Data Analysis
25:1-25:15
Regular Paper
Mathieu
Carrière
Mathieu Carrière
Steve
Oudot
Steve Oudot
10.4230/LIPIcs.SoCG.2017.25
P. Agarwal, K. Fox, A. Nath, A. Sidiropoulos, and Y. Wang. Computing the Gromov-Hausdorff Distance for Metric Trees. In Symp. Algo. Comput., 2015.
M. Alagappan. From 5 to 13: Redefining the Positions in Basketball. MIT Sloan Sports Analytics Conference, 2012.
V. Barra and S. Biasotti. 3D Shape Retrieval and Classification using Multiple Kernel Learning on Extended Reeb graphs. The Visual Computer, 30(11):1247-1259, 2014.
U. Bauer, X. Ge, and Y. Wang. Measuring Distance between Reeb Graphs. In Symp. Comput. Geom., pages 464-473, 2014.
U. Bauer, X. Ge, and Y. Wang. Measuring Distance between Reeb Graphs (v2). CoRR, abs/1307.2839v2, 2016.
U. Bauer, E. Munch, and Y. Wang. Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs. In Symp. Comput. Geom., 2015.
S. Biasotti, D. Giorgi, M. Spagnuolo, and B. Falcidieno. Reeb Graphs for Shape Analysis and Applications. Theo. Comp. Sci., 392(1-3):5-22, 2008.
H. Bjerkevik. Stability of Higher Dimensional Interval Decomposable Persistence Modules. CoRR, abs/1609.02086, 2016.
A. Blumberg, I. Gall, M. Mandell, and M. Pancia. Robust Statistics, Hypothesis Testing, and Confidence Intervals for Persistent Homology on Metric Measure Spaces. CoRR, abs/1206.4581, 2012.
D. Burago, Y. Burago, and S. Ivanov. A Course in Metric Geometry, volume 33 of Graduate Studies in Mathematics. AMS, Providence, RI, 2001.
G. Carlsson, V. de Silva, and D. Morozov. Zigzag Persistent Homology and Real-valued Functions. In Symp. Comput. Geom., pages 247-256, 2009.
M. Carrière and S. Oudot. Structure and Stability of the 1-Dimensional Mapper. In Symp. Comput. Geom., volume 51, pages 1-16, 2016.
Frédéric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The Structure and Stability of Persistence Modules. Springer, 2016.
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math., 9(1):79-103, 2009.
Vin de Silva, Elizabeth Munch, and Amit Patel. Categorified Reeb Graphs. Discr. Comput. Geom., 55:854-906, 2016.
B. di Fabio and C. Landi. The Edit Distance for Reeb Graphs of Surfaces. Discrete Computational Geometry, 55(2):423-461, 2016.
M. Gameiro, Y. Hiraoka, and I. Obayashi. Continuation of Point Clouds via Persistence Diagrams. Physica D, 334:118-132, 2016.
X. Ge, I. Safa, M. Belkin, and Y. Wang. Data Skeletonization via Reeb Graphs. In Neural Inf. Proc. Sys., pages 837-845, 2011.
Alexandr Ivanov, Nadezhda Nikolaeva, and Alexey Tuzhilin. The Gromov-Hausdorff Metric on the Space of Compact Metric Spaces is Strictly Intrinsic. Mathematical Notes, 100(6):947-950, 2016.
P. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson, and G. Carlsson. Extracting insights from the shape of complex data using topology. Scientific Reports, 3(1236), 2013.
W. Mohamed and A. Ben Hamza. Reeb graph path dissimilarity for 3d object matching and retrieval. The Visual Computer, 28(3):305-318, 2012.
T. Mukasa, S. Nobuhara, A. Maki, and T. Matsuyama. Finding Articulated Body in Time-Series Volume Data, pages 395-404. Springer Berlin Heidelberg, 2006.
M. Nicolau, A. Levine, and G. 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 Science, 108(17):7265-7270, 2011.
S. Ohta. Gradient flows on Wasserstein spaces over compact Alexandrov spaces. American Journal Mathematics, 131(2):475-516, 2009.
S. Ohta. Barycenters in Alexandrov spaces of curvature bounded below. Advances Geometry, 12:571-587, 2012.
G. Reeb. Sur les points singuliers d'une forme de pfaff complètement intégrable ou d'une fonction numérique. CR Acad. Sci. Paris, 222:847-849, 1946.
G. Singh, F. Mémoli, and G. Carlsson. Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition. In Symp. PB Graphics, 2007.
J. Tierny, J.-P. Vandeborre, and M. Daoudi. Invariant High Level Reeb Graphs of 3D Polygonal Meshes. Symp. 3D Data Proc. Vis. Trans., pages 105-112, 2006.
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Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry
We apply the polynomial method - specifically, Chebyshev polynomials - to obtain a number of new results on geometric approximation algorithms in low constant dimensions. For example, we give an algorithm for constructing epsilon-kernels (coresets for approximate width and approximate convex hull) in close to optimal time O(n + (1/epsilon)^{(d-1)/2}), up to a small near-(1/epsilon)^{3/2} factor, for any d-dimensional n-point set. We obtain an improved data structure for Euclidean *approximate nearest neighbor search* with close to O(n log n + (1/epsilon)^{d/4} n) preprocessing time and O((1/epsilon)^{d/4} log n) query time. We obtain improved approximation algorithms for discrete Voronoi diagrams, diameter, and bichromatic closest pair in the L_s-metric for any even integer constant s >= 2. The techniques are general and may have further applications.
diameter
coresets
approximate nearest neighbor search
the polynomial method
streaming
26:1-26:15
Regular Paper
Timothy M.
Chan
Timothy M. Chan
10.4230/LIPIcs.SoCG.2017.26
Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. J. ACM, 51(4):606-635, 2004. Preliminary version in SODA'01 and FOCS'01.
Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Geometric approximation via coresets. In Emo Welzl, editor, Current Trends in Combinatorial and Computational Geometry, pages 1-30. Cambridge University Press, 2005.
Pankaj K. Agarwal, Jirí Matoušek, and Subhash Suri. Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Comput. Geom. Theory Appl., 1:189-201, 1991. URL: http://dx.doi.org/10.1016/0925-7721(92)90001-9.
http://dx.doi.org/10.1016/0925-7721(92)90001-9
Josh Alman, Timothy M. Chan, and Ryan Williams. Polynomial representation of threshold functions and algorithmic applications. In Proc. 57th IEEE Symp. Found. Comput. Sci. (FOCS), pages 467-476, 2016.
Alexandr Andoni and Huy L. Nguyen. Width of points in the streaming model. In Proc. 23rd ACM-SIAM Symp. Discrete Algorithms (SODA), pages 447-452, 2012. ACM Trans. Algorithms, to appear.
S. Arya, G. D. da Fonseca, and D. M. Mount. Approximate polytope membership queries. In Proc. 43rd ACM Symp. Theory Comput. (STOC), pages 579-586, 2011. SIAM J. Comput., to appear. URL: http://www.uniriotec.br/~fonseca/polytope_conf.pdf, URL: http://dx.doi.org/10.1145/1993636.1993713.
http://dx.doi.org/10.1145/1993636.1993713
S. Arya, T. Malamatos, and D. M. Mount. Space-time tradeoffs for approximate nearest neighbor searching. J. ACM, 57:1-54, 2009. Preliminary version in SODA'02 and STOC'02. URL: http://dx.doi.org/10.1145/1613676.1613677.
http://dx.doi.org/10.1145/1613676.1613677
S. Arya, D. M. Mount, N. Netanyahu, R. Silverman, and A. Y. Wu. An optimal algorithm for approximate nearest neighbor searching in fixed dimensions. J. ACM, 45:891-923, 1998. Preliminary version in SODA'94.
Sunil Arya and Timothy M. Chan. Better ε-dependencies for offline approximate nearest neighbor search, Euclidean minimum spanning trees, and ε-kernels. In Proc. 30th Symp. Comput. Geom. (SoCG), pages 416-425, 2014. URL: http://dx.doi.org/10.1145/2582112.2582161.
http://dx.doi.org/10.1145/2582112.2582161
Sunil Arya, Guilherme Dias da Fonseca, and David M. Mount. Optimal area-sensitive bounds for polytope approximation. In Proc. 28th Symp. Comput. Geom. (SoCG), pages 363-372, 2012. URL: http://dx.doi.org/10.1145/2261250.2261305.
http://dx.doi.org/10.1145/2261250.2261305
Sunil Arya, Guilherme Dias da Fonseca, and David M. Mount. On the combinatorial complexity of approximating polytopes. In Proc. 32nd Symp. Comput. Geom. (SoCG), pages 11:1-11:15, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.11.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.11
Sunil Arya, Guilherme Dias da Fonseca, and David M. Mount. Near-optimal ε-kernel construction and related problems. In Proc. 33rd Symp. Comput. Geom. (SoCG), 2017.
Sunil Arya, Guilherme Dias da Fonseca, and David M. Mount. Optimal approximate polytope membership. In Proc. 28th ACM-SIAM Symp. Discrete Algorithms (SODA), pages 270-288, 2017.
Gill Barequet and Sariel Har-Peled. Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms, 38(1):91-109, 2001. Preliminary version in SODA'99. URL: http://dx.doi.org/10.1006/jagm.2000.1127.
http://dx.doi.org/10.1006/jagm.2000.1127
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Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM J. Comput., 39(2):546-563, 2009. URL: http://dx.doi.org/10.1137/070683933.
http://dx.doi.org/10.1137/070683933
H. Breu, J. Gil, D. Kirkpatrick, and M. Werman. Linear time Euclidean distance transform algorithms. IEEE Trans. Pattern Analysis and Machine Intelligence, 17:529-533, 1995.
Timothy M. Chan. Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. Int. J. Comput. Geom.Appl., 12(1-2):67-85, 2002. Preliminary version in SoCG'00. URL: http://dx.doi.org/10.1142/S0218195902000748.
http://dx.doi.org/10.1142/S0218195902000748
Timothy M. Chan. Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput. Geom. Theory Appl., 35(1-2):20-35, 2006. Preliminary version in SoCG'04. URL: http://dx.doi.org/10.1016/j.comgeo.2005.10.002.
http://dx.doi.org/10.1016/j.comgeo.2005.10.002
Timothy M. Chan. Dynamic streaming algorithms for ε-kernels. In Proc. 32nd Symp. Comput. Geom. (SoCG), pages 27:1-27:11, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.27.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.27
Timothy M. Chan and Ryan Williams. Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing Razborov-Smolensky. In Proc. 27th ACM-SIAM Symp. Discrete Algorithms (SODA), pages 1246-1255, 2016.
T. M. Chan. Approximate nearest neighbor queries revisited. Discrete Comput. Geom., 20:359-373, 1998. Preliminary version in SoCG'97.
Don Coppersmith. Rapid multiplication of rectangular matrices. SIAM J. Comput., 11(3):467-471, 1982. URL: http://dx.doi.org/10.1137/0211037.
http://dx.doi.org/10.1137/0211037
Sariel Har-Peled. A practical approach for computing the diameter of a point set. In Proc. 17th Symp. Comput. Geom. (SoCG), pages 177-186, 2001. URL: http://dx.doi.org/10.1145/378583.378662.
http://dx.doi.org/10.1145/378583.378662
Xiaohan Huang and Victor Y. Pan. Fast rectangular matrix multiplication and applications. J. Complexity, 14(2):257-299, 1998. URL: http://dx.doi.org/10.1006/jcom.1998.0476.
http://dx.doi.org/10.1006/jcom.1998.0476
R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995.
Otfried Schwarzkopf. Parallel computation of distance transforms. Algorithmica, 6(5):685-697, 1991. URL: http://dx.doi.org/10.1007/BF01759067.
http://dx.doi.org/10.1007/BF01759067
Gregory Valiant. Finding correlations in subquadratic time, with applications to learning parities and the closest pair problem. J. ACM, 62(2):13, 2015. Preliminary version in FOCS'12.
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H. Yu, P. K. Agarwal, R. Poreddy, and K. R. Varadarajan. Practical methods for shape fitting and kinetic data structures using coresets. Algorithmica, 52(3):378-402, 2008. Preliminary version in SoCG'04.
Hamid Zarrabi-Zadeh. An almost space-optimal streaming algorithm for coresets in fixed dimensions. Algorithmica, 60(1):46-59, 2011. Preliminary version in ESA'08. URL: http://dx.doi.org/10.1007/s00453-010-9392-2.
http://dx.doi.org/10.1007/s00453-010-9392-2
Creative Commons Attribution 3.0 Unported license
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Orthogonal Range Searching in Moderate Dimensions: k-d Trees and Range Trees Strike Back
We revisit the orthogonal range searching problem and the exact l_infinity nearest neighbor searching problem for a static set of n points when the dimension d is moderately large. We give the first data structure with near linear space that achieves truly sublinear query time when the dimension is any constant multiple of log n. Specifically, the preprocessing time and space are O(n^{1+delta}) for any constant delta>0, and the expected query time is n^{1-1/O(c log c)} for d = c log n. The data structure is simple and is based on a new "augmented, randomized, lopsided" variant of k-d trees. It matches (in fact, slightly improves) the performance of previous combinatorial algorithms that work only in the case of offline queries [Impagliazzo, Lovett, Paturi, and Schneider (2014) and Chan (SODA'15)]. It leads to slightly faster combinatorial algorithms for all-pairs shortest paths in general real-weighted graphs and rectangular Boolean matrix multiplication.
In the offline case, we show that the problem can be reduced to the Boolean orthogonal vectors problem and thus admits an n^{2-1/O(log c)}-time non-combinatorial algorithm [Abboud, Williams, and Yu (SODA'15)]. This reduction is also simple and is based on range trees.
Finally, we use a similar approach to obtain a small improvement to Indyk's data structure [FOCS'98] for approximate l_infinity nearest neighbor search when d = c log n.
computational geometry
data structures
range searching
nearest neighbor searching
27:1-27:15
Regular Paper
Timothy M.
Chan
Timothy M. Chan
10.4230/LIPIcs.SoCG.2017.27
Amir Abboud, Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Proc. 26th ACM-SIAM Symp. Discrete Algorithms (SODA), pages 218-230, 2015.
Peyman Afshani, Timothy M. Chan, and Konstantinos Tsakalidis. Deterministic rectangle enclosure and offline dominance reporting on the RAM. In Proc. 41st Int'l Colloq. Automata, Languages, and Programming (ICALP), Part I, pages 77-88, 2014.
Josh Alman, Timothy M. Chan, and Ryan Williams. Polynomial representation of threshold functions with applications. In Proc. 57th IEEE Symp. Found. Comput. Sci. (FOCS), pages 467-476, 2016.
Josh Alman and Ryan Williams. Probabilistic polynomials and Hamming nearest neighbors. In Proc. 56th IEEE Symp. Found. Comput. Sci. (FOCS), pages 136-150, 2015.
Alexandr Andoni, Dorian Croitoru, and Mihai M. Pătraşcu. Hardness of nearest neighbor under L_∞. In Proc. 49th IEEE Symp. Found. Comput. Sci. (FOCS), pages 424-433, 2008.
V. Z. Arlazarov, E. A. Dinic, M. A. Kronrod, and I. A. Faradzhev. On economical construction of the transitive closure of a directed graph. Soviet Mathematics Doklady, 11:1209-1210, 1970.
Timothy M. Chan. Geometric applications of a randomized optimization technique. Discrete Comput. Geom., 22(4):547-567, 1999.
Timothy M. Chan. Speeding up the Four Russians algorithm by about one more logarithmic factor. In Proc. 26th ACM-SIAM Symp. Discrete Algorithms (SODA), pages 212-217, 2015.
Timothy M. Chan and Ryan Williams. Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing Razborov-Smolensky. In Proc. 27th ACM-SIAM Symp. Discrete Algorithms (SODA), pages 1246-1255, 2016.
T. M. Chan. All-pairs shortest paths with real weights in O(n³/log n) time. Algorithmica, 50:236-243, 2008.
T. M. Chan. More algorithms for all-pairs shortest paths in weighted graphs. SIAM J. Comput., 39:2075-2089, 2010.
T. M. Chan, K. G. Larsen, and M. Pătraşcu. Orthogonal range searching on the RAM, revisited. In Proc. 27th ACM Symp. Comput. Geom. (SoCG), pages 1-10, 2011.
T. M. Chan and M. Pătraşcu. Counting inversions, offline orthogonal range counting, and related problems. In Proc. 21st ACM-SIAM Symp. Discrete Algorithms (SODA), pages 161-173, 2010.
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Sariel Har-Peled, Piotr Indyk, and Rajeev Motwani. Approximate nearest neighbor: Towards removing the curse of dimensionality. Theory Comput., 8(1):321-350, 2012.
R. Impagliazzo, S. Lovett, R. Paturi, and S. Schneider. 0-1 integer linear programming with a linear number of constraints, 2014.
Piotr Indyk. On approximate nearest neighbors under l_∞ norm. J. Comput. Sys. Sci., 63(4):627-638, 2001.
Kasper Green Larsen and Ryan Williams. Faster online matrix-vector multiplication. In Proc. 28th ACM-SIAM Symp. Discrete Algorithms (SODA), pages 2182-2189, 2017.
François Le Gall. Faster algorithms for rectangular matrix multiplication. In Proc. 53rd IEEE Symposium on Foundations of Computer Science (FOCS), pages 514-523, 2012.
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R. Williams. Faster all-pairs shortest paths via circuit complexity. In Proc. 46th ACM Symp. Theory Comput. (STOC), pages 664-673, 2014.
Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005.
Ryan Williams. Matrix-vector multiplication in sub-quadratic time (some preprocessing required). In Proc. 18th ACM-SIAM Symp. Discrete Algorithms (SODA), pages 995-1001, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283490.
http://dl.acm.org/citation.cfm?id=1283383.1283490
Huacheng Yu. An improved combinatorial algorithm for Boolean matrix multiplication. In Proc. 42nd Int'l Colloq. Automata, Languages, and Programming (ICALP), Part I, pages 1094-1105, 2015.
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Dynamic Orthogonal Range Searching on the RAM, Revisited
We study a longstanding problem in computational geometry: 2-d dynamic orthogonal range reporting. We present a new data structure achieving O(log n / log log n + k) optimal query time and O(log^{2/3+o(1)}n) update time (amortized) in the word RAM model, where n is the number of data points and k is the output size. This is the first improvement in over 10 years of Mortensen's previous result [SIAM J. Comput., 2006], which has O(log^{7/8+epsilon}n) update time for an arbitrarily small constant epsilon.
In the case of 3-sided queries, our update time reduces to O(log^{1/2+epsilon}n), improving Wilkinson's previous bound [ESA 2014] of O(log^{2/3+epsilon}n).
dynamic data structures
range searching
computational geometry
28:1-28:13
Regular Paper
Timothy M.
Chan
Timothy M. Chan
Konstantinos
Tsakalidis
Konstantinos Tsakalidis
10.4230/LIPIcs.SoCG.2017.28
Stephen Alstrup, Gerth Stølting Brodal, and Theis Rauhe. New data structures for orthogonal range searching. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 198-207, 2000. URL: http://dx.doi.org/10.1109/SFCS.2000.892088.
http://dx.doi.org/10.1109/SFCS.2000.892088
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, Nov 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):13, 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
Jon Louis Bentley. Decomposable searching problems. Information Processing Letters, 8(5):244-251, 1979. URL: http://dx.doi.org/10.1016/0020-0190(79)90117-0.
http://dx.doi.org/10.1016/0020-0190(79)90117-0
Gerth Stølting Brodal. External memory three-sided range reporting and top-k queries with sublogarithmic updates. In Proceedings of the 33rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), pages 23:1-23:14, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2016.23.
http://dx.doi.org/10.4230/LIPIcs.STACS.2016.23
Timothy M. Chan. Three problems about dynamic convex hulls. International Journal of Computational Geometry and Applications, 22(4):341-364, 2012. URL: http://dx.doi.org/10.1142/S0218195912600096.
http://dx.doi.org/10.1142/S0218195912600096
Timothy M. Chan, Kasper Green Larsen, and Mihai Pătraşcu. Orthogonal range searching on the RAM, revisited. In Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG), pages 1-10, 2011. URL: http://dx.doi.org/10.1145/1998196.1998198.
http://dx.doi.org/10.1145/1998196.1998198
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
Bernard Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal on Computing, 17(3):427-462, 1988. URL: http://dx.doi.org/10.1137/0217026.
http://dx.doi.org/10.1137/0217026
Paul F. Dietz. Maintaining order in a linked list. In Proceedings of the 14th Annual ACM Symposium on Theory of Computing (STOC), pages 122-127, 1982. URL: http://dx.doi.org/10.1145/800070.802184.
http://dx.doi.org/10.1145/800070.802184
Otfied Fries, Kurt Mehlhorn, Stefan Näher, and Athanasios K. Tsakalidis. A log log n data structure for three-sided range queries. Information Processing Letters, 25(4):269-273, 1987. URL: http://dx.doi.org/10.1016/0020-0190(87)90174-8.
http://dx.doi.org/10.1016/0020-0190(87)90174-8
Vijay Kumar and Eric J. Schwabe. Improved algorithms and data structures for solving graph problems in external memory. In Proceedings of the 8th Annual IEEE Symposium on Parallel and Distributed Processing, pages 169-176, 1996. URL: http://dx.doi.org/10.1109/SPDP.1996.570330.
http://dx.doi.org/10.1109/SPDP.1996.570330
George S. Lueker. A data structure for orthogonal range queries. In Proceedings of the 19th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 28-34, 1978. URL: http://dx.doi.org/10.1109/SFCS.1978.1.
http://dx.doi.org/10.1109/SFCS.1978.1
Edward M. McCreight. Priority search trees. SIAM Journal on Computing, 14(2):257-276, 1985. URL: http://dx.doi.org/10.1137/0214021.
http://dx.doi.org/10.1137/0214021
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. 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
Yakov Nekrich. Space efficient dynamic orthogonal range reporting. Algorithmica, 49(2):94-108, 2007. URL: http://dx.doi.org/10.1007/s00453-007-9030-9.
http://dx.doi.org/10.1007/s00453-007-9030-9
Yakov Nekrich. Orthogonal range searching in linear and almost-linear space. Computational Geometry, 42(4):342-351, 2009. URL: http://dx.doi.org/10.1016/j.comgeo.2008.09.001.
http://dx.doi.org/10.1016/j.comgeo.2008.09.001
Mark H. Overmars. Efficient data structures for range searching on a grid. Journal of Algorithms, 9(2):254-275, 1988. URL: http://dx.doi.org/10.1016/0196-6774(88)90041-7.
http://dx.doi.org/10.1016/0196-6774(88)90041-7
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
Bryan T. Wilkinson. Amortized bounds for dynamic orthogonal range reporting. In Proceedings of the 22th 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
Dan E. Willard. New data structures for orthogonal range queries. SIAM Journal on Computing, 14(1):232-253, 1985. URL: http://dx.doi.org/10.1137/0214019.
http://dx.doi.org/10.1137/0214019
Dan E. Willard. Examining computational geometry, Van Emde Boas trees, and hashing from the perspective of the fusion tree. SIAM Journal on Computing, 29(3):1030-1049, 2000. URL: http://dx.doi.org/10.1137/S0097539797322425.
http://dx.doi.org/10.1137/S0097539797322425
Dan E. Willard and George S. Lueker. Adding range restriction capability to dynamic data structures. Journal of the ACM, 32(3):597-617, 1985. URL: http://dx.doi.org/10.1145/3828.3839.
http://dx.doi.org/10.1145/3828.3839
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On Bend-Minimized Orthogonal Drawings of Planar 3-Graphs
An orthogonal drawing of a graph is a planar drawing where each edge is drawn as a sequence of horizontal and vertical line segments. Finding a bend-minimized orthogonal drawing of a planar graph of maximum degree 4 is NP-hard. The problem becomes tractable for planar graphs of maximum degree 3, and the fastest known algorithm takes O(n^5 log n) time. Whether a faster algorithm exists has been a long-standing open problem in graph drawing. In this paper we present an algorithm that takes only O~(n^{17/7}) time, which is a significant improvement over the previous state of the art.
Bend minimization
graph drawing
orthogonal drawing
planar graph
29:1-29:15
Regular Paper
Yi-Jun
Chang
Yi-Jun Chang
Hsu-Chun
Yen
Hsu-Chun Yen
10.4230/LIPIcs.SoCG.2017.29
Michael A. Bekos, Michael Kaufmann, Stephen G. Kobourov, and Antonios Symvonis. Smooth orthogonal layouts. JGAA, 17(5):575-595, 2013.
Michael A. Bekos, Michael Kaufmann, Robert Krug, Thorsten Ludwig, Stefan Näher, and Vincenzo Roselli. Slanted orthogonal drawings: Model, algorithms and evaluations. JGAA, 18(3):459-489, 2014.
Thomas Bläsius, Ignaz Rutter, and Dorothea Wagner. Optimal orthogonal graph drawing with convex bend costs. ACM Trans. Algorithms, 12(3):33:1-33:32, 2016.
Franz Brandenburg, David Eppstein, Michael T. Goodrich, Stephen Kobourov, Giuseppe Liotta, and Petra Mutzel. Selected open problems in graph drawing. In Proceedings of the 11th International Symposium on Graph Drawing (GD'03), pages 515-539. Springer Berlin Heidelberg, 2004.
Yi-Jun Chang and Hsu-Chun Yen. On orthogonally convex drawings of plane graphs. Computational Geometry, 62:34-51, 2017.
Michael B. Cohen, Aleksander Mądry, Piotr Sankowski, and Adrian Vladu. Negative-weight shortest paths and unit capacity minimum cost flow in Õ(m^10/7log W) time. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'17), pages 752-771. Society for Industrial and Applied Mathematics, 2017.
Sabine Cornelsen and Andreas Karrenbauer. Accelerated bend minimization. JGAA, 16(3):635-650, 2012.
Giuseppe Di Battista, Walter Didimo, Maurizio Patrignani, and Maurizio Pizzonia. Orthogonal and quasi-upward drawings with vertices of prescribed size. In Proceedings of the 7th International Symposium on Graph Drawing (GD'99), pages 297-310. Springer Berlin Heidelberg, 1999.
Giuseppe Di Battista, Giuseppe Liotta, and Francesco Vargiu. Spirality and optimal orthogonal drawings. SIAM Journal on Computing, 27(6):1764-1811, 1998.
Giuseppe Di Battista and Roberto Tamassia. On-line planarity testing. SIAM Journal on Computing, 25(5):956-997, 1996.
Walter Didimo, Giuseppe Liotta, and Maurizio Patrignani. On the complexity of hv-rectilinear planarity testing. In Proceedings of the 22nd International Symposium on Graph Drawing (GD'14), pages 343-354. Springer Berlin Heidelberg, 2014.
Christian A. Duncan and Michael T. Goodrich. Planar orthogonal and polyline drawing algorithms. In Roberto Tamassia, editor, Handbook of Graph Drawing and Visualization, chapter 8. CRC Press, 2013.
Stephane Durocher, Stefan Felsner, Saeed Mehrabi, and Debajyoti Mondal. Drawing hv-restricted planar graphs. In Proceedings of the 11th Latin American Theoretical Informatics Symposium (LATIN'14), pages 156-167. Springer Berlin Heidelberg, 2014.
Ashim Garg and Roberto Tamassia. A new minimum cost flow algorithm with applications to graph drawing. In Proceedings of the Symposium on Graph Drawing (GD'96), pages 201-216. Springer Berlin Heidelberg, 1997.
Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. SIAM Journal on Computing, 31(2):601-625, 2001.
Gunnar W. Klau and Petra Mutzel. Quasi-orthogonal drawing of planar graphs. Technical Report MPI-I-98-1-013, Max-Planck-Institut für Informatik, Saarbrücken, 1998.
Md. Saidur Rahman, Shin-ichi Nakano, and Takao Nishizeki. A linear algorithm for bend-optimal orthogonal drawings of triconnected cubic plane graphs. JGAA, 3(4):31-62, 1999.
Md. Saidur Rahman and Takao Nishizeki. Bend-minimum orthogonal drawings of plane 3-graphs. In Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science (WG'02), pages 367-378. Springer Berlin Heidelberg, 2002.
Md. Saidur Rahman, Takao Nishizeki, and Mahmuda Naznin. Orthogonal drawings of plane graphs without bends. JGAA, 7(4):335-362, 2003.
Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM Journal on Computing, 16(3):421-444, 1987.
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Adaptive Planar Point Location
We present a self-adjusting point location structure for convex subdivisions. Let n be the number of vertices in a convex subdivision S. Our structure for S uses O(n) space and processes any online query sequence sigma in O(n + OPT) time, where OPT is the minimum time required by any linear decision tree for answering point location queries in S to process sigma. The O(n + OPT) time bound includes the preprocessing time. Our result is a two-dimensional analog of the static optimality property of splay trees. For connected subdivisions, we achieve a processing time of O(|sigma| log log n + n + OPT).
point location
planar subdivision
static optimality
30:1-30:15
Regular Paper
Siu-Wing
Cheng
Siu-Wing Cheng
Man-Kit
Lau
Man-Kit Lau
10.4230/LIPIcs.SoCG.2017.30
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High Dimensional Consistent Digital Segments
We consider the problem of digitalizing Euclidean line segments from R^d to Z^d. Christ {et al.} (DCG, 2012) showed how to construct a set of {consistent digital segments} (CDS) for d=2: a collection of segments connecting any two points in Z^2 that satisfies the natural extension of the Euclidean axioms to Z^d. In this paper we study the construction of CDSs in higher dimensions.
We show that any total order can be used to create a set of {consistent digital rays} CDR in Z^d (a set of rays emanating from a fixed point p that satisfies the extension of the Euclidean axioms). We fully characterize for which total orders the construction holds and study their Hausdorff distance, which in particular positively answers the question posed by Christ {et al.}.
Consistent Digital Line Segments
Digital Geometry
Computer Vision
31:1-31:15
Regular Paper
Man-Kwun
Chiu
Man-Kwun Chiu
Matias
Korman
Matias Korman
10.4230/LIPIcs.SoCG.2017.31
Iffat Chowdhury and Matt Gibson. A characterization of consistent digital line segments in ℤ². In Proceedings of the 23rd Annual European Symposium on Algorithms, pages 337-348, 2015.
Iffat Chowdhury and Matt Gibson. Constructing consistent digital line segments. In Proceedings of the 12th Latin American Theoretical Informatics Symposium, pages 263-274, 2016.
Tobias Christ, Dömötör Pálvölgyi, and Miloš Stojaković. Consistent digital line segments. Discrete & Computational Geometry, 47(4):691-710, 2012.
Jinhee Chun, Matias Korman, Martin Nöllenburg, and Takeshi Tokuyama. Consistent digital rays. Discrete and Computational Geometry, 42(3):359-378, 2009.
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 13th Annual Symposium on Computational Geometry, pages 284-293, 1997.
Daniel H. Greene and F. Frances Yao. Finite-resolution computational geometry. In Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pages 143-152, 1986.
M. G. Luby. Grid geometries which preserve properties of Euclidean geometry: A study of graphics line drawing algorithms. In NATO Conference on Graphics/CAD, pages 397-432, 1987.
Kokichi Sugihara. Robust geometric computation based on topological consistency. In Proceedings of the 9th International Conference on Computational Science, pages 12-26, 2001.
Johannes van der Corput. Verteilungsfunktionen I &II (in german). Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 38:813-820, 1058-1066, 1935.
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TSP With Locational Uncertainty: The Adversarial Model
In this paper we study a natural special case of the Traveling Salesman Problem (TSP) with point-locational-uncertainty which we will call the adversarial TSP problem (ATSP). Given a metric space (X, d) and a set of subsets R = {R_1, R_2, ... , R_n} : R_i subseteq X, the goal is to devise an ordering of the regions, sigma_R, that the tour will visit such that when a single point is chosen from each region, the induced tour over those points in the ordering prescribed by sigma_R is as short as possible. Unlike the classical locational-uncertainty-TSP problem, which focuses on minimizing the expected length of such a tour when the point within each region is chosen according to some probability distribution, here, we focus on the adversarial model in which once the choice of sigma_R is announced, an adversary selects a point from each region in order to make the resulting tour as long as possible. In other words, we consider an offline problem in which the goal is to determine an ordering of the regions R that is optimal with respect to the ``worst'' point possible within each region being chosen by an adversary, who knows the chosen ordering. We give a 3-approximation when R is a set of arbitrary regions/sets of points in a metric space. We show how geometry leads to improved constant factor approximations when regions are parallel line segments of the same lengths, and a polynomial-time approximation scheme (PTAS) for the important special case in which R is a set of disjoint unit disks in the plane.
traveling salesperson problem
TSP with neighborhoods
approximation algorithms
uncertainty
32:1-32:16
Regular Paper
Gui
Citovsky
Gui Citovsky
Tyler
Mayer
Tyler Mayer
Joseph S. B.
Mitchell
Joseph S. B. Mitchell
10.4230/LIPIcs.SoCG.2017.32
Esther M. Arkin and Refael Hassin. Approximation algorithms for the geometric covering salesman problem. Discrete Applied Mathematics, 55(3):197-218, 1994.
Dimitris J. Bertsimas, Patrick Jaillet, and Amedeo R. Odoni. A priori optimization. Operations Research, 38(6):1019-1033, 1990.
Gui Citovsky, Tyler Mayer, and Joseph S. B. Mitchell. TSP With Locational Uncertainty: The Adversarial Model, March 2017. arXiv:1705.06180 [cs.CG]. URL: https://arxiv.org/abs/1705.06180.
https://arxiv.org/abs/1705.06180
Reza Dorrigiv, Robert Fraser, Meng He, Shahin Kamali, Akitoshi Kawamura, Alejandro López-Ortiz, and Diego Seco. On minimum-and maximum-weight minimum spanning trees with neighborhoods. Theory of Computing Systems, 56(1):220-250, 2015.
Adrian Dumitrescu and Joseph S. B. Mitchell. Approximation algorithms for TSP with neighborhoods in the plane. Journal of Algorithms, 48:135-159, 2003. Special issue devoted to 12th ACM-SIAM Symposium on Discrete Algorithms, Washington, DC, January, 2001.
Robert Fraser. Algorithms for geometric covering and piercing problems. PhD thesis, University of Waterloo, 2012.
Patrick Jaillet. A priori solution of a traveling salesman problem in which a random subset of the customers are visited. Operations Research, 36(6):929-936, 1988.
Lujun Jia, Guolong Lin, Guevara Noubir, Rajmohan Rajaraman, and Ravi Sundaram. Universal approximations for TSP, Steiner tree, and set cover. In Proc. 37th ACM Symposium on Theory of Computing, pages 386-395. ACM, 2005.
Pegah Kamousi and Subhash Suri. Euclidean traveling salesman tours through stochastic neighborhoods. In International Symposium on Algorithms and Computation, pages 644-654. Springer, 2013.
Chih-Hung Liu and Sandro Montanari. Minimizing the diameter of a spanning tree for imprecise points. In International Symposium on Algorithms and Computation, pages 381-392. Springer, 2015.
Maarten Löffler and Marc van Kreveld. Largest and smallest convex hulls for imprecise points. Algorithmica, 56(2):235-269, 2010.
Joseph S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing, 28(4):1298-1309, 1999.
Joseph S. B. Mitchell. Shortest paths and networks. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry (2nd Edition), chapter 27, pages 607-641. Chapman &Hall/CRC, Boca Raton, FL, 2004.
Joseph S. B. Mitchell. A PTAS for TSP with neighborhoods among fat regions in the plane. In Proc. 18th ACM-SIAM Symposium on Discrete algorithms, pages 11-18. Society for Industrial and Applied Mathematics, 2007. URL: http://www.ams.sunysb.edu/~jsbm/papers/tspn-soda07-rev.pdf.
http://www.ams.sunysb.edu/~jsbm/papers/tspn-soda07-rev.pdf
Sandro Montanari. Computing routes and trees under uncertainty. PhD thesis, Dissertation, ETH-Zürich, 2015, No. 23042, 2015.
Yang Yang, Mingen Lin, Jinhui Xu, and Yulai Xie. Minimum spanning tree with neighborhoods. In International Conference on Algorithmic Applications in Management, pages 306-316. Springer, 2007.
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On Planar Greedy Drawings of 3-Connected Planar Graphs
A graph drawing is greedy if, for every ordered pair of vertices (x,y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed.
In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra.
In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra's result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture.
Greedy drawings
3-connectivity
planar graphs
convex drawings
33:1-33:16
Regular Paper
Giordano
Da Lozzo
Giordano Da Lozzo
Anthony
D'Angelo
Anthony D'Angelo
Fabrizio
Frati
Fabrizio Frati
10.4230/LIPIcs.SoCG.2017.33
S. Alamdari, T. M. Chan, E. Grant, A. Lubiw, and V. Pathak. Self-approaching graphs. In Didimo and Patrignani, editors, GD, volume 7704 of LNCS, pages 260-271, 2012.
P. Angelini, G. Di Battista, and F. Frati. Succinct greedy drawings do not always exist. Networks, 59(3):267-274, 2012.
P. Angelini, E. Colasante, G. Di Battista, F. Frati, and M. Patrignani. Monotone drawings of graphs. J. Graph Algorithms Appl., 16(1):5-35, 2012.
P. Angelini, F. Frati, and L. Grilli. An algorithm to construct greedy drawings of triangulations. J. Graph Algorithms Appl., 14(1):19-51, 2010.
D. Barnette. Trees in polyhedral graphs. Canadian J. Math., 18:731-736, 1966.
P. Bose, P. Morin, I. Stojmenović, and J. Urrutia. Routing with guaranteed delivery in ad hoc wireless networks. Wireless Networks, 7(6):609-616, 2001.
G. Chen and X. Yu. Long cycles in 3-connected graphs. J. Comb. Theory, Ser. B, 86(1):80-99, 2002.
G. Da Lozzo, A. D'Angelo, and F. Frati. On planar greedy drawings of 3-connected planar graphs. CoRR, 2016. URL: http://arxiv.org/abs/1612.09277.
http://arxiv.org/abs/1612.09277
G. Da Lozzo, V. Dujmović, F. Frati, T. Mchedlidze, and V. Roselli. Drawing planar graphs with many collinear vertices. In Hu and Nöllenburg, editors, GD, volume 9801 of LNCS, pages 152-165, 2016.
H. R. Dehkordi, F. Frati, and J. Gudmundsson. Increasing-chord graphs on point sets. J. Graph Algorithms Appl., 19(2):761-778, 2015.
R. Dhandapani. Greedy drawings of triangulations. Discr. Comp. Geom., 43(2):375-392, 2010.
G. Di Battista and R. Tamassia. Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci., 61:175-198, 1988.
D. Eppstein and M. T. Goodrich. Succinct greedy geometric routing using hyperbolic geometry. IEEE Trans. Computers, 60(11):1571-1580, 2011.
S. Felsner, A. Igamberdiev, P. Kindermann, B. Klemz, T. Mchedlidze, and M. Scheucher. Strongly monotone drawings of planar graphs. In Fekete and Lubiw, editors, SoCG, volume 51 of LIPIcs, pages 37:1-37:15, 2016.
H. Frey, S. Rührup, and I. Stojmenović. Routing in wireless sensor networks. In Misra, Woungang, and Misra, editors, Guide to Wireless Sensor Networks, Computer Communications and Networks, chapter 4, pages 81-111. Springer, 2009.
M. T. Goodrich and D. Strash. Succinct greedy geometric routing in the Euclidean plane. In Dong, Du, and Ibarra, editors, ISAAC, volume 5878 of LNCS, pages 781-791, 2009.
X. He and H. Zhang. On succinct greedy drawings of plane triangulations and 3-connected plane graphs. Algorithmica, 68(2):531-544, 2014.
P. Kindermann, A. Schulz, J. Spoerhase, and A. Wolff. On monotone drawings of trees. In Duncan and Symvonis, editors, GD, volume 8871 of LNCS, pages 488-500, 2014.
E. Kranakis, H. Singh, and J. Urrutia. Compass routing on geometric networks. In CCCG, 1999. URL: http://www.cccg.ca/proceedings/1999/c46.pdf.
http://www.cccg.ca/proceedings/1999/c46.pdf
F. Kuhn, R. Wattenhofer, and A. Zollinger. An algorithmic approach to geographic routing in ad hoc and sensor networks. IEEE/ACM Trans. Netw., 16(1):51-62, 2008.
T. Leighton and A. Moitra. Some results on greedy embeddings in metric spaces. Discr. Comp. Geom., 44(3):686-705, 2010.
M. Nöllenburg and R. Prutkin. Euclidean greedy drawings of trees. In Bodlaender and Italiano, editors, ESA, volume 8125 of LNCS, pages 767-778, 2013.
M. Nöllenburg, R. Prutkin, and I. Rutter. On self-approaching and increasing-chord drawings of 3-connected planar graphs. J. Comp. Geom., 7(1):47-69, 2016.
C. H. Papadimitriou and D. Ratajczak. On a conjecture related to geometric routing. In Nikoletseas and Rolim, editors, ALGOSENSORS, volume 3121 of LNCS, pages 9-17, 2004.
C. H. Papadimitriou and D. Ratajczak. On a conjecture related to geometric routing. Theor. Comput. Sci., 344(1):3-14, 2005.
A. Rao, C. H. Papadimitriou, S. Shenker, and I. Stoica. Geographic routing without location information. In Johnson, Joseph, and Vaidya, editors, MOBICOM, pages 96-108, 2003.
C. Siva Ram Murthy and B. S. Manoj. Ad Hoc Wireless Networks: Architectures and Protocols. Prentice Hall, 2004.
C. K. Toh. Ad Hoc Mobile Wireless Networks: Protocols and Systems. Prentice Hall, 2002.
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Origamizer: A Practical Algorithm for Folding Any Polyhedron
It was established at SoCG'99 that every polyhedral complex can be folded from a sufficiently large square of paper, but the known algorithms are extremely impractical, wasting most of the material and making folds through many layers of paper. At a deeper level, these foldings get the topology wrong, introducing many gaps (boundaries) in the surface, which results in flimsy foldings in practice. We develop a new algorithm designed specifically for the practical folding of real paper into complicated polyhedral models. We prove that the algorithm correctly folds any oriented polyhedral manifold, plus an arbitrarily small amount of additional structure on one side of the surface (so for closed manifolds, inside the model). This algorithm is the first to attain the watertight property: for a specified cutting of the manifold into a topological disk with boundary, the folding maps the boundary of the paper to within epsilon of the specified boundary of the surface (in Fréchet distance). Our foldings also have the geometric feature that every convex face is folded seamlessly, i.e., as one unfolded convex polygon of the piece of paper. This work provides the theoretical underpinnings for Origamizer, freely available software written by the second author, which has enabled practical folding of many complex polyhedral models such as the Stanford bunny.
origami
folding
polyhedra
Voronoi diagram
computational geometry
34:1-34:16
Regular Paper
Erik D.
Demaine
Erik D. Demaine
Tomohiro
Tachi
Tomohiro Tachi
10.4230/LIPIcs.SoCG.2017.34
Erik D. Demaine, Martin L. Demaine, and Joseph S. B. Mitchell. Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami. Computational Geometry: Theory and Applications, 16(1):3-21, 2000.
Erik D. Demaine, David Eppstein, Jeff Erickson, George W. Hart, and Joseph O'Rourke. Vertex-unfolding of simplicial manifolds. In Discrete Geometry: In Honor of W. Kuperberg’s 60th Birthday, pages 215-228. Marcer Dekker Inc., 2003.
Erik D. Demaine, Sándor P. Fekete, and Robert J. Lang. Circle packing for origami design is hard. In Origami⁵: Proceedings of the 5th International Conference on Origami in Science, Mathematics and Education, pages 609-626. A K Peters, Singapore, July 2010.
Erik D. Demaine and Joseph O'Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, July 2007.
Erik D. Demaine and Tomohiro Tachi. Origamizer: A practical algorithm for folding any polyhedron. Manuscript, 2017. URL: http://erikdemaine.org/papers/Origamizer/.
http://erikdemaine.org/papers/Origamizer/
Robert J. Lang. A computational algorithm for origami design. In Proceedings of the 12th Annual ACM Symposium on Computational Geometry, pages 98-105, Philadelphia, PA, May 1996.
Robert J. Lang and Erik D. Demaine. Facet ordering and crease assignment in uniaxial bases. In Origami⁴: Proceedings of the 4th International Conference on Origami in Science, Mathematics, and Education, Pasadena, California, September 2006.
Tomohiro Tachi. Software: Origamizer, 2008. URL: http://www.tsg.ne.jp/TT/software/.
http://www.tsg.ne.jp/TT/software/
Tomohiro Tachi. Origamizing polyhedral surfaces. IEEE Transactions on Visualization and Computer Graphics, 16(2):298-311, 2010. URL: http://dx.doi.org/10.1109/TVCG.2009.67.
http://dx.doi.org/10.1109/TVCG.2009.67
W. T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 13:743-767, 1963. URL: http://plms.oxfordjournals.org/cgi/pdf_extract/s3-13/1/743, URL: http://dx.doi.org/doi:10.1112/plms/s3-13.1.743.
http://dx.doi.org/doi:10.1112/plms/s3-13.1.743
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Computing the Geometric Intersection Number of Curves
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most l on a combinatorial surface of complexity n we describe simple algorithms to (1) compute the geometric intersection number of c in O(n+ l^2) time, (2) construct a curve homotopic to c that realizes this geometric intersection number in O(n+l^4) time, (3) decide if the geometric intersection number of c is zero, i.e. if c is homotopic to a simple curve, in O(n+l log^2 l) time.
To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a O(n+g^2l^2) time complexity on a genus g surface without boundary. No polynomial time algorithm was known for problem (2). Interestingly, our solution to problem (3) is the first quasi-linear algorithm since the problem was raised by Poincare more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most l in O(n+ l^2) time.
computational topology
curves on surfaces
combinatorial geodesic
35:1-35:15
Regular Paper
Vincent
Despré
Vincent Despré
Francis
Lazarus
Francis Lazarus
10.4230/LIPIcs.SoCG.2017.35
Chris Arettines. A combinatorial algorithm for visualizing representatives with minimal self-intersection. J. Knot Theor. Ramif., 24(11):1-17, 2015.
Joan S. Birman and Caroline Series. An algorithm for simple curves on surfaces. J. London Math. Soc., 29(2):331-342, 1984.
Hsien-Chih Chang and Jeff Erickson. Untangling planar curves. In Proc. 32nd Int'l Symp. Comput. Geom. (SoCG), volume 51, pages 29:1-15, 2016.
Hsien-Chih Chang, Jeff Erickson, and Chao Xu. Detecting weakly simple polygons. In Proc. 26th ACM-SIAM Symp. Discrete Alg. (SODA), pages 1655-1670, 2015.
Moira Chas. Self-intersection numbers of length-equivalent curves on surfaces. Exp. Math., 23(3):271-276, 2014.
Moira Chas and Steven P. Lalley. Self-intersections in combinatorial topology: statistical structure. Invent. Math., 188(2):429-463, 2012.
David R. J. Chillingworth. Simple closed curves on surfaces. Bull. London Math. Soc., 1(3):310-314, 1969.
David R. J. Chillingworth. Winding numbers on surfaces. II. Math. Ann., 199(3):131-153, 1972.
Marshall Cohen and Martin Lustig. Paths of geodesics and geometric intersection numbers: I. In Combinatorial group theory and topology, volume 111 of Ann. Math. Stud., pages 479-500. Princeton Univ. Press, 1987.
Éric Colin de Verdière and Francis Lazarus. Optimal System of Loops on an Orientable Surface. Discrete Comput. Geom., 33(3):507-534, 2005.
Maurits de Graaf and Alexander Schrijver. Making curves minimally crossing by Reidemeister moves. J. Com. Theory B, 70(1):134-156, 1997.
Pierre De La Harpe. Topologie, théorie des groupes et problèmes de décision: célébration d'un article de max dehn de 1910. Gazette des mathématiciens, 125:41-75, 2010.
Tamal K. Dey and Sumanta Guha. Transforming Curves on Surfaces. J. Comput. and Syst. Sci., 58(2):297-325, 1999.
David Epstein and Derek Holt. The linearity of the conjugacy problem in word-hyperbolic groups. Int'l J. Algebr. Comput., 16(02):287-305, 2006.
Jeff Erickson and Kim Whittelsey. Transforming curves on surfaces redux. In Proc. 24th ACM-SIAM Symp. Discrete Alg. (SODA), 2013.
Benson Farb and Dan Margalit. A primer on mapping class groups. Princeton Univ. Press, 2012.
Steve M. Gersten and Hamish B. Short. Small cancellation theory and automatic groups. Invent. Math., 102:305-334, 1990.
Daciberg L. Gonçalves, Elena Kudryavtseva, and Heiner Zieschang. An algorithm for minimal number of (self-)intersection points of curves on surfaces. In Proc. Seminar on Vector and Tensor Analysis, volume 26, pages 139-167, 2005.
Joel Hass and Peter Scott. Intersections of curves on surfaces. Isr. J. Math., 51(1-2):90-120, 1985.
Joel Hass and Peter Scott. Configurations of curves and geodesics on surfaces. Geometry and Topology Monographs, 2:201-213, 1999.
Donald E. Knuth, James H. Morris, Jr, and Vaughan R. Pratt. Fast pattern matching in strings. SIAM J. Comput., 6(2):323-350, 1977.
Francis Lazarus and Julien Rivaud. On the homotopy test on surfaces. In Proc. 53rd IEEE Symp. Found. Comput. Sci. (FOCS), pages 440-449, 2012.
Martin Lustig. Paths of geodesics and geometric intersection numbers: II. In Combinatorial group theory and topology, volume 111 of Ann. of Math. Stud., pages 501-543. Princeton Univ. Press, 1987.
Maryam Mirzakhani. Growth of the number of simple closed geodesies on hyperbolic surfaces. Ann. Math., pages 97-125, 2008.
Maryam Mirzakhani. Counting mapping class group orbits on hyperbolic surfaces. Preprint http://arxiv.org/pdf/1601.03342, January 2016.
http://arxiv.org/pdf/1601.03342
Bojan Mohar and Carsten Thomassen. Graphs on Surfaces. Studies in the Mathematical Sciences. Johns Hopkins Univ. Press, 2001.
Max Neumann-Coto. A characterization of shortest geodesics on surfaces. Algebr. Geom. Topol., 1:349-368, 2001.
J. M. Paterson. A combinatorial algorithm for immersed loops in surfaces. Topol. Appl., 123(2):205-234, 2002.
Henri Poincaré. Cinquième complément à l'analysis situs. Rendiconti del Circolo Matematico di Palermo, 18(1):45-110, 1904.
Bruce L. Reinhart. Algorithms for Jordan curves on compact surfaces. Ann. Math., pages 209-222, 1962.
Jenya Sapir. Bounds on the number of non-simple closed geodesics on a surface. Preprint http://arxiv.org/pdf/1505.07171, May 2015. URL: http://arxiv.org/abs/1505.07171.
http://arxiv.org/pdf/1505.07171
Marcus Schaefer, Eric Sedgwick, and Daniel Stefankovic. Computing Dehn twists and geometric intersection numbers in polynomial time. In CCCG, pages 111-114, 2008.
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Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers
Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called nerve of a cover of the domain.
In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage.
More specifically, we show that the one-dimensional homology of the nerve complex N(U) of a path-connected cover U of a domain X cannot be richer than that of the domain X itself. Intuitively, this result means that no new H_1-homology class can be "created" under a natural map from X to the nerve complex N(U). Equipping X with a pseudometric d, we further refine this result and characterize the classes of H_1(X) that may survive in the nerve complex using the notion of size of the covering elements in U. These fundamental results about nerve complexes then lead to an analysis of the H_1-homology of Reeb spaces, mappers and multiscale mappers.
The analysis of H_1-homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces.
Topology
Nerves
Mapper
Multiscale Mapper
Reeb Spaces
36:1-36:16
Regular Paper
Tamal K.
Dey
Tamal K. Dey
Facundo
Mémoli
Facundo Mémoli
Yusu
Wang
Yusu Wang
10.4230/LIPIcs.SoCG.2017.36
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Locality-Sensitive Hashing of Curves
We study data structures for storing a set of polygonal curves in R^d such that, given a query curve, we can efficiently retrieve similar curves from the set, where similarity is measured using the discrete Fréchet distance or the dynamic time warping distance. To this end we devise the first locality-sensitive hashing schemes for these distance measures. A major challenge is posed by the fact that these distance measures internally optimize the alignment between the curves. We give solutions for different types of alignments including constrained and unconstrained versions. For unconstrained alignments, we improve over a result by Indyk [SoCG 2002] for short curves. Let n be the number of input curves and let m be the maximum complexity of a curve in the input. In the particular case where m <= (a/(4d)) log n, for some fixed a>0, our solutions imply an approximate near-neighbor data structure for the discrete Fréchet distance that uses space in O(n^(1+a) log n) and achieves query time in O(n^a log^2 n) and constant approximation factor. Furthermore, our solutions provide a trade-off between approximation quality and computational performance: for any parameter k in [m], we can give a data structure that uses space in O(2^(2k) m^(k-1) n log n + nm), answers queries in O( 2^(2k) m^(k) log n) time and achieves approximation factor in O(m/k).
Locality-Sensitive Hashing
Frechet distance
Dynamic Time Warping
37:1-37:16
Regular Paper
Anne
Driemel
Anne Driemel
Francesco
Silvestri
Francesco Silvestri
10.4230/LIPIcs.SoCG.2017.37
S. Arya, D. Mount, A. Vigneron, and J. Xia. Space-time tradeoffs for proximity searching in doubling spaces. In Proc. 16th European Symp. Algorithms (ESA), pages 112-123, 2008.
A. Backurs and A. Sidiropoulos. Constant-distortion embeddings of Hausdorff metrics into constant-dimensional l_p spaces. In Proc. 19th Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), volume 60, pages 1:1-1:15, 2016.
Y. Bartal, L. A. Gottlieb, and O. Neiman. On the impossibility of dimension reduction for doubling subsets of 𝓁_p. In Proc. 13th Symp. on Computational Geometry (SOCG), pages 60:60-60:66, 2014.
K. Bringmann. Why Walking the Dog Takes Time: Fréchet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails. In Proc. 55th Symp. on Foundations of Computer Science (FOCS), pages 661-670, 2014.
J. C. Brown and P. J. O. Miller. Automatic classification of killer whale vocalizations using dynamic time warping. J. of the Acoustical Society of America, 122(2):1201-1207, 2007.
J. Campbell, J. Tremblay, and C. Verbrugge. Clustering player paths. In Proc. 10th Int'l Conf. on the Foundations of Digital Games (FDG), 2015.
M. de Berg, A. F. Cook, and J. Gudmundsson. Fast Fréchet queries. Comput. Geom., 46(6):747-755, 2013.
A. Driemel and S. Har-Peled. Jaywalking your dog: Computing the Fréchet distance with shortcuts. SIAM J. Computing, 42(5):1830-1866, 2013.
A. Driemel, A. Krivošija, and C. Sohler. Clustering time series under the Fréchet distance. In Proc. 27th Symp. on Discrete Algorithms (SODA), pages 766-785, 2016.
A. Driemel and F. Silvesstri. Locality-sensitive hashing of curves. Arxiv:1703.04040, 2017.
G. Forestier, F. Lalys, L. Riffaud, B. Trelhu, and P. Jannin. Classification of surgical processes using dynamic time warping. J. Biomedical Informatics, 45(2):255-264, 2012.
J. Gudmundsson and N. Valladares. A GPU approach to subtrajectory clustering using the Fréchet distance. IEEE Trans. on Parallel and Distributed Systems, 26(4):924-937, 2015.
Anupam Gupta, Robert Krauthgamer, and James R Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proc. 44th Symp. Found. Comp. Science (FOCS), pages 534-543, 2003.
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.
B. Huang and W. Kinsner. ECG frame classification using dynamic time warping. In Proc. Canadian Conf. on Electrical and Computer Engineering, volume 2, pages 1105-1110, 2002.
P. Indyk. On approximate nearest neighbors in non-euclidean spaces. In Proc. 39th Symp. on Foundations of Computer Science, pages 148-155, 1998.
P. Indyk. Approximate nearest neighbor algorithms for Fréchet distance via product metrics. In Proc. 18th Symp. on Computational Geometry (SOCG), pages 102-106, 2002.
P. Indyk and J. Matoušek. Low-distortion embeddings of finite metric spaces. In Handbook of Discrete and Computational Geometry, pages 177-196. CRC Press, 2004.
P. Indyk and R. Motwani. Approximate nearest neighbors: Towards removing the curse of dimensionality. In Proc. 30th Symp. Theory of Computing (STOC), pages 604-613, 1998.
R. J. Kenefic. Track clustering using Fréchet distance and minimum description length. J. of Aerospace Information Systems, 11(8):512-524, 2014.
E. Keogh and C. A. Ratanamahatana. Exact indexing of dynamic time warping. Knowledge and Information Systems, 7(3):358-386, 2005.
Z. M. Kovacs-Vajna. A fingerprint verification system based on triangular matching and dynamic time warping. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11):1266-1276, 2000.
B. Legrand, C. S. Chang, S. H. Ong, S. Y. Neo, and N. Palanisamy. Chromosome classification using dynamic time warping. Pattern Recognition Letters, 29(3):215-222, 2008.
Q. Lv, W. Josephson, Z. Wang, M. Charikar, and K. Li. Multi-probe lsh: Efficient indexing for high-dimensional similarity search. In Proc. 33rd Int'l Conf. on Very Large Data Bases, VLDB'07, pages 950-961. VLDB Endowment, 2007.
J. Matoušek. Embedding finite metric spaces into euclidean spaces. In Lectures on Discrete Geometry, chapter 15. Springer, 2002.
T. Rakthanmanon, B. Campana, A. Mueen, G. Batista, B. Westover, Q. Zhu, J. Zakaria, and E. Keogh. Searching and mining trillions of time series subsequences under dynamic time warping. In Proc. 18th Conf. Knowl. Disc. and Data Mining, pages 262-270, 2012.
C. A. Ratanamahatana and E. Keogh. Three myths about dynamic time warping data mining. In Proc. SIAM Conf. on Data Mining (SDM), pages 506-510, 2005.
G. Shakhnarovich, T. Darrell, and P. Indyk, editors. Nearest-Neighbor Methods in Learning and Vision: Theory and Practice. MIT Press, 2006.
A. Shrivastava and P. Li. Asymmetric LSH (ALSH) for sublinear time maximum inner product search (MIPS). In Proc. 27th Conf. on Neural Information Processing Systems (NIPS), pages 2321-2329, 2014.
G. K. D. Vries. Kernel methods for vessel trajectories. PhD thesis, Univ. Amsterdam, 2012.
H. Zhu, J. Luo, H. Yin, X. Zhou, J. Z. Huang, and F. B. Zhan. Mining trajectory corridors using Fréchet distance and meshing grids. In Proc. 14th Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD), pages 228-237, 2010.
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Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning
The packing lemma of Haussler states that given a set system (X,R) with bounded VC dimension, if every pair of sets in R have large symmetric difference, then R cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity.
In this paper we present several new results and applications related to packings:
* an optimal lower bound for shallow packings,
* improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry,
* we observe that Mnets provide a general, more powerful framework from which the state-of-the-art unweighted epsilon-net results follow immediately, and
* simplifying and generalizing one of the main technical tools in [Fox et al. , J. of the EMS, to appear].
Epsilon-nets
Haussler's packing lemma
Mnets
shallow-cell complexity
shallow packing lemma
38:1-38:15
Regular Paper
Kunal
Dutta
Kunal Dutta
Arijit
Ghosh
Arijit Ghosh
Bruno
Jartoux
Bruno Jartoux
Nabil H.
Mustafa
Nabil H. Mustafa
10.4230/LIPIcs.SoCG.2017.38
P. K. Agarwal, J. Pach, and M. Sharir. State of the Union (of Geometric Objects): A Review. In J. Goodman, J. Pach, and R. Pollack, editors, Computational Geometry: Twenty Years Later, pages 9-48. American Mathematical Society, 2008.
B. Aronov, M. de Berg, E. Ezra, and M. Sharir. Improved Bounds for the Union of Locally Fat Objects in the Plane. SIAM J. Comput., 43(2):543-572, 2014.
S. Arya, G. D. da Fonseca, and D. M. Mount. Optimal Area-Sensitive Bounds for Polytope Approximation. In Proc. 28th Annual Symposium on Computational Geometry (SoCG), pages 363-372, 2012.
S. Arya, G. D. da Fonseca, and D. M. Mount. On the Combinatorial Complexity of Approximating Polytopes. In Proc. 32nd International Symposium on Computational Geometry (SoCG), volume 51, pages 11:1-11:15, 2016.
S. Basu, R. Pollack, and M. F. Roy. Algorithms in Real Algebraic Geometry. Springer-Verlag, 2003.
T. M. Chan, E. Grant, J. Könemann, and M. Sharpe. Weighted Capacitated, Priority, and Geometric Set Cover via Improved Quasi-Uniform Sampling. In Proc. 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1576-1585, 2012.
B. Chazelle. A note on Haussler’s packing lemma. See Section 5.3 from Geometric Discrepancy: An Illustrated Guide by J. Matoušek, 1992.
B. Chazelle. The Discrepancy Method: Randomness and Complexity. Cambridge University Press, Cambridge, New York, 2000.
K. Dutta, E. Ezra, and A. Ghosh. Two Proofs for Shallow Packings. Discrete &Computational Geometry, 56(4):910-939, 2016. Extended abstract appeared in Proc. 31st International Symposium on Computational Geometry (SoCG), pages 96-110, 2015.
A. Ene, S. Har-Peled, and B. Raichel. Geometric Packing under Non-uniform Constraints. In Proc. 28th Annual Symposium on Computational Geometry (SoCG), pages 11-20, 2012.
E. Ezra. A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension. SIAM J. Comput., 45(1):84-101, 2016. Extended abstract appeared in Proc. 25th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1378-1388, 2014.
E. Ezra, B. Aronov, and S. Sharir. Improved Bound for the Union of Fat Triangles. In Proc. 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1778-1785, 2011.
J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl. A Semi-Algebraic Version of Zarankiewicz’s Problem. J. of the European Mathematical Society, to appear.
L. Guth and N. H. Katz. On the Erdös distinct distances problem in the plane. Annals of Math., 181(1):155-190, 2015.
D. Haussler. Sphere Packing Numbers for Subsets of the Boolean n-Cube with Bounded Vapnik-Chervonenkis Dimension. J. Comb. Theory, Ser. A, 69(2):217-232, 1995.
D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete &Computational Geometry, 2:127-151, 1987.
A. Kupavskii, N. H. Mustafa, and J. Pach. Near-Optimal Lower Bounds for ε-nets for Half-spaces and Low Complexity Set Systems. In M. Loebl, J. Nešetřil, and R. Thomas, editors, A Journey Through Discrete Mathematics: A Tribute to Jiří Matoušek. Springer, 2017. Extended abstract with the title "New Lower Bounds for epsilon-Nets" appeared in Proc. 32nd International Symposium on Computational Geometry (SoCG), 54:1-54:16, 2016.
Yi Li, Philip M. Long, and Aravind Srinivasan. Improved bounds on the sample complexity of learning. J. of Computer and System Sciences, 62(3):516-527, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1741.
http://dx.doi.org/10.1006/jcss.2000.1741
A. M. Macbeath. A theorem on non-homogeneous lattices. Annals of Math., 56:269-293, 1952.
J. Matoušek. Geometric Discrepancy: An Illustrated Guide. Algorithms and Combinatorics. Springer, Berlin, New York, 1999.
J. Matoušek. Lectures in Discrete Geometry. Springer-Verlag, New York, NY, 2002.
J. Matoušek, J. Pach, M. Sharir, S. Sifrony, and E. Welzl. Fat Triangles Determine Linearly Many Holes. SIAM J. Comput., 23(1):154-169, 1994.
J. Matoušek and Z. Patáková. Multilevel Polynomial Partitions and Simplified Range Searching. Discrete &Computational Geometry, 54(1):22-41, 2015.
N. H. Mustafa. A Simple Proof of the Shallow Packing Lemma. Discrete &Computational Geometry, 55(3):739-743, 2016.
N. H. Mustafa, K. Dutta, and A. Ghosh. A Simple Proof of Optimal Epsilon-nets. Combinatorica, to appear.
N. H. Mustafa and S. Ray. ε -Mnets: Hitting Geometric Set Systems with Subsets. Discrete &Computational Geometry, 57(3):625-640, 2017. Extended abstract with the title "Near-Optimal Generalisations of a Theorem of Macbeath" appeared in Proc. 31st Symposium on Theoretical Aspects of Computer Science (STACS), pages 578-589, 2014.
N. H. Mustafa and K. Varadarajan. Epsilon-approximations and Epsilon-nets. In J. E. Goodman, J. O'Rourke, and C. D. Tóth, editors, Handbook of Discrete and Computational Geometry. CRC Press LLC, 2017.
J. Pach and P. K. Agarwal. Combinatorial Geometry. John Wiley &Sons, New York, NY, 1995.
N. Sauer. On the Density of Families of Sets. J. Comb. Theory, Ser. A, 13(1):145-147, 1972.
S. Shelah. A Combinatorial Problem, Stability and Order for Models and Theories in Infinitary Languages. Pacific J. of Mathematics, 41:247-261, 1972.
K. R. Varadarajan. Weighted Geometric Set Cover via Quasi-Uniform Sampling. In Proc. 42nd Symposium on Theory of Computing (STOC), pages 641-648, 2010.
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Topological Data Analysis with Bregman Divergences
We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.
Topological data analysis
Bregman divergences
persistent homology
proximity complexes
algorithms
39:1-39:16
Regular Paper
Herbert
Edelsbrunner
Herbert Edelsbrunner
Hubert
Wagner
Hubert Wagner
10.4230/LIPIcs.SoCG.2017.39
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Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension
We consider the problem of finding a small hitting set in an infinite range space F=(Q,R) of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta>0, a set of size O(s_F(z^*_F)) that hits (1-delta)-fraction of R (with respect to a given measure) in time proportional to log(1/delta), where s_F(1/epsilon) is the size of the smallest epsilon-net the range space admits, and z^*_F is the value of the fractional optimal solution. This exponentially improves upon previous results which achieve the same approximation guarantees with running time proportional to poly(1/delta). Our assumptions hold, for instance, in the case when the range space represents the visibility regions of a polygon in the plane, giving thus a deterministic polynomial-time O(log z^*_F)-approximation algorithm for guarding (1-delta)-fraction of the area of any given simple polygon, with running time proportional to polylog(1/delta).
VC-dimension
approximation algorithms
fractional covering
multiplicative weights update
art gallery problem
polyhedral separators
geometric cove
40:1-40:15
Regular Paper
Khaled
Elbassioni
Khaled Elbassioni
10.4230/LIPIcs.SoCG.2017.40
P. K. Agarwal and J. Pan. Near-linear algorithms for geometric hitting sets and set covers. In SoCG'14, pages 271-279, 2014.
N. Alon and J. H. Spencer. The Probabilistic Method. Wiley Series in Discrete Mathematics and Optimization. Wiley, 2008.
B. Aronov, E. Ezra, and M. Sharir. Small-size ε-nets for axis-parallel rectangles and boxes. SIAM Journal on Computing, 39(7):3248-3282, 2010.
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É. Bonnet and T. Miltzow. An approximation algorithm for the art gallery problem. In EuroCG'16, also available online as: https://arxiv.org/abs/1607.05527, 2016.
H. Brönnimann, B. Chazelle, and J. Matoušek. Product range spaces, sensitive sampling, and derandomization. SIAM Journal on Computing, 28(5):1552-1575, 1999.
H. Brönnimann and M. T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete &Computational Geometry, 14(4):463-479, 1995.
T. M. Chan, E. Grant, J. Könemann, and M. Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In SODA'12, pages 1576-1585, 2012.
B. Chazelle. The Discrepancy Method: Randomness and Complexity. Cambridge University Press, New York, NY, USA, 2000.
B. Chazelle and J.Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension. Journal of Algorithms, 21(3):579-597, 1996.
O. Cheong, A. Efrat, and S. Har-Peled. Finding a guard that sees most and a shop that sells most. Discrete & Computational Geometry, 37(4):545-563, 2007.
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K. L. Clarkson. Algorithms for polytope covering and approximation. In WADS'93, pages 246-252, 1993.
K. L. Clarkson and K. Varadarajan. Improved approximation algorithms for geometric set cover. Discrete &Computational Geometry, 37(1):43-58, 2006.
A. Deshpande, T. Kim, E. D. Demaine, and S. E. Sarma. A pseudopolynomial time O(log n)-approximation algorithm for art gallery problems. In WADS'07, pages 163-174, 2007.
I. Dinur and D. Steurer. Analytical approach to parallel repetition. In STOC'14, pages 624-633, 2014.
A. Efrat and S. Har-Peled. Guarding galleries and terrains. Inf. Process. Lett., 100(6):238-245, 2006.
G. Even, D. Rawitz, and S. (M.) Shahar. Hitting sets when the VC-dimension is small. Inf. Process. Lett., 95(2):358-362, 2005.
S. K. Ganjugunte. Geometric Hitting Sets and Their Variants. PhD thesis, Duke University, USA, 2011.
N. Garg and J. Könemann. Faster and simpler algorithms for multicommodity flow and other fractional packing problems. SIAM J. Comput., 37(2):630-652, 2007.
S. K. Ghosh. Approximation algorithms for art gallery problems in polygons. Discrete Applied Mathematics, 158(6):718-722, 2010.
A. Gilbers and R. Klein. A new upper bound for the VC-dimension of visibility regions. Computational Geometry, 47(1):61-74, 2014.
R. Glück. Covering polygons with rectangles. In EuroCG'16, 2016.
D. Grigoriev and N. Vorobjov. Solving systems of polynomial inequalities in subexponential time. J. Symb. Comput., 5(1/2):37-64, 1988.
S. Har-Peled and M. Sharir. Relative (p, ε)-approximations in geometry. Discrete & Computational Geometry, 45(3):462-496, 2011.
D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete & Computational Geometry, 2:127-151, 1987.
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J. King and D. G. Kirkpatrick. Improved approximation for guarding simple galleries from the perimeter. Discrete & Computational Geometry, 46(2):252-269, 2011.
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S. Laue. Geometric set cover and hitting sets for polytopes in ℝ³. In STACS'08, pages 479-490, 2008.
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J. Matoušek, R. Seidel, and E. Welzl. How to net a lot with little: Small ε-nets for disks and halfspaces. In SoCG'90, pages 16-22, 1990.
J. S. B. Mitchell and S. Suri. Separation and approximation of polyhedral objects. Computational Geometry, 5(2):95-114, 1995.
S. Ntafos and M. Tsoukalas. Optimum placement of guards. Information Sciences, 76(1-2):141-150, 1994.
J. Pach and G. Woeginger. Some new bounds for Epsilon-nets. In SoCG'90, pages 10-15, 1990.
E. Pyrga and S. Ray. New existence proofs ε-nets. In SoCG'08, pages 199-207, 2008.
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P. Valtr. Guarding galleries where no point sees a small area. Israel Journal of Mathematics, 104(1):1-16, 1998.
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K. Varadarajan. Epsilon nets and union complexity. In SoCG'09, pages 11-16, 2009.
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A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM
We show that the k-SUM problem can be solved by a linear decision tree of depth O(n^2 log^2 n),improving the recent bound O(n^3 log^3 n) of Cardinal et al. Our bound depends linearly on k, and allows us to conclude that the number of linear queries required to decide the n-dimensional Knapsack or SubsetSum problems is only O(n^3 log n), improving the currently best known bounds by a factor of n. Our algorithm extends to the RAM model, showing that the k-SUM problem can be solved in expected polynomial time, for any fixed k, with the above bound on the number of linear queries. Our approach relies on a new point-location mechanism, exploiting "Epsilon-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions.
A major side result of the analysis in this paper is a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension). We hope that this study will reveal further structural properties of vertical decompositions in hyperplane arrangements.
k-SUM and k-LDT
linear decision tree
hyperplane arrangements
point-location
vertical decompositions
Epsilon-cuttings
41:1-41:15
Regular Paper
Esther
Ezra
Esther Ezra
Micha
Sharir
Micha Sharir
10.4230/LIPIcs.SoCG.2017.41
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Computing the Fréchet Gap Distance
Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Frechet distance is one of the most well studied similarity measures. Informally, the Fréchet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study a variant called the Fréchet gap distance. In the man and dog analogy, the Fréchet gap distance minimizes the difference of the longest and smallest leash lengths used over the entire walk. This measure in some ways better captures our intuitive notions of curve similarity, for example giving distance zero to translated copies of the same curve.
The Fréchet gap distance was originally introduced by Filtser and Katz (2015) in the context of the discrete Fréchet distance. Here we study the continuous version, which presents a number of additional challenges not present in discrete case. In particular, the continuous nature makes bounding and searching over the critical events a rather difficult task.
For this problem we give an O(n^5 log(n)) time exact algorithm and a more efficient O(n^2 log(n) + (n^2/epsilon) log(1/epsilon)) time (1+epsilon)-approximation algorithm, where n is the total number of vertices of the input curves. Note that for (small enough) constant epsilon and ignoring logarithmic factors, our approximation has quadratic running time, matching the lower bound, assuming SETH (Bringmann 2014), for approximating the standard Fréchet distance for general curves.
Frechet Distance
Approximation
Polygonal Curves
42:1-42:16
Regular Paper
Chenglin
Fan
Chenglin Fan
Benjamin
Raichel
Benjamin Raichel
10.4230/LIPIcs.SoCG.2017.42
P. Agarwal, R. Avraham, H. Kaplan, and M. Sharir. Computing the discrete Fréchet distance in subquadratic time. SIAM Journal on Computing, 43(2):429-449, 2014.
H. Alt and M. Buchin. Can we compute the similarity between surfaces? Discrete &Computational Geometry, 43(1):78-99, 2010.
H. Alt, A. Efrat, G. Rote, and C. Wenk. Matching planar maps. In Proc. of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 589-598, 2003.
H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geometry Appl., 5:75-91, 1995.
H. Alt, C. Knauer, and C. Wenk. Matching polygonal curves with respect to the Fréchet distance. In Annual Symp. on Theo. Aspects of Comp. Sci. (STACS), pages 63-74, 2001.
R. Avraham, O. Filtser, H. Kaplan, M. Katz, and M. Sharir. The discrete and semicontinuous Fréchet distance with shortcuts via approximate distance counting and selection. ACM Trans. Algorithms, 11(4):29, 2015.
R. Avraham, H. Kaplan, and M. Sharir. A faster algorithm for the discrete Fréchet distance under translation. CoRR, abs/1501.03724, 2015.
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K. Bringmann. Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless seth fails. In Symp. on Found. of Comp. Sci. (FOCS), pages 661-670. IEEE, 2014.
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K. Buchin, M. Buchin, W. Meulemans, and W. Mulzer. Four soviets walk the dog - with an application to alt’s conjecture. In Proc. of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1399-1413, 2014.
K. Buchin, M. Buchin, and C. Wenk. Computing the Fréchet distance between simple polygons in polynomial time. In 22nd Annual Symp. Comput. Geom., pages 80-87, 2006.
M. Buchin, A. Driemel, and B. Speckmann. Computing the Fréchet distance with shortcuts is np-hard. In 30th Annual Symp. Comput. Geom. (SoCG), page 367, 2014.
A. Driemel and S. Har-Peled. Jaywalking your dog: computing the Fréchet distance with shortcuts. SIAM Journal on Computing, 42(5):1830-1866, 2013.
A. Driemel, S. Har-Peled, and C. Wenk. Approximating the Fréchet distance for realistic curves in near linear time. Discrete &Computational Geometry, 48(1):94-127, 2012.
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C. Fan and B. Raichel. Computing the Fréchet gap distance. URL: http://www.utdallas.edu/~bar150630/gap.pdf.
http://www.utdallas.edu/~bar150630/gap.pdf
O. Filtser and M. Katz. The discrete Fréchet distance gap. arXiv:1506.04861, 2015.
S. Har-Peled and B. Raichel. The Fréchet distance revisited and extended. ACM Transactions on Algorithms (TALG), 10(1):3, 2014.
M. Kim, S. Kim, and M. Shin. Optimization of subsequence matching under time warping in time-series databases. In Proc. ACM Symp. on Applied Computing, pages 581-586, 2005.
G. Rote. Computing the Fréchet distance between piecewise smooth curves. Computational Geometry, 37(3):162-174, 2007.
J. Serrà, E. Gómez, P. Herrera, and X. Serra. Chroma binary similarity and local alignment applied to cover song identifica. Audio, Speech & Lang. Proc., 16(6):1138-1151, 2008.
C. Wenk, R. Salas, and D. Pfoser. Addressing the need for map-matching speed: Localizing global curve-matching algorithms. In Sci. Statis. Database Manag., pages 879-888, 2006.
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Erdös-Hajnal Conjecture for Graphs with Bounded VC-Dimension
The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e^{(log n)^{1 - o(1)}}. The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, e^{c sqrt{log n}}, due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erdos-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e^{Omega(log n)}. Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties.
Our main tool is a partitioning result found by Lovasz-Szegedy and Alon-Fischer-Newman, which is called the "ultra-strong regularity lemma" for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/epsilon)^{O(d)}, improving the original bound of (1/epsilon)^{O(d^2)} in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(n^k)-time algorithm for finding a partition meeting the requirements in the k-uniform setting.
VC-dimension
Ramsey theory
regularity lemma
43:1-43:15
Regular Paper
Jacob
Fox
Jacob Fox
János
Pach
János Pach
Andrew
Suk
Andrew Suk
10.4230/LIPIcs.SoCG.2017.43
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Implementing Delaunay Triangulations of the Bolza Surface
The CGAL library offers software packages to compute Delaunay triangulations of the (flat) torus of genus one in two and three dimensions. To the best of our knowledge, there is no available software for the simplest possible extension, i.e., the Bolza surface, a hyperbolic manifold homeomorphic to a torus of genus two.
In this paper, we present an implementation based on the theoretical results and the incremental algorithm proposed last year at SoCG by Bogdanov, Teillaud, and Vegter. We describe the representation of the triangulation, we detail the different steps of the algorithm, we study predicates, and report experimental results.
hyperbolic surface
Fuchsian group
arithmetic issues
Dehn's algorithm
CGAL
44:1-44:15
Regular Paper
Iordan
Iordanov
Iordan Iordanov
Monique
Teillaud
Monique Teillaud
10.4230/LIPIcs.SoCG.2017.44
N. L. Balazs and A. Voros. Chaos on the pseudosphere. Physics Reports, 143(3):109-240, 1986. URL: http://dx.doi.org/10.1016/0370-1573(86)90159-6.
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Mikhail Bogdanov, Monique Teillaud, and Gert Vegter. Delaunay triangulations on orientable surfaces of low genus. In Proceedings of the Thirty-second International Symposium on Computational Geometry, pages 20:1-20:15, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.20.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.20
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Manuel Caroli and Monique Teillaud. 3D periodic triangulations. In CGAL User and Reference Manual. CGAL Editorial Board, 3.5 (and further) edition, 2009-. URL: http://doc.cgal.org/latest/Manual/packages.html#PkgPeriodic3Triangulation3Summary.
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Olivier Devillers, Sylvain Pion, and Monique Teillaud. Walking in a triangulation. International Journal of Foundations of Computer Science, 13:181-199, 2002. URL: https://hal.inria.fr/inria-00102194.
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Olivier Devillers and Monique Teillaud. Perturbations for Delaunay and weighted Delaunay 3D Triangulations. Computational Geometry: Theory and Applications, 44:160-168, 2011. URL: http://dx.doi.org/10.1016/j.comgeo.2010.09.010.
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Lower Bounds for Differential Privacy from Gaussian Width
We study the optimal sample complexity of a given workload of linear queries under the constraints of differential privacy. The sample complexity of a query answering mechanism under error parameter alpha is the smallest n such that the mechanism answers the workload with error at most alpha on any database of size n. Following a line of research started by Hardt and Talwar [STOC 2010], we analyze sample complexity using the tools of asymptotic convex geometry. We study the sensitivity polytope, a natural convex body associated with a query workload that quantifies how query answers can change between neighboring databases. This is the information that, roughly speaking, is protected by a differentially private algorithm, and, for this reason, we expect that a "bigger" sensitivity polytope implies larger sample complexity. Our results identify the mean Gaussian width as an appropriate measure of the size of the polytope, and show sample complexity lower bounds in terms of this quantity. Our lower bounds completely characterize the workloads for which the Gaussian noise mechanism is optimal up to constants as those having asymptotically maximal Gaussian width.
Our techniques also yield an alternative proof of Pisier's Volume Number Theorem which also suggests an approach to improving the parameters of the theorem.
differential privacy
convex geometry
lower bounds
sample complexity
45:1-45:16
Regular Paper
Assimakis
Kattis
Assimakis Kattis
Aleksandar
Nikolov
Aleksandar Nikolov
10.4230/LIPIcs.SoCG.2017.45
Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman. Asymptotic geometric analysis. Part I, volume 202 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015. URL: http://dx.doi.org/10.1090/surv/202.
http://dx.doi.org/10.1090/surv/202
Aditya Bhaskara, Daniel Dadush, Ravishankar Krishnaswamy, and Kunal Talwar. Unconditional differentially private mechanisms for linear queries. In Proceedings of the 44th Symposium on Theory of Computing, STOC'12, pages 1269-1284, New York, NY, USA, 2012. ACM. URL: http://dx.doi.org/10.1145/2213977.2214089.
http://dx.doi.org/10.1145/2213977.2214089
Mark Bun, Jonathan Ullman, and Salil Vadhan. Fingerprinting codes and the price of approximate differential privacy. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 1-10. ACM, 2014.
Irit Dinur and Kobbi Nissim. Revealing information while preserving privacy. In Proceedings of the 22nd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pages 202-210. ACM, 2003.
Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Theory of Cryptography Conference, pages 265-284. Springer, 2006.
Cynthia Dwork, Aleksandar Nikolov, and Kunal Talwar. Using convex relaxations for efficiently and privately releasing marginals. In 30th Annual Symposium on Computational Geometry, SOCG'14, Kyoto, Japan, June 08-11, 2014, page 261. ACM, 2014. URL: http://dx.doi.org/10.1145/2582112.2582123.
http://dx.doi.org/10.1145/2582112.2582123
Cynthia Dwork and Kobbi Nissim. Privacy-preserving datamining on vertically partitioned databases. In Annual International Cryptology Conference, pages 528-544. Springer, 2004.
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Moritz Hardt and Kunal Talwar. On the geometry of differential privacy. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC'10, pages 705-714, New York, NY, USA, 2010. ACM. URL: http://dx.doi.org/10.1145/1806689.1806786.
http://dx.doi.org/10.1145/1806689.1806786
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Aleksandar Nikolov. An improved private mechanism for small databases. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, volume 9134 of Lecture Notes in Computer Science, pages 1010-1021. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_82.
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Aleksandar Nikolov, Kunal Talwar, and Li Zhang. The geometry of differential privacy: the sparse and approximate cases. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 351-360. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488652.
http://dx.doi.org/10.1145/2488608.2488652
Alain Pajor and Nicole Tomczak-Jaegermann. Subspaces of small codimension of finite-dimensional Banach spaces. Proc. Amer. Math. Soc., 97(4):637-642, 1986. URL: http://dx.doi.org/10.2307/2045920.
http://dx.doi.org/10.2307/2045920
Christos H. Papadimitriou and Mihalis Yannakakis. On limited nondeterminism and the complexity of the V-C dimension. J. Comput. Syst. Sci., 53(2):161-170, 1996. URL: http://dx.doi.org/10.1006/jcss.1996.0058.
http://dx.doi.org/10.1006/jcss.1996.0058
Allan Pinkus. n-widths in approximation theory, volume 7 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1985. URL: http://dx.doi.org/10.1007/978-3-642-69894-1.
http://dx.doi.org/10.1007/978-3-642-69894-1
G. Pisier. Sur les espaces de Banach K-convexes. In Seminar on Functional Analysis, 1979-1980 (French), pages Exp. No. 11, 15. École Polytech., Palaiseau, 1980.
Gilles Pisier. The volume of convex bodies and Banach space geometry, volume 94 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1989. URL: http://dx.doi.org/10.1017/CBO9780511662454.
http://dx.doi.org/10.1017/CBO9780511662454
Thomas Steinke and Jonathan Ullman. Between pure and approximate differential privacy. CoRR, abs/1501.06095, 2015. URL: http://arxiv.org/abs/1501.06095.
http://arxiv.org/abs/1501.06095
R. Vershynin. Lectures in geometric functional analysis. 2009. URL: http://www-personal.umich.edu/~romanv/papers/GFA-book.pdf.
http://www-personal.umich.edu/~romanv/papers/GFA-book.pdf
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Constrained Triangulations, Volumes of Polytopes, and Unit Equations
Given a polytope P in R^d and a subset U of its vertices, is there a triangulation of P using d-simplices that all contain U? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of P. Our proof relates triangulations of P to triangulations of its "shadow", a projection to a lower-dimensional space determined by U. In particular, we obtain a formula relating the volume of P with the volume of its shadow. This leads to an exact formula for the volume of a polytope arising in the theory of unit equations.
constrained triangulations
simplotopes
volumes of polytopes
projections of polytopes
unit equations
S-integers
46:1-46:15
Regular Paper
Michael
Kerber
Michael Kerber
Robert
Tichy
Robert Tichy
Mario
Weitzer
Mario Weitzer
10.4230/LIPIcs.SoCG.2017.46
C. Athanasiadis. Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley. J. Reine Angew. Math., 583:163-174, 2005.
A. Baker and G. Wüstholz. Logarithmic forms and group varieties. J. Reine Angew. Math., 442:19-62, 1993.
F. Barroero, C. Frei, and R. Tichy. Additive unit representations in rings over global fields - a survey. Publ. Math. Debrecen, 79(3-4):291-307, 2011.
M. Beck and R. Sanyal. Combinatorial Reciprocity Theorems. American Mathematical Society, 2016. In preparation, available at URL: http://math.sfsu.edu/beck/crt.html.
http://math.sfsu.edu/beck/crt.html
W. Bruns and T. Römer. h-vectors of Gorenstein polytopes. J. Combin. Theory Ser. A, 114:65-76, 2007.
H. Croft, K. Falconer, and R. Guy. Unsolved Problems in Geometry. Springer, 1991.
J. de Loera, F. Liu, and R. Yoshida. A generating function for all semi-magic squares and the volume of the Birkhoff polytope. J. Algebraic Combin., pages 113-139, 2009.
J. de Loera, J. Rambau, and F. Santos. Triangulations. Springer, 2010.
T. de Wolff. Polytopes with special simplices. arXiv:1009.6158.
H. Edelsbrunner and M. Kerber. Dual complexes of cubical subdivisions of ℝⁿ. Discrete Comput. Geom., 47(2):393-414, 2012.
I. Emiris and V. Fisikopoulos. Efficient random-walk methods for approximating polytope volume. In Proc. of the 13th Annual Symp. on Comp. Geom., SOCG'14, pages 318:318-318:327, 2014.
G. R. Everest. A "Hardy-Littlewood" approach to the S-unit equation. Compos. Math., 70(2):101-118, 1989.
G. R. Everest. Counting the values taken by sums of S-units. J. Number Theory, 35(3):269-286, 1990.
J. Evertse and H. P. Schlickewei. A quantitative version of the absolute subspace theorem. J. Reine Angew. Math., 548:21-127, 2002.
C. Frei, R. Tichy, and V. Ziegler. On sums of S-integers of bounded norm. Monatsh. Math., 175(2):241-247, 2014.
H. Freudenthal. Simplizialzerlegung beschränkter Flachheit. Ann. of Math., pages 580-582, 1942.
R. Freund. Combinatorial theorems on the simplotope that generalize results on the simplex and cube. Math. Oper. Res., 11(1):169-179, 1986.
I. Gelfand, M. Kapranov, and A. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, 2008.
K. Győry and K. Yu. Bounds for the solutions of S-unit equations and decomposable form equations. Acta Arith., 123(1):9-41, 2006.
H. Hadwiger. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer, 1957.
T. Hibi and H. Ohsugi. Special simplices and Gorenstein toric rings. J. Combin. Theory Ser. A, 113:718-725, 2006.
R. Hughes and M. Anderson. Simplexity of the cube. Discrete Math., 158:99-150, 1996.
M. Kerber, R. Tichy, and M. Weitzer. Constrained triangulations, volumes of polytopes, and unit equations. arXiv, 1609.05017, 2016.
J. Matoušek. Lectures in Discrete Geometry. Springer, 2002.
J. Neukirch. Algebraic number theory, volume 322 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, 1999.
K. Nishioka. Algebraic independence by Mahler’s method and S-unit equations. Compositio Math., 92(1):87-110, 1994.
I. Pak. Four questions on Birkhoff polytope. Ann. Comb., 4:83-90, 2000.
V. Reiner and V. Welker. On the Charney-Davis and Neggers-Stanley conjectures. J. Combin. Theory Ser. A, 109(2):247-280, 2005.
F. Santos. A counterexample to the Hirsch conjecture. Ann. of Math., 176:383-412, 2012.
H. P. Schlickewei. S-unit equations over number fields. Invent. Math., 102(1):95-108, 1990.
J. Shewchuk. General-dimensional constrained Delaunay and constrained regular triangulations, I: Combinatorial properties. Discrete Comput. Geom., 39:580-637, 2008.
G. van der Laan and A. Talman. On the computation of fixed points in the product space of unit simplices and an application to noncooperative n person games. Math. Oper. Res., 7(1):1-13, 1982.
G. Ziegler. Lectures on Polytopes. Springer, 2007.
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Proper Coloring of Geometric Hypergraphs
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions.
discrete geometry
decomposition of multiple coverings
geometric hypergraph coloring
47:1-47:15
Regular Paper
Balázs
Keszegh
Balázs Keszegh
Dömötör
Pálvölgyi
Dömötör Pálvölgyi
10.4230/LIPIcs.SoCG.2017.47
Eyal Ackerman, Balázs Keszegh, and Máté Vizer. Coloring points with respect to squares. In Sándor P. Fekete and Anna Lubiw, editors, 32nd International Symposium on Computational Geometry, SoCG 2016, June 14-18, 2016, Boston, MA, USA, volume 51 of LIPIcs, pages 5:1-5:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016.
Greg Aloupis, Jean Cardinal, Sébastien Collette, Stefan Langerman, and Shakhar Smorodinsky. Coloring geometric range spaces. Discrete & Computational Geometry, 41(2):348-362, 2009.
Andrei Asinowski, Jean Cardinal, Nathann Cohen, Sébastien Collette, Thomas Hackl, Michael Hoffmann, Kolja B. Knauer, Stefan Langerman, Michal Lason, Piotr Micek, Günter Rote, and Torsten Ueckerdt. Coloring hypergraphs induced by dynamic point sets and bottomless rectangles. In Frank Dehne, Roberto Solis-Oba, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures - 13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings, volume 8037 of Lecture Notes in Computer Science, pages 73-84. Springer, 2013.
Prosenjit Bose, Paz Carmi, Sébastien Collette, and Michiel H. M. Smid. On the stretch factor of convex delaunay graphs. J. of Computational Geometry, 1(1):41-56, 2010.
Peter Brass, William O. J. Moser, and János Pach. Research problems in discrete geometry. Springer, 2005.
Sarit Buzaglo, Rom Pinchasi, and Günter Rote. Topological hypergraphs. In János Pach, editor, Thirty Essays on Geometric Graph Theory, pages 71-81. Springer New York, 2013. URL: http://dx.doi.org/10.1007/978-1-4614-0110-0_6.
http://dx.doi.org/10.1007/978-1-4614-0110-0_6
Jean Cardinal, Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. Making triangles colorful. J. of Computational Geometry, 4:240-246, 2013.
Jean Cardinal, Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. Making octants colorful and related covering decomposition problems. SIAM J. on Discrete Math., 28(4):1948-1959, 2014.
Jean Cardinal and Matias Korman. Coloring planar homothets and three-dimensional hypergraphs. Computational Geometry, 46(9):1027-1035, 2013.
Xiaomin Chen, János Pach, Mario Szegedy, and Gábor Tardos. Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles. Random Struct. Algorithms, 34(1):11-23, 2009. URL: http://dx.doi.org/10.1002/rsa.20246.
http://dx.doi.org/10.1002/rsa.20246
Louis Esperet and Gwenaël Joret. Colouring planar graphs with three colours and no large monochromatic components. Combinatorics, Probability & Computing, 23(4):551-570, 2014.
Radoslav Fulek. Coloring geometric hypergraph defined by an arrangement of half-planes. In Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, Winnipeg, Manitoba, Canada, August 9-11, 2010, pages 71-74, 2010.
Matt Gibson and Kasturi R. Varadarajan. Decomposing coverings and the planar sensor cover problem. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 159-168. IEEE Computer Society, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.54.
http://dx.doi.org/10.1109/FOCS.2009.54
Wayne Goddard. Acyclic colorings of planar graphs. Discrete Math., 91(1):91-94, 1991.
B. Gonska and A. Padrol. Neighborly inscribed polytopes and delaunay triangulations. Advances in Geometry, 16(3):349-360, 2016.
A. W. Hales and R. I. Jewett. Regularity and positional games. Trans. Amer. Math. Soc., 106:222-229, 1963.
Balázs Keszegh. Coloring half-planes and bottomless rectangles. Computational Geometry, 45(9):495-507, 2012.
Balázs Keszegh, Nathan Lemons, and Dömötör Pálvölgyi. Online and quasi-online colorings of wedges and intervals. Order, 33(3):389-409, 2016.
Balázs Keszegh and Dömötör Pálvölgyi. Octants are cover-decomposable. Discrete &Computational Geometry, 47(3):598-609, 2012.
Balázs Keszegh and Dömötör Pálvölgyi. Convex polygons are self-coverable. Discrete &Computational Geometry, 51(4):885-895, 2014.
Balázs Keszegh and Dömötör Pálvölgyi. Octants are cover-decomposable into many coverings. Computational Geometry, 47(5):585-588, 2014.
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.
Balázs Keszegh and Dömötör Pálvölgyi. More on decomposing coverings by octants. J. of Computational Geometry, 6(1):300-315, 2015.
Rolf Klein. Concrete and abstract Voronoi diagrams, volume 400. Springer Science &Business Media, 1989.
Jon M. Kleinberg, Rajeev Motwani, Prabhakar Raghavan, and Suresh Venkatasubramanian. Storage management for evolving databases. In 38th Annual Symposium on Foundations of Computer Science, FOCS'97, Miami Beach, Florida, USA, October 19-22, 1997, pages 353-362. IEEE Computer Society, 1997.
István Kovács. Indecomposable coverings with homothetic polygons. Discrete &Computational Geometry, 53(4):817-824, 2015.
L. Ma. Bisectors and Voronoi Diagrams for Convex Distance Functions. PhD thesis, FernUniversität Hagen, Germany, 2000.
János Pach. Decomposition of multiple packing and covering. In 2. Kolloquium über Diskrete Geometrie, pages 169-178. Institut für Mathematik der Universität Salzburg, 1980.
János Pach. Covering the plane with convex polygons. Discrete & Computational Geometry, 1:73-81, 1986. URL: http://dx.doi.org/10.1007/BF02187684.
http://dx.doi.org/10.1007/BF02187684
János Pach and Dömötör Pálvölgyi. Unsplittable coverings in the plane. Advances in Mathematics, 302:433-457, 2016.
János Pach, Dömötör Pálvölgyi, and Géza Tóth. Survey on decomposition of multiple coverings. In Imre Bárány, Károly J. Böröczky, Gábor Fejes Tóth, and János Pach, editors, Geometry - Intuitive, Discrete, and Convex, volume 24 of Bolyai Society Mathematical Studies, pages 219-257. Springer Berlin Heidelberg, 2013.
János Pach and Gábor Tardos. Coloring axis-parallel rectangles. J. of Combinatorial Theory, Series A, 117(6):776-782, 2010. URL: http://dx.doi.org/10.1016/j.jcta.2009.04.007.
http://dx.doi.org/10.1016/j.jcta.2009.04.007
János Pach, Gábor Tardos, and Géza Tóth. Indecomposable coverings. Canadian mathematical bulletin, 52(3):451-463, 2009.
János Pach and Géza Tóth. Decomposition of multiple coverings into many parts. In Proceedings of the twenty-third annual symposium on Computational geometry, pages 133-137. ACM, 2007.
Dömötör Pálvölgyi. Decomposition of geometric set systems and graphs, phd thesis. arXiv preprint arXiv:1009.4641, 2010.
Dömötör Pálvölgyi. Indecomposable coverings with concave polygons. Discrete & Computational Geometry, 44(3):577-588, 2010. URL: http://dx.doi.org/10.1007/s00454-009-9194-y.
http://dx.doi.org/10.1007/s00454-009-9194-y
Dömötör Pálvölgyi and Géza Tóth. Convex polygons are cover-decomposable. Discrete &Computational Geometry, 43(3):483-496, 2010.
K. S. Poh. On the linear vertex-arboricity of a planar graph. J. of Graph Theory, 14(1):73-75, 1990.
Shakhar Smorodinsky. On the chromatic number of geometric hypergraphs. SIAM J. on Discrete Math., 21(3):676-687, 2007.
Shakhar Smorodinsky and Yelena Yuditsky. Polychromatic coloring for half-planes. J. of Combinatorial Theory, Series A, 119(1):146-154, 2012.
Gábor Tardos and Géza Tóth. Multiple coverings of the plane with triangles. Discrete & Computational Geometry, 38(2):443-450, 2007. URL: http://dx.doi.org/10.1007/s00454-007-1345-4.
http://dx.doi.org/10.1007/s00454-007-1345-4
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Computing Representative Networks for Braided Rivers
Drainage networks on terrains have been studied extensively from an algorithmic perspective. However, in drainage networks water flow cannot bifurcate and hence they do not model braided rivers (multiple channels which split and join, separated by sediment bars). We initiate the algorithmic study of braided rivers by employing the descending quasi Morse-Smale complex on the river bed (a polyhedral terrain), and extending it with a certain ordering of bars from the one river bank to the other. This allows us to compute a graph that models a representative channel network, consisting of lowest paths. To ensure that channels in this network are sufficiently different we define a sand function that represents the volume of sediment separating them. We show that in general the problem of computing a maximum network of non-crossing channels which are delta-different from each other (as measured by the sand function) is NP-hard. However, using our ordering between the river banks, we can compute a maximum delta-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.
braided rivers
Morse-Smale complex
persistence
network extraction
polyhedral terrain
48:1-48:16
Regular Paper
Maarten
Kleinhans
Maarten Kleinhans
Marc
van Kreveld
Marc van Kreveld
Tim
Ophelders
Tim Ophelders
Willem
Sonke
Willem Sonke
Bettina
Speckmann
Bettina Speckmann
Kevin
Verbeek
Kevin Verbeek
10.4230/LIPIcs.SoCG.2017.48
Pankaj Agarwal, Mark de Berg, Prosenjit Bose, Katrin Dobrint, Marc van Kreveld, Mark Overmars, Marko de Groot, Thomas Roos, Jack Snoeyink, and Sidi Yu. The complexity of rivers in triangulated terrains. In Proc. 8th Canadian Conference on Computational Geometry CCCG'96, pages 325-330, 1996.
Lars Arge, Jeffrey S. Chase, Patrick Halpin, Laura Toma, Jeffrey S. Vitter, Dean Urban, and Rajiv Wickremesinghe. Efficient flow computation on massive grid terrain datasets. GeoInformatica, 7(4):283-313, 2003.
Peter Ashmore. Channel morphology and bed load pulses in braided, gravel-bed streams. Geografiska Annaler: Series A, Physical Geography, 73(1):37-52, 1991.
Mark de Berg, Otfried Cheong, Herman Haverkort, Jung-Gun Lim, and Laura Toma. The complexity of flow on fat terrains and its I/O-efficient computation. Computational Geometry, 43(4):331-356, 2010.
Mark de Berg and Constantinos Tsirogiannis. Exact and approximate computations of watersheds on triangulated terrains. In Proc. 19th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pages 74-83. ACM, 2011.
Walter Bertoldi, Luca Zanoni, and Marco Tubino. Planform dynamics of braided streams. Earth Surface Processes and Landforms, 34:547–557, 2009.
Hamish Carr, Jack Snoeyink, and Ulrike Axen. Computing contour trees in all dimensions. Computational Geometry, 24(2):75-94, 2003.
Yi-Jen Chiang, Tobias Lenz, Xiang Lu, and Günter Rote. Simple and optimal output-sensitive construction of contour trees using monotone paths. Computational Geometry, 30(2):165-195, 2005.
Wout M. van Dijk, Wietse I. van de Lageweg, and Maarten G. Kleinhans. Formation of a cohesive floodplain in a dynamic experimental meandering river. Earth Surface Processes and Landforms, 38, 2013.
Herbert Edelsbrunner, John Harer, and Afra Zomorodian. Hierarchical Morse complexes for piecewise linear 2-manifolds. In Proc. 17th Annual ACM Symposium on Computational Geometry, pages 70-79, 2001.
D. Murray Hicks, Maurice J. Duncan, and Jeremy M. Walsh. New views of the morphodynamics of large braided rivers from high-resolution topographic surveys and time-lapse video. In The Structure, Function and Management Implications of Fluvial Sedimentary Systems (Proceedings), pages 373-380. IAHS Publ. no. 276, 2002.
Alan D. Howard, Mary E. Keetch, and C. Linwood Vincent. Topological and geometrical properties of braided streams. Water Resources Research, 6(6), 1970.
Maarten G. Kleinhans. Flow discharge and sediment transport models for estimating a minimum timescale of hydrological activity and channel and delta formation on Mars. Journal of Geophysical Research, 110, 2005.
Maarten G. Kleinhans, Robert I. Ferguson, Stuart N. Lane, and Richard J. Hardy. Splitting rivers at their seams: bifurcations and avulsion. Earth Surface Processes and Landforms, 38(1):47-61, 2013.
Thierry de Kok, Marc van Kreveld, and Maarten Löffler. Generating realistic terrains with higher-order Delaunay triangulations. Computational Geometry, 36(1):52-65, 2007.
Marc van Kreveld and Rodrigo I. Silveira. Embedding rivers in triangulated irregular networks with linear programming. International Journal of Geographical Information Science, 25(4):615-631, 2011.
Yuanxin Liu and Jack Snoeyink. Flooding triangulated terrain. In Developments in Spatial Data Handling, pages 137-148. Springer, 2005.
Wouter A. Marra, Maarten G. Kleinhans, and Elisabeth A. Addink. Network concepts to describe channel importance and change in multichannel systems: test results for the Jamuna river, Bangladesh. Earth Surface Processes and Landforms, 39(6):766-778, 2014.
Michael McAllister and Jack Snoeyink. Extracting consistent watersheds from digital river and elevation data. In Proc. ASPRS/ACSM Annu. Conf, volume 138, 1999.
Gary Parker. On the cause and characteristic scales of meandering and braiding in rivers. Journal of Fluid Mechanics, 76(3):457-480, 1976.
Günter Rote. Lexicographic Fréchet matchings. In Abstracts of the 30th European Workshop on Computational Geometry, 2014.
Filip Schuurman, Maarten G. Kleinhans, and Hans Middelkoop. Network response to disturbances in large sand-bed braided rivers. Earth Surface Dynamics, 4(1):25-45, 2016.
Filip Schuurman, Wouter A. Marra, and Maarten G. Kleinhans. Physics-based modeling of large braided sand-bed rivers: Bar pattern formation, dynamics, and sensitivity. Journal of Geophysical Research: Earth Surface, 118(4):2509-2527, 2013.
Nithin Shivashankar, Senthilnathan M, and Vijay Natarajan. Parallel computation of 2D Morse-Smale complexes. IEEE Transactions on Visualization and Computer Graphics, 18(10):1757-1770, 2012.
Rodrigo I. Silveira and René van Oostrum. Flooding countries and destroying dams. International Journal of Computational Geometry &Applications, 20(3):361-380, 2010.
Sidi Yu, Marc van Kreveld, and Jack Snoeyink. Drainage queries in TINs: from local to global and back again. In Advances in GIS Research II: Proc. 7th International Symposium on Spatial Data Handling, pages 829-842, 1997.
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A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations
Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n^7) on the length of the flip sequence.
Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.
triangulations
reconfiguration
flip
constrained triangulations
Delaunay triangulation
shellability
piecewise linear balls
49:1-49:15
Regular Paper
Anna
Lubiw
Anna Lubiw
Zuzana
Masárová
Zuzana Masárová
Uli
Wagner
Uli Wagner
10.4230/LIPIcs.SoCG.2017.49
Oswin Aichholzer, Wolfgang Mulzer, and Alexander Pilz. Flip distance between triangulations of a simple polygon is NP-complete. Discrete &Computational Geometry, 54(2):368-389, 2015. URL: http://dx.doi.org/10.1007/s00454-015-9709-7.
http://dx.doi.org/10.1007/s00454-015-9709-7
Gabriela Araujo-Pardo, Isabel Hubard, Deborah Oliveros, and Egon Schulte. Colorful associahedra and cyclohedra. Journal of Combinatorial Theory, Series A, 129:122-141, 2015. URL: http://dx.doi.org/10.1016/j.jcta.2014.09.001.
http://dx.doi.org/10.1016/j.jcta.2014.09.001
Marshall Bern and David Eppstein. Mesh generation and optimal triangulation. In Ding-Zhu Du and Frank Hwang, editors, Computing in Euclidean geometry, volume 1 of Lecture Notes Series on Computing, pages 23-90. World Scientific, 1992. URL: http://dx.doi.org/10.1142/9789814355858_0002.
http://dx.doi.org/10.1142/9789814355858_0002
R. H. Bing. Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. In Lectures on modern mathematics, Vol. II, pages 93-128. Wiley, New York, 1964.
Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler. Oriented Matroids, volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2nd edition, 1999. URL: http://dx.doi.org/10.1017/CBO9780511586507.
http://dx.doi.org/10.1017/CBO9780511586507
Prosenjit Bose and Ferran Hurtado. Flips in planar graphs. Computational Geometry Theory and Applications, 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
Prosenjit Bose, Anna Lubiw, Vinayak Pathak, and Sander Verdonschot. Flipping edge-labelled triangulations. arXiv:1310.1166, 2013. To appear in Computational Geometry. URL: http://arxiv.org/abs/1310.1166.
http://arxiv.org/abs/1310.1166
Prosenjit Bose and Sander Verdonschot. Flips in edge-labelled pseudo-triangulations. Computational Geometry, 60:45-54, 2017.
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A Spectral Gap Precludes Low-Dimensional Embeddings
We prove that if an n-vertex O(1)-expander embeds with average distortion D into a finite dimensional normed space X, then necessarily the dimension of X is at least n^{c/D} for some universal constant c>0. This is sharp up to the value of the constant c, and it improves over the previously best-known estimate dim(X)> c(log n)^2/D^2 of Linial, London and Rabinovich, strengthens a theorem of Matousek, and answers a question of Andoni, Nikolov, Razenshteyn and Waingarten.
Metric embeddings
dimensionality reduction
expander graphs
nonlinear spectral gaps
nearest neighbor search
complex interpolation
Markov type.
50:1-50:16
Regular Paper
Assaf
Naor
Assaf Naor
10.4230/LIPIcs.SoCG.2017.50
Noga Alon, Peter Frankl, and Vojtech Rödl. Geometrical realization of set systems and probabilistic communication complexity. In 26th Annual Symposium on Foundations of Computer Science, pages 277-280, 1985. URL: http://dx.doi.org/10.1109/SFCS.1985.30.
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Subhash Khot and Assaf Naor. Nonembeddability theorems via Fourier analysis. Math. Ann., 334(4):821-852, 2006. URL: http://dx.doi.org/10.1007/s00208-005-0745-0.
http://dx.doi.org/10.1007/s00208-005-0745-0
R. Krauthgamer, J. R. Lee, M. Mendel, and A. Naor. Measured descent: a new embedding method for finite metrics. Geom. Funct. Anal., 15(4):839-858, 2005. URL: http://dx.doi.org/10.1007/s00039-005-0527-6.
http://dx.doi.org/10.1007/s00039-005-0527-6
James R. Lee, Manor Mendel, and Assaf Naor. Metric structures in L₁: dimension, snowflakes, and average distortion. European J. Combin., 26(8):1180-1190, 2005. URL: http://dx.doi.org/10.1016/j.ejc.2004.07.002.
http://dx.doi.org/10.1016/j.ejc.2004.07.002
J. R. Lee and A. Naor. Embedding the diamond graph in L_p and dimension reduction in L₁. Geom. Funct. Anal., 14(4):745-747, 2004. URL: http://dx.doi.org/10.1007/s00039-004-0473-8.
http://dx.doi.org/10.1007/s00039-004-0473-8
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http://dx.doi.org/10.1007/BF01200757
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Jiří Matoušek. On embedding expanders into l_p spaces. Israel J. Math., 102:189-197, 1997. URL: http://dx.doi.org/10.1007/BF02773799.
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Manor Mendel and Assaf Naor. Markov convexity and local rigidity of distorted metrics. J. Eur. Math. Soc. (JEMS), 15(1):287-337, 2013. URL: http://dx.doi.org/10.4171/JEMS/362.
http://dx.doi.org/10.4171/JEMS/362
Manor Mendel and Assaf Naor. Spectral calculus and Lipschitz extension for barycentric metric spaces. Anal. Geom. Metr. Spaces, 1:163-199, 2013. URL: http://dx.doi.org/10.2478/agms-2013-0003.
http://dx.doi.org/10.2478/agms-2013-0003
Manor Mendel and Assaf Naor. Nonlinear spectral calculus and super-expanders. Publ. Math. Inst. Hautes Études Sci., 119:1-95, 2014. URL: http://dx.doi.org/10.1007/s10240-013-0053-2.
http://dx.doi.org/10.1007/s10240-013-0053-2
Manor Mendel and Assaf Naor. Expanders with respect to Hadamard spaces and random graphs. Duke Math. J., 164(8):1471-1548, 2015. URL: http://dx.doi.org/10.1215/00127094-3119525.
http://dx.doi.org/10.1215/00127094-3119525
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Assaf Naor. An introduction to the Ribe program. Jpn. J. Math., 7(2):167-233, 2012. URL: http://dx.doi.org/10.1007/s11537-012-1222-7.
http://dx.doi.org/10.1007/s11537-012-1222-7
Assaf Naor. On the Banach-space-valued Azuma inequality and small-set isoperimetry of Alon-Roichman graphs. Combin. Probab. Comput., 21(4):623-634, 2012. URL: http://dx.doi.org/10.1017/S0963548311000757.
http://dx.doi.org/10.1017/S0963548311000757
Assaf Naor. Comparison of metric spectral gaps. Anal. Geom. Metr. Spaces, 2:1-52, 2014. URL: http://dx.doi.org/10.2478/agms-2014-0001.
http://dx.doi.org/10.2478/agms-2014-0001
Assaf Naor, Yuval Peres, Oded Schramm, and Scott Sheffield. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1):165-197, 2006. URL: http://dx.doi.org/10.1215/S0012-7094-06-13415-4.
http://dx.doi.org/10.1215/S0012-7094-06-13415-4
Assaf Naor and Lior Silberman. Poincaré inequalities, embeddings, and wild groups. Compos. Math., 147(5):1546-1572, 2011. URL: http://dx.doi.org/10.1112/S0010437X11005343.
http://dx.doi.org/10.1112/S0010437X11005343
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http://dx.doi.org/10.1515/9783110264012
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http://dx.doi.org/10.1090/S0065-9266-10-00601-0
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http://dx.doi.org/10.1007/s00454-007-9047-5
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Dynamic Geodesic Convex Hulls in Dynamic Simple Polygons
We consider the geodesic convex hulls of points in a simple polygonal region in the presence of non-crossing line segments (barriers) that subdivide the region into simply connected faces. We present an algorithm together with data structures for maintaining the geodesic convex hull of points in each face in a sublinear update time under the fully-dynamic setting where both input points and barriers change by insertions and deletions. The algorithm processes a mixed update sequence of insertions and deletions of points and barriers. Each update takes O(n^2/3 log^2 n) time with high probability, where n is the total number of the points and barriers at the moment. Our data structures support basic queries on the geodesic convex hull, each of which takes O(polylog n) time. In addition, we present an algorithm together with data structures for geodesic triangle counting queries under the fully-dynamic setting. With high probability, each update takes O(n^2/3 log n) time, and each query takes O(n^2/3 log n) time.
Dynamic geodesic convex hull
dynamic simple polygons
51:1-51:15
Regular Paper
Eunjin
Oh
Eunjin Oh
Hee-Kap
Ahn
Hee-Kap Ahn
10.4230/LIPIcs.SoCG.2017.51
Julien Basch, Jeff Erickson, Leonidas J. Guibas, John Hershberger, and Li Zhang. Kinetic collision detection between two simple polygons. Computational Geometry, 27(3):211-235, 2004.
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Gerth Stølting Brodal and Riko Jacob. Dynamic planar convex hull. In Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2002), pages 617-626, 2002.
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Timothy M. Chan and Yakov Nekrich. Towards an optimal method for dynamic planar point location. In Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS 2015), pages 390-409, 2015.
Bernard Chazelle. Lower bounds on the complexity of polytope range searching. Journal of the American Mathematical Society, 2(4):637-666, 1989.
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Bernard Chazelle, Micha Sharir, and Emo Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica, 8(1):407-429, 1992.
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Michael T. Goodrich and Roberto Tamassia. Dynamic ray shooting and shortest paths in planar subdivisions via balanced geodesic triangulations. Journal of Algorithms, 23(1):51-73, 1997.
Mashhood Ishaque and Csaba D. Tóth. Relative convex hulls in semi-dynamic arrangements. Algorithmica, 68(2):448-482, 2014.
David Kirkpatrick and Jack Snoeyink. Computing common tangents without a separating line. In Proceedings of the 4th International Workshop on Algorithms and Data Structures (WADS 1995), pages 183-193, 1995.
Jiří Matousěk. Efficient partition trees. Discrete &Computational Geometry, 8(3):315-334, 1992.
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Ileana Streinu. Pseudo-triangulations, rigidity and motion planning. Discrete and Computational Geometry, 34:587-635, 2005.
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Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon
Given a set of sites in a simple polygon, a geodesic Voronoi diagram partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for m < n/polylog n. Moreover, the algorithms for the nearest-point and farthest-point Voronoi diagrams are optimal for m < n/polylog n. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.
Simple polygons
Voronoi diagrams
geodesic distance
52:1-52:15
Regular Paper
Eunjin
Oh
Eunjin Oh
Hee-Kap
Ahn
Hee-Kap Ahn
10.4230/LIPIcs.SoCG.2017.52
Hee-Kap Ahn, Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Matias Korman, and Eunjin Oh. A linear-time algorithm for the geodesic center of a simple polygon. Discrete &Computational Geometry, 56(4):836-859, 2016.
Boris Aronov. On the geodesic Voronoi diagram of point sites in a simple polygon. Algorithmica, 4(1):109-140, 1989.
Boris Aronov, Steven Fortune, and Gordon Wilfong. The furthest-site geodesic Voronoi diagram. Discrete &Computational Geometry, 9(1):217-255, 1993.
Jon Louis Bentley and James B. Saxe. Decomposable searching problems 1: Static-to-dynamic transformations. Journal of Algorithms, 1(4):297-396, 1980.
Bernard Chazelle, Herbert Edelsbrunner, Michelangelo Grigni, Leonidas Guibas, John Hershberger, Micha Sharir, and Jack Snoeyink. Ray shooting in polygons using geodesic triangulations. Algorithmica, 12(1):54-68, 1994.
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.
Steven Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2(1):153-174, 1987.
Leonidas 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):209-233, 1987.
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.
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Chih-Hung Liu and D. T. Lee. Higher-order geodesic Voronoi diagrams in a polygonal domain with holes. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), pages 1633-1645, 2013.
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Joseph S. B. Mitchell. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, pages 633-701. Elsevier, 2000.
Eunjin Oh, Luis Barba, and Hee-Kap Ahn. The farthest-point geodesic voronoi diagram of points on the boundary of a simple polygon. In Proceedings of the 32nd International Symposium on Computational Geometry (SoCG 2016), pages 56:1-56:15, 2016.
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, 1998(4):319-352, 1998.
Richard Pollack, Micha Sharir, and Günter Rote. Computing the geodesic center of a simple polygon. Discrete &Computational Geometry, 4(6):611-626, 1989.
Maksym Zavershynskyi and Evanthia Papadopoulou. A sweepline algorithm for higher order voronoi diagrams. In Proceedings of the 10th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2013), pages 16-22, 2013.
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A Quest to Unravel the Metric Structure Behind Perturbed Networks
Graphs and network data are ubiquitous across a wide spectrum of scientific and application domains. Often in practice, an input graph can be considered as an observed snapshot of a (potentially
continuous) hidden domain or process. Subsequent analysis, processing, and inferences are then performed on this observed graph. In this paper we advocate the perspective that an observed graph is often a noisy version of some discretized 1-skeleton of a hidden domain, and specifically we will consider the following natural network model: We assume that there is a true graph G^* which is a certain proximity graph for points sampled from a hidden domain X; while the observed graph G is an Erdos-Renyi type perturbed version of G^*.
Our network model is related to, and slightly generalizes, the much-celebrated small-world network model originally proposed by Watts and Strogatz. However, the main question we aim to answer is orthogonal to the usual studies of network models (which often focuses on characterizing / predicting behaviors and properties of real-world networks). Specifically, we aim to recover the metric structure of G^* (which reflects that of the hidden space X as we will show) from the observed graph G. Our main result is that a simple filtering process based on the Jaccard index can recover this metric within a multiplicative factor of 2 under our network model. Our work makes one step towards the general question of inferring structure of a hidden space from its observed noisy graph representation. In addition, our results also provide a theoretical understanding for Jaccard-Index-based denoising approaches.
metric structure
Erdös-Rényi perturbation
graphs
doubling measure
53:1-53:16
Regular Paper
Srinivasan
Parthasarathy
Srinivasan Parthasarathy
David
Sivakoff
David Sivakoff
Minghao
Tian
Minghao Tian
Yusu
Wang
Yusu Wang
10.4230/LIPIcs.SoCG.2017.53
Morteza Alamgir and Ulrike V. Luxburg. Shortest path distance in random k-nearest neighbor graphs. In 29th Intl. Conf. Machine Learning (ICML), pages 1031-1038, 2012.
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S. Parthasarathy, D. Sivakoff, M. Tian, and Y. Wang. A quest to unravel the metric structure behind perturbed networks. ArXiv e-prints, March 2017. URL: http://arxiv.org/abs/1703.05475.
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Mathew Penrose. Random geometric graphs. Oxford University Press, 2003.
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A. Singer and H.-T. Wu. Two-dimensional tomography from noisy projections taken at unknown random directions. SIAM journal on imaging sciences, 6(1):136-175, 2013.
H. F. Song and X.-J. Wang. Simple, distance-dependent formulation of the Watts-Strogatz model for directed and undirected small-world networks. Phys. Rev. E, 90:062801, 2014.
Xiao Fan Wang and Guanrong Chen. Complex networks: small-world, scale-free and beyond. Circuits and Systems Magazine, IEEE, 3(1):6-20, 2003.
Duncan J. Watts, Peter Sheridan Dodds, and M. E. J. Newman. Identity and search in social networks. Science, 296(5571):1302-1305, 2002. URL: http://dx.doi.org/10.1126/science.1070120.
http://dx.doi.org/10.1126/science.1070120
Duncan J. Watts and Steven H. Strogatz. Collective dynamics of ‘small-world’networks. nature, 393(6684):440-442, 1998.
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https://creativecommons.org/licenses/by/3.0/legalcode
From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices
A set P = H cup {w} of n+1 points in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on P, it suffices to know the so-called frequency vector of P. While there are roughly 2^n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2^{n/2}.
We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, w-embracing triangles, and many more. Based on these formulas, the corresponding numbers of graphs can be computed efficiently.
Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of simplices spanned by H that contain w. While the concept of frequency vectors does not generalize easily, we show how to apply similar methods in higher dimensions. The result is an O(n^{d-1}) time algorithm for computing the simplicial depth of a point w in a set H of n d-dimensional points, improving on the previously best bound of O(n^d log n).
Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n=d+k points in general position in R^d in time O(n^max(omega,k-2)) where omega = 2.373, even though the asymptotic number of facets may be as large as n^k.
Geometric Graph
Wheel Set
Simplicial Depth
Gale Transform
Polytope
54:1-54:16
Regular Paper
Alexander
Pilz
Alexander Pilz
Emo
Welzl
Emo Welzl
Manuel
Wettstein
Manuel Wettstein
10.4230/LIPIcs.SoCG.2017.54
Bernardo M. Ábrego and Silvia Fernández-Merchant. A lower bound for the rectilinear crossing number. Graphs Combin., 21(3):293-300, 2005. URL: http://dx.doi.org/10.1007/s00373-005-0612-5.
http://dx.doi.org/10.1007/s00373-005-0612-5
Peyman Afshani, Donald R. Sheehy, and Yannik Stein. Approximating the simplicial depth. CoRR, abs/1512.04856, 2015.
Oswin Aichholzer, Thomas Hackl, Clemens Huemer, Ferran Hurtado, Hannes Krasser, and Birgit Vogtenhuber. On the number of plane geometric graphs. Graphs Combin., 23:67-84, 2007. URL: http://dx.doi.org/10.1007/s00373-007-0704-5.
http://dx.doi.org/10.1007/s00373-007-0704-5
Esther M. Arkin, Samir Khuller, and Joseph S. B. Mitchell. Geometric knapsack problems. Algorithmica, 10(5):399-427, 1993. URL: http://dx.doi.org/10.1007/BF01769706.
http://dx.doi.org/10.1007/BF01769706
Andries E. Brouwer. The enumeration of locally transitive tournaments. Technical Report ZW 138/80, Mathematisch Centrum, Amsterdam, 1980.
Andrew Y. Cheng and Ming Ouyang. On algorithms for simplicial depth. In Proc. 13superscriptth Canadian Conference on Computational Geometry, pages 53-56, 2001.
Serge Dulucq and Jean-Guy Penaud. Cordes, arbres et permutations. Discr. Math., 117(1):89-105, 1993. URL: http://dx.doi.org/10.1016/0012-365X(93)90326-O.
http://dx.doi.org/10.1016/0012-365X(93)90326-O
Herbert Edelsbrunner and Ernst P. Mücke. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph., 9(1):66-104, 1990. URL: http://dx.doi.org/10.1145/77635.77639.
http://dx.doi.org/10.1145/77635.77639
Herbert Edelsbrunner, Joseph O'Rourke, and Raimund Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput., 15(2):341-363, 1986. URL: http://dx.doi.org/10.1137/0215024.
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David Eppstein, Mark H. Overmars, Günter Rote, and Gerhard J. Woeginger. Finding minimum area k-gons. Discr. Comput. Geom., 7:45-58, 1992. URL: http://dx.doi.org/10.1007/BF02187823.
http://dx.doi.org/10.1007/BF02187823
Philippe Flajolet and Marc Noy. Analytic combinatorics of non-crossing configurations. Discr. Math., 204(1-3):203-229, 1999. URL: http://dx.doi.org/10.1016/S0012-365X(98)00372-0.
http://dx.doi.org/10.1016/S0012-365X(98)00372-0
Joseph Gil, William L. Steiger, and Avi Wigderson. Geometric medians. Discr. Math., 108(1-3):37-51, 1992. URL: http://dx.doi.org/10.1016/0012-365X(92)90658-3.
http://dx.doi.org/10.1016/0012-365X(92)90658-3
Jacob E. Goodman and Richard Pollack. Multidimensional sorting. SIAM J. Comput., 12(3):484-507, 1983.
Jacob E. Goodman and Richard Pollack. Semispaces of configurations, cell complexes of arrangements. J. Combin. Theory Ser. A, 37(3):257-293, 1984.
Branko Grünbaum. Convex Polytopes. Springer, 2nd edition, 2003.
Samir Khuller and Joseph S. B. Mitchell. On a triangle counting problem. Inf. Process. Lett., 33(6):319-321, 1990. URL: http://dx.doi.org/10.1016/0020-0190(90)90217-L.
http://dx.doi.org/10.1016/0020-0190(90)90217-L
Svante Linusson. The number of M-sequences and f-vectors. Combinatorica, 19(2):255-266, 1999. URL: http://dx.doi.org/10.1007/s004930050055.
http://dx.doi.org/10.1007/s004930050055
R. Y. Liu. On a notion of data depth based on random simplices. Annals of Statistics, 18:405-414, 1990.
László Lovász, Katalin Vesztergombi, Uli Wagner, and Emo Welzl. Convex quadrilaterals and k-sets. In Towards a Theory of Geometric Graphs, pages 139-148. AMS, Providence, 2004.
Jiří Matoušek. Lectures on Discrete Geometry. Springer, 2002.
Juan José Montellano-Ballesteros and Ricardo Strausz. Counting polytopes via the Radon complex. J. Comb. Theory, Ser. A, 106(1):109-121, 2004. URL: http://dx.doi.org/10.1016/j.jcta.2004.01.005.
http://dx.doi.org/10.1016/j.jcta.2004.01.005
Theodore S. Motzkin. Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products. Bull. Amer. Math. Soc., 54(4):352-360, 1948.
Edgar M. Palmer and Robert W. Robinson. Enumeration of self-dual configurations. Pacific J. Math., 110(1):203-221, 1984.
Dana Randall, Günter Rote, Francisco Santos, and Jack Snoeyink. Counting triangulations and pseudo-triangulations of wheels. In Proc. 13superscriptth Canadian Conference on Computational Geometry, pages 149-152, 2001.
Peter J. Rousseeuw and Ida Ruts. Bivariate location depth. J. Royal Stat. Soc. Ser. C, 45(4):516-526, 1996.
Andres J. Ruiz-Vargas and Emo Welzl. Crossing-free perfect matchings in wheel point sets. Unpublished manuscript, September 2015.
Micha Sharir and Adam Sheffer. Counting triangulations of planar point sets. Electr. J. Combin., 18(1), 2011.
Micha Sharir, Adam Sheffer, and Emo Welzl. Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleyn’s technique. J. Comb. Theory Ser. A, 120(4):777-794, 2013. URL: http://dx.doi.org/10.1016/j.jcta.2013.01.002.
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Micha Sharir and Emo Welzl. On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput., 36(3):695-720, 2006. URL: http://dx.doi.org/10.1137/050636036.
http://dx.doi.org/10.1137/050636036
Uli Wagner. On the rectilinear crossing number of complete graphs. In Proc. 14superscriptth Annual Symposium on Discrete Algorithms, pages 583-588. ACM/SIAM, 2003.
Uli Wagner and Emo Welzl. A continuous analogue of the Upper Bound Theorem. Discr. Comput. Geom., 26(2):205-219, 2001. URL: http://dx.doi.org/10.1007/s00454-001-0028-9.
http://dx.doi.org/10.1007/s00454-001-0028-9
Emo Welzl. Entering and leaving j-facets. Discr. Comput. Geom., 25(3):351-364, 2001. URL: http://dx.doi.org/10.1007/s004540010085.
http://dx.doi.org/10.1007/s004540010085
Günter M. Ziegler. Lectures on Polytopes. Springer, 1995.
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https://creativecommons.org/licenses/by/3.0/legalcode
Approximate Range Counting Revisited
We study range-searching for colored objects, where one has to count (approximately) the number of colors present in a query range. The problems studied mostly involve orthogonal range-searching in two and three dimensions, and the dual setting of rectangle stabbing by points. We present optimal and near-optimal solutions for these problems. Most of the results are obtained via reductions to the approximate uncolored version, and improved data-structures for them. An additional contribution of this work is the introduction of nested shallow cuttings.
orthogonal range searching
rectangle stabbing
colors
approximate count
geometric data structures
55:1-55:15
Regular Paper
Saladi
Rahul
Saladi Rahul
10.4230/LIPIcs.SoCG.2017.55
Peyman Afshani and Timothy M. Chan. On approximate range counting and depth. Discrete & Computational Geometry, 42(1):3-21, 2009.
Peyman Afshani, Chris H. Hamilton, and Norbert Zeh. A general approach for cache-oblivious range reporting and approximate range counting. Computational Geometry: Theory and Applications, 43(8):700-712, 2010.
Boris Aronov and Sariel Har-Peled. On approximating the depth and related problems. SIAM Journal of Computing, 38(3):899-921, 2008.
Jon Louis Bentley and James B. Saxe. Decomposable searching problems I: Static-to-dynamic transformation. Journal of Algorithms, 1(4):301-358, 1980.
Timothy M. Chan and Bryan T. Wilkinson. Adaptive and approximate orthogonal range counting. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 241-251, 2013.
Bernard Chazelle. Filtering search: A new approach to query-answering. SIAM Journal of Computing, 15(3):703-724, 1986.
Michael L. Fredman and Dan E. Willard. Surpassing the information theoretic bound with fusion trees. Journal of Computer and System Sciences (JCSS), 47(3):424-436, 1993.
Prosenjit Gupta, Ravi Janardan, and Michiel H. M. Smid. Computational geometry: Generalized intersection searching. In Handbook of Data Structures and Applications. 2004.
Haim Kaplan, Edgar Ramos, and Micha Sharir. Range minima queries with respect to a random permutation, and approximate range counting. Discrete & Computational Geometry, 45(1):3-33, 2011.
Haim Kaplan, Natan Rubin, Micha Sharir, and Elad Verbin. Counting colors in boxes. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 785-794, 2007.
Yakov Nekrich. Efficient range searching for categorical and plain data. ACM Transactions on Database Systems (TODS), 39(1):9, 2014.
Qingmin Shi and Joseph JáJá. Optimal and near-optimal algorithms for generalized intersection reporting on pointer machines. Information Processing Letters (IPL), 95(3):382-388, 2005.
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https://creativecommons.org/licenses/by/3.0/legalcode
Coloring Curves That Cross a Fixed Curve
We prove that for every integer t greater than or equal to 1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is chi-bounded. This is essentially the strongest chi-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers k > 1 and t > 0, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has O(n log n) edges.
String graphs
chi-boundedness
k-quasi-planar graphs
56:1-56:15
Regular Paper
Alexandre
Rok
Alexandre Rok
Bartosz
Walczak
Bartosz Walczak
10.4230/LIPIcs.SoCG.2017.56
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Barcodes of Towers and a Streaming Algorithm for Persistent Homology
A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and can be efficiently computed by a streaming algorithm, both in theory and in practice. Furthermore, we show that our approach can be combined with a streaming algorithm to compute the barcode of the tower via matrix reduction. The space complexity of the algorithm does not depend on the length of the tower, but the maximal size of any subcomplex within the tower. Experimental evaluations show that our approach can efficiently handle towers with billions of complexes.
Persistent Homology
Topological Data Analysis
Matrix reduction
Streaming algorithms
Simplicial Approximation
57:1-57:16
Regular Paper
Michael
Kerber
Michael Kerber
Hannah
Schreiber
Hannah Schreiber
10.4230/LIPIcs.SoCG.2017.57
U. Bauer, M. Kerber, and J. Reininghaus. Clear and Compress: Computing Persistent Homology in Chunks. In Topological Methods in Data Analysis and Visualization III, Mathematics and Visualization, pages 103-117. Springer, 2014.
U. Bauer, M. Kerber, and J. Reininghaus. Distributed Computation of Persistent Homology. In Workshop on Algorithm Engineering and Experiments (ALENEX), pages 31-38, 2014.
U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner. Phat - Persistent Homology Algorithms Toolbox. Journal of Symbolic Computation, 78:76-90, 2017.
J.-D. Boissonnat, T. Dey, and C. Maria. The Compressed Annotation Matrix: An Efficient Data Structure for Computing Persistent Cohomology. In European Symp. on Algorithms (ESA), pages 695-706, 2013.
M. Botnan and G. Spreemann. Approximating Persistent Homology in Euclidean space through collapses. Applied Algebra in Engineering, Communication and Computing, 26:73-101, 2015.
G. Carlsson. Topology and Data. Bulletin of the AMS, 46:255-308, 2009.
G. Carlsson, V. de Silva, and D. Morozov. Zigzag Persistent Homology and Real-valued Functions. In ACM Symp. on Computational Geometry (SoCG), pages 247-256, 2009.
C. Chen and M. Kerber. Persistent Homology Computation With a Twist. In European Workshop on Computational Geometry (EuroCG), pages 197-200, 2011.
C. Chen and M. Kerber. An output-sensitive algorithm for persistent homology. Computational Geometry: Theory and Applications, 46:435-447, 2013.
A. Choudhary, M. Kerber, and S. Raghvendra. Polynomial-Sized Topological Approximations Using The Permutahedron. In 32nd Int. Symp. on Computational Geometry (SoCG), pages 31:1-31:16, 2016.
T. Cormen, C. Leiserson, R. Rivest, and C. Stein. Introduction to algorithms. The MIT press, 3rd edition, 2009.
V. de Silva, D. Morozov, and M. Vejdemo-Johansson. Dualities in persistent (co)homology. Inverse Problems, 27:124003, 2011.
T. Dey, F. Fan, and Y. Wang. Computing Topological Persistence for Simplicial Maps. In ACM Symp. on Computational Geometry (SoCG), pages 345-354, 2014.
T. Dey, D. Shi, and Y. Wang. SimBa: An efficient tool for approximating Rips-filtration persistence via Simplicial Batch-collapse. In European Symp. on Algorithms (ESA), pages 35:1-35:16, 2016.
H. Edelsbrunner and J. Harer. Computational Topology: an introduction. American Mathematical Society, 2010.
H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological Persistence and Simplification. Discrete &Computational Geometry, 28:511-533, 2002.
H. Edelsbrunner and S. Parsa. On the Computational Complexity of Betti Numbers: Reductions from Matrix Rank. In ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 152-160, 2014.
M. Kerber. Persistent Homology: State of the art and challenges. Internationale Mathematische Nachrichten, 231:15-33, 2016.
M. Kerber and H. Schreiber. Barcodes of Towers and a Streaming Algorithm for Persistent Homology. arXiv, abs/1701.02208, 2017. URL: http://arxiv.org/abs/1701.02208.
http://arxiv.org/abs/1701.02208
M. Kerber and R. Sharathkumar. Approximate Čech Complex in Low and High Dimensions. In Int. Symp. on Algortihms and Computation (ISAAC), pages 666-676, 2013.
C. Maria, J.-D. Boissonnat, M. Glisse, and M. Yvinec. The Gudhi Library: Simplicial Complexes and Persistent Homology. In Int. Congress on Mathematical Software (ICMS), volume 8592 of Lecture Notes in Computer Science, pages 167-174, 2014.
C. Maria and S. Oudot. Zigzag Persistence via Reflections and Transpositions. In ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 181-199, 2015.
N. Milosavljevic, D. Morozov, and P. Skraba. Zigzag persistent homology in matrix multiplication time. In ACM Symp. on Computational Geometry (SoCG), pages 216-225, 2011.
N. Otter, M. Porter, U. Tillmann, P. Grindrod, and H. Harrington. A roadmap for the computation of persistent homology. arXiv, abs/1506.08903, 2015.
S. Oudot. Persistence theory: From Quiver Representation to Data Analysis, volume 209 of Mathematical Surveys and Monographs. American Mathematical Society, 2015.
D. Sheehy. Linear-size approximation to the Vietoris-Rips Filtration. Discrete &Computational Geometry, 49:778-796, 2013.
A. Zomorodian and G. Carlsson. Computing Persistent Homology. Discrete & Computational Geometry, 33:249-274, 2005.
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Algorithmic Interpretations of Fractal Dimension
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces which agrees with standard notions used to empirically estimate the fractal dimension of various sets. We define the fractal dimension of some metric space to be the infimum delta>0, such that for any eps>0, for any ball B of radius r >= 2eps, and for any eps-net N, we have |B cap N|=O((r/eps)^delta).
Using this definition we obtain faster algorithms for a plethora of classical problems on sets of low fractal dimension in Euclidean space. Our results apply to exact and fixed-parameter algorithms, approximation schemes, and spanner constructions. Interestingly, the dependence of the performance of these algorithms on the fractal dimension nearly matches the currently best-known dependence on the standard Euclidean dimension. Thus, when the fractal dimension is strictly smaller than the ambient dimension, our results yield improved solutions in all of these settings.
We remark that our definition of fractal definition is equivalent up to constant factors to the well-studied notion of doubling dimension.
However, in the problems that we consider, the dimension appears in the exponent of the running time, and doubling dimension is not precise enough for capturing the best possible such exponent for subsets of Euclidean space. Thus our work is orthogonal to previous results on spaces of low doubling dimension; while algorithms on spaces of low doubling dimension seek to extend results from the case of low dimensional Euclidean spaces to more general metric spaces, our goal is to obtain faster algorithms for special pointsets in Euclidean space.
fractal dimension
exact algorithms
fixed parameter tractability
approximation schemes
spanners
58:1-58:16
Regular Paper
Anastasios
Sidiropoulos
Anastasios Sidiropoulos
Vijay
Sridhar
Vijay Sridhar
10.4230/LIPIcs.SoCG.2017.58
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.
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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.
T. H. Hubert Chan and Anupam Gupta. Small hop-diameter sparse spanners for doubling metrics. Discrete &Computational Geometry, 41(1):28-44, 2009.
T. H. Hubert Chan and Anupam Gupta. Approximating tsp on metrics with bounded global growth. SIAM Journal on Computing, 41(3):587-617, 2012.
T. H. Hubert 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.
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.
Kenneth Falconer. Fractal geometry: mathematical foundations and applications. John Wiley &Sons, 2004.
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, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Foundations of Computer Science, 2003. Proceedings. 44th Annual IEEE Symposium on, pages 534-543. IEEE, 2003.
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. Geometric approximation algorithms, volume 173. American Mathematical Society, Providence, 2011.
Sariel Har-Peled. A simple proof of the existence of a planar separator. arXiv preprint arXiv:1105.0103, 2011.
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.
Juha Heinonen. Lectures on analysis on metric spaces. Springer Science &Business Media, 2012.
Dorit S. Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and VLSI. Journal of the ACM (JACM), 32(1):130-136, 1985.
David R. Karger and Matthias Ruhl. Finding nearest neighbors in growth-restricted metrics. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 741-750. ACM, 2002.
Marc Khoury and Rephael Wenger. On the fractal dimension of isosurfaces. IEEE Transactions on Visualization and Computer Graphics, 16(6):1198-1205, 2010.
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 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, page 67. ACM, 2014.
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.
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.
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C. T. Zahn. Black box maximization of circular coverage. Journal of Research of the National Bureau of Standards B, 66:181-216, 1962.
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Disjointness Graphs of Segments
The disjointness graph G=G(S) of a set of segments S in R^d, d>1 is a graph whose vertex set is S and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies chi(G)<=omega(G)^4+omega(G)^3 where omega(G) denotes the clique number of G. It follows, that S has at least cn^{1/5} pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments.
We show that computing omega(G) and chi(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of G in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (omega(G)=2), but whose chromatic numbers are arbitrarily large.
disjointness graph
chromatic number
clique number
chi-bounded
59:1-59:15
Regular Paper
János
Pach
János Pach
Gábor
Tardos
Gábor Tardos
Géza
Tóth
Géza Tóth
10.4230/LIPIcs.SoCG.2017.59
Paola Alimonti and Viggo Kann. Some APX-completeness results for cubic graphs. Theoretical Computer Science, 237(1-2):123-134, 2000.
Edgar Asplund and Branko Grünbaum. On a coloring problem. Mathematica Scandinavica, 8(1):181-188, 1960.
Claude Berge. Färbung von Graphen, deren sämtliche bzw. ungerade Kreise starr sind (Zusammenfassung). Wiss. Z. Martin-Luther-Univ. Halle Wittenberg Math. Natur. Reihe, 114, 1961.
Béla Bollobás. Modern Graph Theory, Graduate Texts in Mathematics vol. 184. Springer-Verlag, New York, 1998.
James P. Burling. On coloring problems of families of prototypes. (PhD thesis), University of Colorado, Boulder, 1965.
Sergio Cabello, Jean Cardinal, and Stefan Langerman. The clique problem in ray intersection graphs. Discrete &computational geometry, 50(3):771-783, 2013.
Robert P. Dilworth. A decomposition theorem for partially ordered sets. Annals of Mathematics, pages 161-166, 1950.
Gabriel Andrew Dirac. On rigid circuit graphs. In Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, volume 25, pages 71-76. Springer, 1961.
Gideon Ehrlich, Shimon Even, and Robert Endre Tarjan. Intersection graphs of curves in the plane. Journal of Combinatorial Theory, Series B, 21(1):8-20, 1976.
Paul Erdős and András Hajnal. Some remarks on set theory. ix: Combinatorial problems in measure theory and set theory. Michigan Math. J, 11(2):107-127, 1964.
Hillel Furstenberg and Yitzhak Katznelson. A density version of the hales-jewett theorem. Journal d’Analyse Mathématique, 57(1):64-119, 1991.
Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin, 1988.
András Gyárfás. On the chromatic number of multiple interval graphs and overlap graphs. Discrete mathematics, 55(2):161-166, 1985.
András Gyárfás. Corrigendum. Discrete mathematics, 62(3):333, 1986.
András Gyárfás. Problems from the world surrounding perfect graphs. Applicationes Mathematicae, 19(3-4):413-441, 1987.
András Gyárfás and Jenő Lehel. Hypergraph families with bounded edge cover or transversal number. Combinatorica, 3(3-4):351-358, 1983.
András Gyárfás and Jenő Lehel. Covering and coloring problems for relatives of intervals. Discrete Mathematics, 55(2):167-180, 1985.
András Hajnal and János Surányi. Über die auflösung von graphen in vollständige teilgraphen. Ann. Univ. Sci. Budapest, Eötvös Sect. Math, 1:113-121, 1958.
György Hajós. Über eine Art von Graphen. Internationale Mathematische Nachrichten, 11:65, 1957.
Ian Holyer. The NP-completeness of edge-coloring. SIAM Journal on Computing, 10(4):718-720, 1981.
Gyula Károlyi. On point covers of parallel rectangles. Periodica Mathematica Hungarica, 23:105-107, 1991.
Gyula Károlyi, János Pach, and Géza Tóth. Ramsey-type results for geometric graphs, i. Discrete &Computational Geometry, 18(3):247-255, 1997.
Alexandr Kostochka. Coloring intersection graphs of geometric figures with a given clique number. Contemporary Mathematics, 342:127-138, 2004.
Alexandr Kostochka and Jan Kratochvíl. Covering and coloring polygon-circle graphs. Discrete Mathematics, 163(1):299-305, 1997.
Alexandr V. Kostochka. Upper bounds for the chromatic numbers of graphs. Modeli i Metody Optim. (Russian), 10:204-226, 1988.
Jan Kratochvíl and Jaroslav Nešetřil. Independent set and clique problems in intersection-defined classes of graphs. Commentationes Mathematicae Universitatis Carolinae, 31(1):85-93, 1990.
Jan Kynčl. Ramsey-type constructions for arrangements of segments. European Journal of Combinatorics, 33(3):336-339, 2012.
David Larman, Jiří Matoušek, János Pach, and Jenő Törőcsik. A Ramsey-type result for convex sets. Bulletin of the London Mathematical Society, 26(2):132-136, 1994.
László Lovász. Normal hypergraphs and the perfect graph conjecture. Discrete Mathematics, 2(3):253-267, 1972.
László Lovász. Kneser’s conjecture, chromatic number, and homotopy. Journal of Combinatorial Theory, Series A, 25(3):319-324, 1978.
László Lovász. Combinatorial problems and exercises. American Mathematical Soc., 1993.
Torsten Mütze, Bartosz Walczak, and Veit Wiechert. Realization of shift graphs as disjointness graphs of 1-intersecting curves in the plane. Manuscript, 2017.
Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Michał Lasoń, Piotr Micek, William T. Trotter, and Bartosz Walczak. Triangle-free intersection graphs of line segments with large chromatic number. Journal of Combinatorial Theory, Series B, 105:6-10, 2014.
Svatopluk Poljak. A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae, 15(2):307-309, 1974.
Noam Solomon, Michael S. Payne, and Jean Cardinal. Ramsey-type theorems for lines in 3-space. Discrete Mathematics &Theoretical Computer Science, 18, 2016.
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Bicriteria Rectilinear Shortest Paths among Rectilinear Obstacles in the Plane
Given a rectilinear domain P of h pairwise-disjoint rectilinear obstacles with a total of n vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points s and t in P. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when h is relatively small. For example, for computing a minimum-link shortest s-t path, the previous algorithm runs in O(n log^{3/2} n) time while the time of our new algorithm is O(n + h log^{3/2} h).
rectilinear paths
shortest paths
minimum-link paths
bicriteria paths
rectilinear polygons
60:1-60:16
Regular Paper
Haitao
Wang
Haitao Wang
10.4230/LIPIcs.SoCG.2017.60
R. Bar-Yehuda and B. Chazelle. Triangulating disjoint Jordan chains. International Journal of Computational Geometry and Applications, 4(4):475-481, 1994.
D. Z. Chen, O. Daescu, and K. S. Klenk. On geometric path query problems. International Journal of Computational Geometry and Applications, 11(6):617-645, 2001.
D. Z. Chen, R. Inkulu, and H. Wang. Two-point L₁ shortest path queries in the plane. In Proc. of the 30th Annual Symposium on Computational Geometry, pages 406-415, 2014.
D. Z. Chen, K. S. Klenk, and H.-Y. T. Tu. Shortest path queries among weighted obstacles in the rectilinear plane. SIAM Journal on Computing, 29(4):1223-1246, 2000.
D. Z. Chen and H. Wang. A nearly optimal algorithm for finding L₁ shortest paths among polygonal obstacles in the plane. In Proc. of the 19th European Symposium on Algorithms, pages 481-492, 2011.
D. Z. Chen and H. Wang. L₁ shortest path queries among polygonal obstacles in the plane. In Proc. of 30th Symp. on Theoretical Aspects of Computer Science, pages 293-304, 2013.
K. Clarkson, S. Kapoor, and P. Vaidya. Rectilinear shortest paths through polygonal obstacles in O(n log² n) time. In Proc. of the 3rd Annual Symposium on Computational Geometry, pages 251-257, 1987.
K. Clarkson, S. Kapoor, and P. Vaidya. Rectilinear shortest paths through polygonal obstacles in O(n log^2/3 n) time. Manuscript, 1988.
G. Das and G. Narasimhan. Geometric searching and link distance. In Proc. of the 2nd Workshop of Algorithms and Data Structures, pages 261-272, 1991.
M. de Berg. On rectilinear link distance. Computational Geometry: Theory and Applications, 1:13-34, 1991.
J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Computational Geometry: Theory and Applications, 4(2):63-97, 1994.
H. Imai and T. Asano. Efficient algorithms for geometric graph search problems. SIAM Journal on Computing, 15(2):478-494, 1986.
D. T. Lee, C. D. Yang, and T. H. Chen. Shortest rectilinear paths among weighted obstacles. International Journal of Computational Geometry and Applications, 1(2):109-124, 1991.
J. S. B. Mitchell. An optimal algorithm for shortest rectilinear paths among obstacles. Abstracts of the 1st Canadian Conference on Computational Geometry, 1989.
J. S. B. Mitchell. L₁ shortest paths among polygonal obstacles in the plane. Algorithmica, 8(1):55-88, 1992.
J. S. B. Mitchell, V. Polishchuk, and M. Sysikaski. Minimum-link paths revisited. CGTA, 47:651-667, 2014.
J. S. B. Mitchell, V. Polishchuk, M. Sysikaski, and H. Wang. An optimal algorithm for minimum-link rectilinear paths in triangulated rectilinear domains. In Proc. of the 42nd International Colloquium on Automata, Languages and Programming, pages 947-959, 2015.
J. S. B. Mitchell, G. Rote, and G. Woeginger. Minimum-link paths among obstacles in the plane. Algorithmica, 8:431-459, 1992.
V. Polishchuk and J. S. B. Mitchell. k-Link rectilinear shortest paths among rectilinear obstacles in the plane. In Proc. of the 17th Canadian Conference on Computational Geometry (CCCG), pages 101-104, 2005.
M. Sato, J. Sakanaka, and T. Ohtsuki. A fast line-search method based on a tile plane. In Proc. of the IEEE International Symposium on Circuits and Systems, pages 588-597, 1987.
S. Schuierer. An optimal data structure for shortest rectilinear path queries in a simple rectilinear polygon. International Journal of Computational Geometry and Applications, 6:205-226, 1996.
H. Wang. Bicriteria rectilinear shortest paths among rectilinear obstacles in the plane. arXiv:1703.04466, 2017.
Y.-F. Wu, P. Widmayer, M. D. F. Schlag, and C. K. Wong. Rectilinear shortest paths and minimum spanning trees in the presence of rectilinear obstacles. IEEE Transactions on Computers, 36:321-331, 1987.
C. D. Yang, D. T. Lee, and C. K. Wong. On bends and lengths of rectilinear paths: A graph-theoretic approach. Int. J. Comput. Geom. Appl., 02:61-74, 1992.
C. D. Yang, D. T. Lee, and C. K. Wong. Rectilinear path problems among rectilinear obstacles revisited. SIAM Journal on Computing, 24:457-472, 1995.
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Quickest Visibility Queries in Polygonal Domains
Let s be a point in a polygonal domain P of h-1 holes and n vertices. We consider the following quickest visibility query problem. Given a query point q in P, the goal is to find a shortest path in P to move from s to see q as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size O(n^2 2^alpha(n) log n) that can answer each query in O(K log^2 n) time, where alpha(n) is the inverse Ackermann function and K is the size of the visibility polygon of q in P (and K can be Theta(n) in the worst case). In this paper, we present a new data structure of size O(n log h + h^2) that can answer each query in O(h log h log n) time. Our result improves the previous work when h is relatively small. In particular, if h is a constant, then our result even matches the best result for the simple polygon case (i.e., h = 1), which is optimal. As a by-product, we also have a new algorithm for the following shortest-path-to-segment query problem. Given a query line segment tau in P, the query seeks a shortest path from s to all points of tau. Previously, Arkin et al. gave a data structure of size O(n^2 2^alpha(n) log n) that can answer each query in O(log^2 n) time, and another data structure of size O(n^3 log n) with O(log n) query time. We present a data structure of size O(n) with query time O(h log n/h), which favors small values of h and is optimal when h = O(1).
shortest paths
visibility
quickest visibility queries
shortest path to segments
polygons with holes
61:1-61:16
Regular Paper
Haitao
Wang
Haitao Wang
10.4230/LIPIcs.SoCG.2017.61
E. M. Arkin, A. Efrat, C. Knauer, J. S. B. Mitchell, V. Polishchuk, G. Rote, L. Schlipf, and T. Talvitie. Shortest path to a segment and quickest visibility queries. Journal of Computational Geometry, 7:77-100, 2016.
B. Chazelle, H. Edelsbrunner, M. Grigni, L. Guibas, J. Hershberger, M. Sharir, and J. Snoeyink. Ray shooting in polygons using geodesic triangulations. Algorithmica, 12(1):54-68, 1994.
D. Z. Chen and H. Wang. L₁ shortest path queries among polygonal obstacles in the plane. In Proc. of 30th Symposium on Theoretical Aspects of Computer Science, pages 293-304, 2013.
D. Z. Chen and H. Wang. Visibility and ray shooting queries in polygonal domains. Computational Geometry: Theory and Applications, 48:31-41, 2015.
D. Z. Chen and H. Wang. Computing the visibility polygon of an island in a polygonal domain. Algorithmica, 77:40-64, 2017.
Y. K. Cheung and O. Daescu. Approximate point-to-face shortest paths in ℛ³. arXiv:1004.1588, 2010.
Y.-J. Chiang and R. Tamassia. Optimal shortest path and minimum-link path queries between two convex polygons in the presence of obstacles. International Journal of Computational Geometry and Applications, 7:85-121, 1997.
L. J. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987.
J. Hershberger and S. Suri. A pedestrian approach to ray shooting: Shoot a ray, take a walk. Journal of Algorithms, 18(3):403-431, 1995.
J. Hershberger and S. Suri. An optimal algorithm for Euclidean shortest paths in the plane. SIAM Journal on Computing, 28(6):2215-2256, 1999.
R. Khosravi and M. Ghodsi. The fastest way to view a query point in simple polygons. In Proc. of the 24th European Workshop on Computational Geometry, pages 187-190, 2005.
E. Melissaratos and D. Souvaine. Shortest paths help solve geometric optimization problems in planar regions. SIAM Journal on Computing, 21(4):601-638, 1992.
J. S. B. Mitchell. A new algorithm for shortest paths among obstacles in the plane. Annals of Mathematics and Artificial Intelligence, 3(1):83-105, 1991.
J. S. B. Mitchell. Shortest paths among obstacles in the plane. International Journal of Computational Geometry and Applications, 6(3):309-332, 1996.
H. Wang. Quickest visibility queries in polygonal domains. arXiv:1703.03048, 2017.
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https://creativecommons.org/licenses/by/3.0/legalcode
Zapping Zika with a Mosquito-Managing Drone: Computing Optimal Flight Patterns with Minimum Turn Cost (Multimedia Contribution)
We present results arising from the problem of sweeping a mosquito-infested area with an Un-manned Aerial Vehicle (UAV) equipped with an electrified metal grid. This is related to the Traveling Salesman Problem, the Lawn Mower Problem and, most closely, Milling with TurnCost. Planning a good trajectory can be reduced to considering penalty and budget variants of covering a grid graph with minimum turn cost. On the theoretical side, we show the solution of a problem from The Open Problems Project that had been open for more than 15 years, and hint at approximation algorithms. On the practical side, we describe an exact method based on Integer Programming that is able to compute provably optimal instances with over 500 pixels. These solutions are actually used for practical trajectories, as demonstrated in the video.
Covering tours
turn cost
complexity
exact optimization
62:1-62:5
Multimedia Contribution
Aaron T.
Becker
Aaron T. Becker
Mustapha
Debboun
Mustapha Debboun
Sándor P.
Fekete
Sándor P. Fekete
Dominik
Krupke
Dominik Krupke
An
Nguyen
An Nguyen
10.4230/LIPIcs.SoCG.2017.62
Alok Aggarwal, Don Coppersmith, Sanjeev Khanna, Rajeev Motwani, and Baruch Schieber. The angular-metric Traveling Salesman Problem. SIAM Journal on Computing, 29(3):697-711, 2000.
David L. Applegate, Robert E. Bixby, Vašek Chvátal, and William J. Cook. The Traveling Salesman Problem: A computational study. Princeton University Press, 2011.
Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Sándor P. Fekete, Joseph S. B. Mitchell, and Saurabh Sethia. Optimal covering tours with turn costs. In Proc. 12th Ann. ACM-SIAM Symp. Disc. Algorithms (SODA 2001), pages 138-147. SIAM, 2001.
Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Sándor P. Fekete, Joseph S. B. Mitchell, and Saurabh Sethia. Optimal covering tours with turn costs. SIAM Journal on Computing, 35(3):531-566, 2005.
Esther M. Arkin, Sándor P. Fekete, and Joseph S. B. Mitchell. The lawnmower problem. In Proc. 5th Canad. Conf. Sympos. Geom. (CCCG93), pages 461-466, 1993.
Esther M. Arkin, Sándor P. Fekete, and Joseph S. B. Mitchell. Approximation algorithms for lawn mowing and milling. Computational Geometry, 17(1):25-50, 2000.
William Cook. In pursuit of the traveling salesman: Mathematics at the limits of computation. Princeton University Press, 2012.
Erik D. Demaine, Joseph S. B. Mitchell, and O'Rourke Joseph. The open problems project. URL: http://cs.smith.edu/~orourke/TOPP/.
http://cs.smith.edu/~orourke/TOPP/
Sándor P. Fekete and Dominik Krupke. Covering tours and cycle covers with turn costs: Hardness and approximation. Manuscript, 2017.
Sándor P. Fekete and Gerhard J. Woeginger. Angle-restricted tours in the plane. Computational Geometry, 8(4):195-218, 1997.
Dominik Krupke. Algorithmic methods for complex dynamic sweeping problems. Master’s thesis, Department of Computer Science, TU Braunschweig, Braunschweig, Germany, 2016.
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Ruler of the Plane - Games of Geometry (Multimedia Contribution)
Ruler of the Plane is a set of games illustrating concepts from combinatorial and computational geometry. The games are based on the art gallery problem, ham-sandwich cuts, the Voronoi game, and geometric network connectivity problems like the Euclidean minimum spanning tree and traveling salesperson problem.
art gallery problem
ham-sandwich cuts
Voronoi game
traveling sales-person problem
63:1-63:5
Multimedia Contribution
Sander
Beekhuis
Sander Beekhuis
Kevin
Buchin
Kevin Buchin
Thom
Castermans
Thom Castermans
Thom
Hurks
Thom Hurks
Willem
Sonke
Willem Sonke
10.4230/LIPIcs.SoCG.2017.63
Hee-Kap Ahn, Siu-Wing Cheng, Otfried Cheong, Mordecai Golin, and Rene Van Oostrum. Competitive facility location: the Voronoi game. Theoretical Computer Science, 310(1-3):457-467, 2004.
Martin Aigner and Günter M. Ziegler. Proofs from THE BOOK. Springer, 4th edition, 2009.
Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geom. Appl., 5:75-91, 1995.
Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete &Computational Geometry, 9(1):81-100, 1993.
Yoav Amit, Joseph S. B. Mitchell, and Eli Packer. Locating guards for visibility coverage of polygons. Int. J. Comput. Geom. Appl., 20:601-630, 2010.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer, 3rd edition, 2008.
Nicos Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, 1976.
Der-Tsai Lee and Arthur K. Lin. Computational complexity of art gallery problems. IEEE Transactions on Information Theory, 32(2):276-282, 1986.
Chi-Yuan Lo, Jiří Matoušek, and William Steiger. Algorithms for ham-sandwich cuts. Discrete &Computational Geometry, 11(4):433-452, 1994.
Matthias Müller-Hannemann and Stefan Schirra, editors. Algorithm engineering: bridging the gap between algorithm theory and practice, volume 5971 of LNCS. Springer, 2010.
Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, New York, NY, USA, 2007.
Jonathan Richard Shewchuk. Lecture notes on geometric robustness, 2013.
Kevin Weiler and Peter Atherton. Hidden surface removal using polygon area sorting. ACM SIGGRAPH computer graphics, 11(2):214-222, 1977.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Folding Free-Space Diagrams: Computing the Fréchet Distance between 1-Dimensional Curves (Multimedia Contribution)
By folding the free-space diagram for efficient preprocessing, we show that the Frechet distance between 1D curves can be computed in O(nk log n) time, assuming one curve has ply k.
Frechet distance
ply
k-packed curves
64:1-64:5
Multimedia Contribution
Kevin
Buchin
Kevin Buchin
Jinhee
Chun
Jinhee Chun
Maarten
Löffler
Maarten Löffler
Aleksandar
Markovic
Aleksandar Markovic
Wouter
Meulemans
Wouter Meulemans
Yoshio
Okamoto
Yoshio Okamoto
Taichi
Shiitada
Taichi Shiitada
10.4230/LIPIcs.SoCG.2017.64
H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves. IJCGA, 5(1-2):78-99, 1995.
K. Bringmann. Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless SETH fails. In Proc. 55th FOCS, pages 661-670, 2014.
K. Bringmann and M. Künnemann. Improved approximation for Fréchet distance on c-packed curves matching conditional lower bounds. In Proc. 26th ISAAC, pages 517-528, 2015.
K. Bringmann and W. Mulzer. Approximability of the discrete fréchet distance. JoCG, 7(2):46-76, 2016.
K. Buchin, M. Buchin, W. Meulemans, and W. Mulzer. Four Soviets walk the dog: improved bounds for computing the Fréchet distance. DCG, 2017.
K. Buchin, M. Buchin, R. van Leusden, W. Meulemans, and W. Mulzer. Computing the Fréchet distance with a retractable leash. DCG, 56(2):315-336, 2016.
A. Driemel, S. Har-Peled, and C. Wenk. Approximating the Fréchet distance for realistic curves in near linear time. In Proc. 26th SoCG, pages 365-374, 2010.
Creative Commons Attribution 3.0 Unported license
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Cardiac Trabeculae Segmentation: an Application of Computational Topology (Multimedia Contribution)
In this video, we present a research project on cardiac trabeculae segmentation. Trabeculae are fine muscle columns within human ventricles whose both ends are attached to the wall. Extracting these structures are very challenging even with state-of-the-art image segmentation techniques. We observed that these structures form natural topological handles. Based on such observation, we developed a topological approach, which employs advanced computational topology methods and achieve high quality segmentation results.
image segmentation
trabeculae
persistent homology
homology localization
65:1-65:4
Multimedia Contribution
Chao
Chen
Chao Chen
Dimitris
Metaxas
Dimitris Metaxas
Yusu
Wang
Yusu Wang
Pengxiang
Wu
Pengxiang Wu
10.4230/LIPIcs.SoCG.2017.65
Yuri Boykov, Olga Veksler, and Ramin Zabih. Fast approximate energy minimization via graph cuts. IEEE Transactions on pattern analysis and machine intelligence, 23(11):1222-1239, 2001.
Oleksiy Busaryev, Sergio Cabello, Chao Chen, Tamal K. Dey, and Yusu Wang. Annotating simplices with a homology basis and its applications. In Scandinavian Workshop on Algorithm Theory, pages 189-200. Springer Berlin Heidelberg, 2012.
Erin W. Chambers, Jeff Erickson, and Amir Nayyeri. Minimum cuts and shortest homologous cycles. In Proceedings of the 25th Annual Symposium on Computational Geometry, pages 377-385. ACM, 2009.
Chao Chen and Daniel Freedman. Hardness results for homology localization. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1594-1604, 2010.
Chao Chen and Daniel Freedman. Measuring and computing natural generators for homology groups. Computational Geometry, 43(2):169-181, 2010.
H. Edelsbrunner and J. Harer. Computational topology: an introduction. American Mathematical Society, 2010.
Jr. Edwin P. Ewing. Gross pathology of idiopathic cardiomyopathy - Wikipedia, the free encyclopedia, 2016. [Online; accessed 09-December-2016].
Jeff Erickson and Amir Nayyeri. Minimum cuts and shortest non-separating cycles via homology covers. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1166-1176. Society for Industrial and Applied Mathematics, 2011.
Mingchen Gao, Chao Chen, Shaoting Zhang, Zhen Qian, Dimitris Metaxas, and Leon Axel. Segmenting the papillary muscles and the trabeculae from high resolution cardiac ct through restoration of topological handles. In Information Processing in Medical Imaging (IPMI), 2013.
Scott Kulp, Mingchen Gao, Shaoting Zhang, Zhen Qian, Szilard Voros, Dimitris Metaxas, and Leon Axel. Using high resolution cardiac CT data to model and visualize patient-specific interactions between trabeculae and blood flow. In MICCAI, LNCS, pages 468-475. 2011. URL: http://dx.doi.org/10.1007/978-3-642-23623-5_59.
http://dx.doi.org/10.1007/978-3-642-23623-5_59
Pengxiang Wu, Chao Chen, Yusu Wang, Shaoting Zhang, Changhe Yuan, Zhen Qian, Dimitris Metaxas, and Leon Axel. Optimal topological cycles and their application in cardiac trabeculae restoration. In Information Processing in Medical Imaging (IPMI), 2017.
Xiantong Zhen, Heye Zhang, Ali Islam, Mousumi Bhaduri, Ian Chan, and Shuo Li. Direct and simultaneous estimation of cardiac four chamber volumes by multioutput sparse regression. Medical Image Analysis, 2016.
Yefeng Zheng, A. Barbu, B. Georgescu, M. Scheuering, and D. Comaniciu. Four-chamber heart modeling and automatic segmentation for 3D cardiac CT volumes using marginal space learning and steerable features. TMI, 27(11):1668-1681, nov. 2008.
S. C. Zhu, T. S. Lee, and A. L. Yuille. Region competition: unifying snakes, region growing, energy/Bayes/MDL for multi-band image segmentation. In ICCV, pages 416-423, June 1995.
Creative Commons Attribution 3.0 Unported license
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MatchTheNet - An Educational Game on 3-Dimensional Polytopes (Multimedia Contribution)
We present an interactive game which challenges a single player to match 3-dimensional polytopes to their planar nets. It is open source, and it runs in standard web browsers.
three-dimensional convex polytopes
unfoldings
66:1-66:5
Multimedia Contribution
Michael
Joswig
Michael Joswig
Georg
Loho
Georg Loho
Benjamin
Lorenz
Benjamin Lorenz
Rico
Raber
Rico Raber
10.4230/LIPIcs.SoCG.2017.66
Marshall Bern, Erik D. Demaine, David Eppstein, Eric Kuo, Andrea Mantler, and Jack Snoeyink. Ununfoldable polyhedra with convex faces. Comput. Geom., 24(2):51-62, 2003. Special issue on the Fourth CGC Workshop on Computational Geometry (Baltimore, MD, 1999). URL: http://dx.doi.org/10.1016/S0925-7721(02)00091-3.
http://dx.doi.org/10.1016/S0925-7721(02)00091-3
Erik D. Demaine and Joseph O'Rourke. Geometric folding algorithms. Cambridge University Press, Cambridge, 2007. Linkages, origami, polyhedra. URL: http://dx.doi.org/10.1017/CBO9780511735172.
http://dx.doi.org/10.1017/CBO9780511735172
Ewgenij Gawrilow and Michael Joswig. polymake: a framework for analyzing convex polytopes. In Polytopes - combinatorics and computation (Oberwolfach, 1997), volume 29 of DMV Sem., pages 43-73. Birkhäuser, Basel, 2000.
Norman W. Johnson. Convex polyhedra with regular faces. Canad. J. Math., 18:169-200, 1966. URL: http://dx.doi.org/10.4153/CJM-1966-021-8.
http://dx.doi.org/10.4153/CJM-1966-021-8
Geoffrey C. Shephard. Convex polytopes with convex nets. Math. Proc. Cambridge Philos. Soc., 78(3):389-403, 1975. URL: http://dx.doi.org/10.1017/S0305004100051860.
http://dx.doi.org/10.1017/S0305004100051860
Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. URL: http://dx.doi.org/10.1007/978-1-4613-8431-1.
http://dx.doi.org/10.1007/978-1-4613-8431-1
Creative Commons Attribution 3.0 Unported license
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On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution)
Hilbert geometry is a metric geometry that extends the hyperbolic Cayley-Klein geometry. In this video, we explain the shape of balls and their properties in a convex polygonal Hilbert geometry. First, we study the combinatorial properties of Hilbert balls, showing that the shapes of Hilbert polygonal balls depend both on the center location and on the complexity of the Hilbert domain but not on their radii. We give an explicit description of the Hilbert ball for any given center and radius. We then study the intersection of two Hilbert balls. In particular, we consider the cases of empty intersection and internal/external tangencies.
Projective geometry
Hilbert geometry
balls
67:1-67:4
Multimedia Contribution
Frank
Nielsen
Frank Nielsen
Laetitia
Shao
Laetitia Shao
10.4230/LIPIcs.SoCG.2017.67
Curtis T. McMullen. Coxeter groups, Salem numbers and the Hilbert metric. Publications mathématiques de l'IHÉS, 95:151-183, 2002.
Frank Nielsen, Boris Muzellec, and Richard Nock. Classification with mixtures of curved Mahalanobis metrics. In IEEE International Conference on Image Processing (ICIP), pages 241-245. IEEE, 2016.
Frank Nielsen, Boris Muzellec, and Richard Nock. Large margin nearest neighbor classification using curved Mahalanobis distances. CoRR, abs/1609.07082, 2016. URL: http://arxiv.org/abs/1609.07082.
http://arxiv.org/abs/1609.07082
Frank Nielsen and Ke Sun. Clustering in Hilbert simplex geometry. ArXiv 1704.00454, April 2017.
Jürgen Richter-Gebert. Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry. Springer Publishing Company, Incorporated, 1st edition, 2011.
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