32nd International Symposium on Computational Geometry (SoCG 2016), SoCG 2016, June 14-18, 2016, Boston, USA
SoCG 2016
June 14-18, 2016
Boston, USA
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
Sándor
Fekete
Sándor Fekete
Anna
Lubiw
Anna Lubiw
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
51
2016
978-3-95977-009-5
https://www.dagstuhl.de/dagpub/978-3-95977-009-5
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
Sándor
Fekete
Sándor Fekete
Anna
Lubiw
Anna Lubiw
10.4230/LIPIcs.SoCG.2016.0
Creative Commons Attribution 3.0 Unported license
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Toward Pervasive Robots (Invited Talk)
The digitization of practically everything coupled with the mobile Internet, the automation of knowledge work, and advanced robotics promises a future with democratized use of machines and wide-spread use of robots and customization. However, pervasive use of robots remains a hard problem. Where are the gaps that we need to address in order to advance toward a future where robots are common in the world and they help reliably with physical tasks? What is the role of geometric reasoning along this trajectory?
In this talk I will discuss challenges toward pervasive use of robots and recent developments in geometric algorithms for customizing robots. I will focus on a suite of gemetric algorithms for automatically designing, fabricating, and tasking robots using a print-and-fold approach. I will also describe how geometric reasoning can play a role in creating robots more capable of reasoning in the world. By enabling on-demand creation of programmable robots, we can begin to imagine a world with one robot for every physical task.
rus@csail.mit.edu
1:1-1:1
Invited Talk
Daniela
Rus
Daniela Rus
10.4230/LIPIcs.SoCG.2016.1
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Discrete Geometry, Algebra, and Combinatorics (Invited Talk)
Many problems in discrete and computational geometry can be viewed as finding patterns in graphs or hypergraphs which arise from geometry or algebra. Famous Ramsey, Turán, and Szemerédi-type results prove the existence of certain patterns in graphs and hypergraphs under mild assumptions. We survey recent results which show much stronger/larger patterns for graphs and hypergraphs that arise from geometry or algebra. We further discuss whether the stronger results in these settings are due to geometric, algebraic, combinatorial, or topological properties of the graphs.
discrete geometry
extremal combinatorics
regularity lemmas
Ramsey theory
2:1-2:1
Invited Talk
Jacob
Fox
Jacob Fox
10.4230/LIPIcs.SoCG.2016.2
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Who Needs Crossings? Hardness of Plane Graph Rigidity
We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the global rigidity case, edge lengths in {1,2}). We show that all nine of these questions are complete for the class Exists-R, defined by the Existential Theory of the Reals, or its complement Forall-R; in particular, each problem is (co)NP-hard.
One of these nine results - that realization of unit-distance graphs is Exists-R-complete - was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class Exists-R. Global rigidity of graphs with edge lengths in {1,2} was known to be coNP-hard (Saxe 1979); we show it is Forall-R-complete.
The majority of the paper is devoted to proving an analog of Kempe's Universality Theorem - informally, "there is a linkage to sign your name" - for globally noncrossing linkages. In particular, we show that any polynomial curve phi(x,y)=0 can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the nontrivial regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions. Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well.
Graph Drawing
Graph Rigidity Theory
Graph Global Rigidity
Linkages
Complexity Theory
Computational Geometry
3:1-3:15
Regular Paper
Zachary
Abel
Zachary Abel
Erik D.
Demaine
Erik D. Demaine
Martin L.
Demaine
Martin L. Demaine
Sarah
Eisenstat
Sarah Eisenstat
Jayson
Lynch
Jayson Lynch
Tao B.
Schardl
Tao B. Schardl
10.4230/LIPIcs.SoCG.2016.3
Timothy Good Abbott. Generalizations of Kempe’s Universality Theorem. Master’s thesis, Massachusetts Institute of Technology, June 2008. Joint work with Reid W. Barton and Erik D. Demaine. URL: http://web.mit.edu/tabbott/www/papers/mthesis.pdf.
http://web.mit.edu/tabbott/www/papers/mthesis.pdf
Kenneth Appel and Wolfgang Haken. Every planar map is four colorable, Part I: Discharging. Illinois Journal of Mathematics, 21:429-490, 1977.
Sergio Cabello, Erik D. Demaine, and Günter Rote. Planar embeddings of graphs with specified edge lengths. Journal of Graph Algorithms and Applications, 11(1):259-276, 2007.
John Canny. Some algebraic and geometric computations in PSPACE. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, pages 460-469, Chicago, Illinois, May 1988. URL: http://www.acm.org/pubs/citations/proceedings/stoc/62212/p460-canny/.
http://www.acm.org/pubs/citations/proceedings/stoc/62212/p460-canny/
Erik D. Demaine and Joseph O'Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, July 2007.
Peter Eades and Nicholas C. Wormald. Fixed edge-length graph drawing is NP-hard. Discrete Applied Mathematics, 28(2):111-134, August 1990. URL: http://dx.doi.org/10.1016/0166-218X(90)90110-X.
http://dx.doi.org/10.1016/0166-218X(90)90110-X
D. Jordan and M. Steiner. Configuration spaces of mechanical linkages. Discrete &Computational Geometry, 22:297-315, 1999.
Michael Kapovich and John J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41(6):1051-1107, 2002. URL: http://arxiv.org/abs/math.AG/9803150.
http://arxiv.org/abs/math.AG/9803150
A. B. Kempe. On a general method of describing plane curves of the n^th degree by linkwork. Proceedings of the London Mathematical Society, 7:213-216, 1876.
A. B. Kempe. How to Draw a Straight Line: A Lecture on Linkages. Macmillan and co., London, 1877.
A. B. Kempe. On the geographical problem of the four colours. American Journal of Mathematics, 2:183-200, 1879.
Henry C. King. Planar linkages and algebraic sets. Turkish Journal of Mathematics, 23(1):33-56, 1999. URL: http://arxiv.org/abs/math.AG/9807023.
http://arxiv.org/abs/math.AG/9807023
Pascal Koiran. The complexity of local dimensions for constructible sets. Journal of Complexity, 16(1):311-323, March 2000. URL: http://dx.doi.org/10.1006/jcom.1999.0536.
http://dx.doi.org/10.1006/jcom.1999.0536
N. E. Mnev. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Oleg Viro and Anatoly Vershik, editors, Topology and Geometry - Rohlin Seminar, volume 1346 of Lecture Notes in Mathematics, pages 527-543. Springer, 1988. URL: http://dx.doi.org/10.1007/BFb0082792.
http://dx.doi.org/10.1007/BFb0082792
Neil Robertson, Daniel Sanders, and Paul Seymour. The four-colour theorem. Journal of Combinatorial Theory, Series B, 70:2-44, 1997. URL: http://dx.doi.org/10.1006/jctb.1997.1750.
http://dx.doi.org/10.1006/jctb.1997.1750
James B. Saxe. Embeddability of weighted graphs in k-space is strongly NP-hard. In Proceedings of the 17th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, October 1979. URL: https://www.cs.duke.edu/brd/Teaching/Bio/asmb/current/Readings3/saxe-embeddability.pdf.
https://www.cs.duke.edu/brd/Teaching/Bio/asmb/current/Readings3/saxe-embeddability.pdf
Marcus Schaefer. Realizability of graphs and linkages. In János Pach, editor, Thirty Essays on Geometric Graph Theory, chapter 23. Springer, 2013. URL: http://ovid.cs.depaul.edu/documents/realizability.pdf.
http://ovid.cs.depaul.edu/documents/realizability.pdf
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Finding the Maximum Subset with Bounded Convex Curvature
We describe an algorithm for solving an important geometric problem arising in computer-aided manufacturing. When machining a pocket in a solid piece of material such as steel using a rough tool in a milling machine, sharp convex corners of the pocket cannot be done properly, but have to be left for finer tools that are more expensive to use. We want to determine a tool path that maximizes the use of the rough tool. Mathematically, this boils down to the following problem. Given a simply-connected set of points P in the plane such that the boundary of P is a curvilinear polygon consisting of n line segments and circular arcs of arbitrary radii, compute the maximum subset Q of P consisting of simply-connected sets where the boundary of each set is a curve with bounded convex curvature. A closed curve has bounded convex curvature if, when traversed in counterclockwise direction, it turns to the left with curvature at most 1. There is no bound on the curvature where it turns to the right. The difference in the requirement to left- and right-curvature is a natural consequence of different conditions when machining convex and concave areas of the pocket. We devise an algorithm to compute the unique maximum such set Q. The algorithm runs in O(n log n) time and uses O(n) space.
For the correctness of our algorithm, we prove a new generalization of the Pestov-Ionin Theorem. This is needed to show that the output Q of our algorithm is indeed maximum in the sense that if Q' is any subset of P with a boundary of bounded convex curvature, then Q' is a subset of Q.
planar computational geometry
bounded curvature
pocket machining
4:1-4:17
Regular Paper
Mikkel
Abrahamsen
Mikkel Abrahamsen
Mikkel
Thorup
Mikkel Thorup
10.4230/LIPIcs.SoCG.2016.4
M. Abrahamsen and M. Thorup. Finding the maximum subset with bounded convex curvature, 2016. URL: http://arxiv.org/abs/1603.02080.
http://arxiv.org/abs/1603.02080
H.-K. Ahn, O. Cheong, J. Matoušek, and A. Vigneron. Reachability by paths of bounded curvature in a convex polygon. Computational Geometry, 45(1):21-32, 2012.
G.E. Blelloch. Space-efficient dynamic orthogonal point location, segment intersection, and range reporting. In Proceedings of SODA, pages 894-903, 2008.
S.-W. Cheng, O. Cheong, H. Everett, and R. van Oostrum. Hierarchical decompositions and circular ray shooting in simple polygons. Discrete and Computational Geometry, 32:401-415, 2004.
H.S. Choy and K.W. Chan. A corner-looping based tool path for pocket milling. Computer-Aided Design, 35(2):155-166, 2003.
X. Han and L. Tang. Precise prediction of forces in milling circular corners. International Journal of Machine Tools and Manufacture, 88:184-193, 2015.
M. Held. Voronoi diagrams and offset curves of curvilinear polygons. Computer-Aided Design, 30(4):287-300, 1998.
R. Howard and A. Treibergs. A reverse isoperimetric inequality, stability and extremal theorems for plane-curves with bounded curvature. Rocky Mountain Journal of Mathematics, 25(2):635-684, 1995.
H. Iwabe, Y. Fujii, K. Saito, and T. Kishinami. Study on corner cut by end mill. International Journal of the Japan Society for Precision Engineering, 28(3):218-223, 1994.
S.C. Park and Y.C. Chung. Mitered offset for profile machining. Computer-Aided Design, 35(5):501-505, 2003.
V. Pateloup, E. Duc, and P. Ray. Corner optimization for pocket machining. International Journal of Machine Tools and Manufacture, 44(12):1343-1353, 2004.
G. Pestov and V. Ionin. The largest possible circle imbedded in a given closed curve. Doklady Akademii Nauk SSSR, 127(6):1170-1172, 1959.
C.K. Yap. An O(nlog n) algorithm for the voronoi diagram of a set of simple curve segments. Discrete and Computational Geometry, 2(1):365-393, 1987.
Z.Y. Zhao, C.Y. Wang, H.M. Zhou, and Z. Qin. Pocketing toolpath optimization for sharp corners. Journal of Materials Processing Technology, 192:175-180, 2007.
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Coloring Points with Respect to Squares
We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram.
Geometric hypergraph coloring
Polychromatic coloring
Homothets
Cover-decomposability
5:1-5:16
Regular Paper
Eyal
Ackerman
Eyal Ackerman
Balázs
Keszegh
Balázs Keszegh
Máté
Vizer
Máté Vizer
10.4230/LIPIcs.SoCG.2016.5
Eyal Ackerman and Rom Pinchasi. On coloring points with respect to rectangles. J. Comb. Theory, Ser. A, 120(4):811-815, 2013. URL: http://dx.doi.org/10.1016/j.jcta.2013.01.005.
http://dx.doi.org/10.1016/j.jcta.2013.01.005
Deepak Ajwani, Khaled M. Elbassioni, Sathish Govindarajan, and Saurabh Ray. Conflict-free coloring for rectangle ranges using O(n^.382) colors. Discrete & Computational Geometry, 48(1):39-52, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9425-5.
http://dx.doi.org/10.1007/s00454-012-9425-5
Greg Aloupis, Jean Cardinal, Sébastien Collette, Stefan Langerman, David Orden, and Pedro Ramos. Decomposition of multiple coverings into more parts. Discrete &Computational Geometry, 44(3):706-723, 2010. URL: http://dx.doi.org/10.1007/s00454-009-9238-3.
http://dx.doi.org/10.1007/s00454-009-9238-3
Prosenjit Bose, Paz Carmi, Sébastien Collette, and Michiel H. M. Smid. On the stretch factor of convex delaunay graphs. JoCG, 1(1):41-56, 2010. URL: http://jocg.org/index.php/jocg/article/view/5.
http://jocg.org/index.php/jocg/article/view/5
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. arXiv preprint arXiv:1212.2346, 2012.
Jean Cardinal, Kolja Knauer, Piotr Micek, and Torsten Ueckerdt. Making octants colorful and related covering decomposition problems. SIAM journal on discrete mathematics, 28(4):1948-1959, 2014.
Jean Cardinal, Kolja B. Knauer, Piotr Micek, and Torsten Ueckerdt. Making triangles colorful. J. of Computational Geometry, 4(1):240-246, 2013. URL: http://jocg.org/index.php/jocg/article/view/136.
http://jocg.org/index.php/jocg/article/view/136
Jean Cardinal, Kolja B. Knauer, Piotr Micek, and Torsten Ueckerdt. Making octants colorful and related covering decomposition problems. SIAM J. Discrete Math., 28(4):1948-1959, 2014. URL: http://dx.doi.org/10.1137/140955975.
http://dx.doi.org/10.1137/140955975
Timothy M. Chan. Conflict-free coloring of points with respect to rectangles and approximation algorithms for discrete independent set. In Tamal K. Dey and Sue Whitesides, editors, Symposuim on Computational Geometry 2012, SoCG '12, Chapel Hill, NC, USA, June 17-20, 2012, pages 293-302. ACM, 2012. URL: http://dx.doi.org/10.1145/2261250.2261293.
http://dx.doi.org/10.1145/2261250.2261293
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
Matt Gibson and Kasturi Varadarajan. Optimally decomposing coverings with translates of a convex polygon. Discrete &Computational Geometry, 46(2):313-333, 2011. URL: http://dx.doi.org/10.1007/s00454-011-9353-9.
http://dx.doi.org/10.1007/s00454-011-9353-9
Sariel Har-Peled and Shakhar Smorodinsky. Conflict-free coloring of points and simple regions in the plane. Discrete & Computational Geometry, 34(1):47-70, 2005. URL: http://dx.doi.org/10.1007/s00454-005-1162-6.
http://dx.doi.org/10.1007/s00454-005-1162-6
Balázs Keszegh. Coloring half-planes and bottomless rectangles. Computational Geometry, 45(9):495-507, 2012.
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. More on decomposing coverings by octants. Journal of Computational Geometry, 6(1):300-315, 2015.
Rolf Klein. Concrete and abstract Voronoi diagrams, volume 400. Springer Science &Business Media, 1989.
István Kovács. Indecomposable coverings with homothetic polygons. Discrete & Computational Geometry, 53(4):817-824, 2015. URL: http://dx.doi.org/10.1007/s00454-015-9687-9.
http://dx.doi.org/10.1007/s00454-015-9687-9
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, number DCG-CONF-2008-001, 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. Indecomposable coverings. In Proc. 41st Int. Workshop on Graph-Theoretic Concepts in Computer Science (WG), to appear.
János Pach, Dömötör Pálvölgyi, and Géza Toth. 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. Journal 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. In Jin Akiyama, William Y. C. Chen, Mikio Kano, Xueliang Li, and Qinglin Yu, editors, Discrete Geometry, Combinatorics and Graph Theory, 7th China-Japan Conference, CJCDGCGT 2005, Tianjin, China, November 18-20, 2005, Xi'an, China, November 22-24, 2005, Revised Selected Papers, volume 4381 of Lecture Notes in Computer Science, pages 135-148. Springer, 2005. URL: http://dx.doi.org/10.1007/978-3-540-70666-3_15.
http://dx.doi.org/10.1007/978-3-540-70666-3_15
János Pach, Gábor Tardos, and Géza Tóth. Indecomposable coverings. Canadian mathematical bulletin, 52(3):451-463, 2009.
János Pach and Géza Tóth. Conflict-free colorings. In Boris Aronov, Saugata Basu, János Pach, and Micha Sharir, editors, Discrete and Computational Geometry, volume 25 of Algorithms and Combinatorics, pages 665-671. Springer Berlin Heidelberg, 2003. URL: http://dx.doi.org/10.1007/978-3-642-55566-4_30.
http://dx.doi.org/10.1007/978-3-642-55566-4_30
János Pach and Géza Tóth. Decomposition of multiple coverings into many parts. Comput. Geom., 42(2):127-133, 2009. URL: http://dx.doi.org/10.1016/j.comgeo.2008.08.002.
http://dx.doi.org/10.1016/j.comgeo.2008.08.002
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. Indecomposable coverings with unit discs. CoRR, abs/1310.6900, 2013. URL: http://arxiv.org/abs/1310.6900.
http://arxiv.org/abs/1310.6900
Dömötör Pálvölgyi and Géza Tóth. Convex polygons are cover-decomposable. Discrete & Computational Geometry, 43(3):483-496, 2010. URL: http://dx.doi.org/10.1007/s00454-009-9133-y.
http://dx.doi.org/10.1007/s00454-009-9133-y
Deniz Sariöz. Generalized delaunay graphs with respect to any convex set are plane graphs. CoRR, abs/1012.4881, 2010. URL: http://arxiv.org/abs/1012.4881.
http://arxiv.org/abs/1012.4881
Shakhar Smorodinsky. Conflict-free coloring and its applications. 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 331-389. Springer Berlin Heidelberg, 2013. URL: http://dx.doi.org/10.1007/978-3-642-41498-5_12.
http://dx.doi.org/10.1007/978-3-642-41498-5_12
Shakhar Smorodinsky and Yelena Yuditsky. Polychromatic coloring for half-planes. Journal 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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Approximating Dynamic Time Warping and Edit Distance for a Pair of Point Sequences
We present the first subquadratic algorithms for computing similarity between a pair of point sequences in R^d, for any fixed d > 1, using dynamic time warping (DTW) and edit distance, assuming that the point sequences are drawn from certain natural families of curves. In particular, our algorithms compute (1 + eps)-approximations of DTW and ED in near-linear time for point sequences drawn from k-packed or k-bounded curves, and subquadratic time for backbone sequences. Roughly speaking, a curve is k-packed if the length of its intersection with any ball of radius r is at most kr, and it is k-bounded if the sub-curve between two curve points does not go too far from the two points compared to the distance between the two points. In backbone sequences, consecutive points are spaced at approximately equal distances apart, and no two points lie very close together. Recent results suggest that a subquadratic algorithm for DTW or ED is unlikely for an arbitrary pair of point sequences even for d = 1.
The commonly used dynamic programming algorithms for these distance measures reduce the problem to computing a minimum-weight path in a grid graph. Our algorithms work by constructing a small set of rectangular regions that cover the grid vertices. The weights of vertices inside each rectangle are roughly the same, and we develop efficient procedures to compute the approximate minimum-weight paths through these rectangles.
Dynamic time warping
Edit distance
Near-linear-time algorithm
Dynamic programming
Well-separated pair decomposition
6:1-6:16
Regular Paper
Pankaj K.
Agarwal
Pankaj K. Agarwal
Kyle
Fox
Kyle Fox
Jiangwei
Pan
Jiangwei Pan
Rex
Ying
Rex Ying
10.4230/LIPIcs.SoCG.2016.6
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In Proc. 56th Annu. IEEE Sympos. on Found. Comp. Sci., pages 59-78, 2015.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Proc. 41st Int. Colloq. on Auto., Lang., and Program., pages 39-51, 2014.
Helmut Alt, Christian Knauer, and Carola Wenk. Comparison of distance measures for planar curves. Algorithmica, 38(1):45-58, 2004.
Alexandr Andoni, Robert Krauthgamer, and Krzysztof Onak. Polylogarithmic approximation for edit distance and the asymmetric query complexity. In Proc. 51st IEEE Sympos. on Found. Comp. Sci., pages 377-386, 2010.
Boris Aronov, Sariel Har-Peled, Christian Knauer, Yusu Wang, and Carola Wenk. Fréchet distance for curves, revisited. In Proc. 14th Annu. Euro. Sympos. Algo., pages 52-63, 2006.
Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In Proc. 47th Annu. ACM Sympos. Theory of Comput., pages 51-58, 2015.
Karl Bringmann. Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails. In Proc. 55th. Annu. IEEE Sympos. on Found. of Comp. Sci., pages 661-670, 2014.
Karl Bringmann and Marvin Künnemann. Improved approximation for Fréchet distance on c-packed curves matching conditional lower bounds. In Proc. 26th Annu. Int. Sympos. Algo. Comp., pages 517-528, 2015.
Karl Bringmann and Marvin Künnemann. Quadratic conditional lower bounds for string problems and dynamic time warping. In Proc. 56th Annu. IEEE Sympos. on Found. of Comp. Sci., pages 79-97, 2015.
Karl Bringmann and Wolfgang Mulzer. Approximability of the discrete Fréchet distance. J. Comput. Geom., 7(2):46-76, 2016.
Thomas Brox and Jitendra Malik. Object segmentation by long term analysis of point trajectories. In Proc. 11th Annu. Euro. Conf. Comp. Vis., pages 282-295, 2010.
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.
Cartesian tree. URL: https://en.wikipedia.org/wiki/Cartesian_tree.
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Alexander De Luca, Alina Hang, Frederik Brudy, Christian Lindner, and Heinrich Hussmann. Touch me once and i know it’s you!: implicit authentication based on touch screen patterns. In Proc. SIGCHI Conf. Human Factors in Comp. Sys., pages 987-996, 2012.
Anne Driemel, Sariel Har-Peled, and Carola Wenk. Approximating the Fréchet distance for realistic curves in near linear time. Discrete Comp. Geom., 48(1):94-127, 2012.
Richard Durbin, Sean R. Eddy, Anders Krogh, and Graeme Mitchison. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Camridge University Press, New York, 1998.
Johannes Fischer and Volker Heun. Theoretical and practical improvements on the RMQ-problem, with applications to LCA and LCE. In Proc. 17th Annu. Sympos. Combin. Pattern Match., pages 36-48, 2006.
T. Gasser and K. Wang. Alignment of curves by dynamic time warping. Annals of Statistics, 25(3):1251-1276, 1997.
M. C. Gonzalez, C. A. Hidalgo, and A.-L. Barabasi. Understanding individual human mobility patterns. Nature, 453(7196):779-782, 2008.
Sariel Har-Peled. Geometric Approximation Algorithms. American Mathematical Society Providence, 2011.
Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comp. Sys. Sci., 62(2):367-375, 201.
Jon Kleinberg and Eva Tardos. Algorithm Design. Addison-Wesley Longman Publishing Co., Inc., 2005.
Rachel Kolodny, Patrice Koehl, and Michael Levitt. Comprehensive evaluation of protein structure alignment methods: Scoring by geometric measures. J. Mol. Bio., 346(4):1173-1188, 2005.
J.-G. Lee, J. Han, and K.-Y. Whang. Trajectory clustering: a partition-and-group framework. In Proc. ACM SIGMOD Int. Conf. Manag. Data, pages 593-604, 2007.
Meinard Müller. Information Retrieval for Music and Motion. Springer-Verlag Berlin Heidelberg, 2007.
Mario E. Munich and Pietro Perona. Continuous dynamic time warping for translation-invariant curve alignment with applications to signature verification. In Proc. 7th Annu. Int. Conf. Comp. Vis., pages 108-115, 1999.
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Range minimum queries: Part two. URL: http://web.stanford.edu/class/archive/cs/cs166/cs166.1146/lectures/01/Slides01.pdf.
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Time series matching with dynamic time warping. URL: https://systematicinvestor.wordpress.com/2012/01/20/time-series-matching-with-dynamic-time-warping/.
https://systematicinvestor.wordpress.com/2012/01/20/time-series-matching-with-dynamic-time-warping/
Xiaoyue Wang, Abdullah Mueen, Hui Ding, Goce Trajcevski, Peter Scheuermann, and Eamonn Keogh. Experimental comparison of representation methods and distance measures for time series data. Data Mining and Knowledge Discovery, 26(2):275-309, 2013.
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An Improved Lower Bound on the Minimum Number of Triangulations
Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Omega(2.631^n) geometric triangulations. Our result improves the previously best general lower bound of Omega(2.43^n) and also covers the previously best lower bound of Omega(2.63^n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest:
(1) Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations.
(2) Generalized configurations of points that minimize the number of triangulations have at most n/2 points on their convex hull.
(3) We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.
Combinatorial geometry
Order types
Triangulations
7:1-7:16
Regular Paper
Oswin
Aichholzer
Oswin Aichholzer
Victor
Alvarez
Victor Alvarez
Thomas
Hackl
Thomas Hackl
Alexander
Pilz
Alexander Pilz
Bettina
Speckmann
Bettina Speckmann
Birgit
Vogtenhuber
Birgit Vogtenhuber
10.4230/LIPIcs.SoCG.2016.7
Oswin Aichholzer, Franz Aurenhammer, and Hannes Krasser. Enumerating order types for small point sets with applications. Order, 19(3):265-281, 2002. URL: http://dx.doi.org/10.1023/A:1021231927255.
http://dx.doi.org/10.1023/A:1021231927255
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
Oswin Aichholzer, Ferran Hurtado, and Marc Noy. A lower bound on the number of triangulations of planar point sets. Comput. Geom., 29(2):135-145, 2004.
Oswin Aichholzer and Hannes Krasser. Abstract order type extension and new results on the rectilinear crossing number. Comput. Geom., 36(1):2-15, 2007.
Selim G. Akl. A lower bound on the maximum number of crossing-free Hamiltonian cycles in a rectilinear drawing of K_n. Ars Combin., 7:7-18, 1979.
Victor Alvarez, Karl Bringmann, Radu Curticapean, and Saurabh Ray. Counting triangulations and other crossing-free structures via onion layers. Discrete Comput. Geom., 53(4):675-690, 2015.
Victor Alvarez and Raimund Seidel. A simple aggregative algorithm for counting triangulations of planar point sets and related problems. In Proc. 29th Annual Symposium on Computational Geometry (SoCG 2013), pages 1-8. ACM, 2013.
Adrian Dumitrescu, André Schulz, Adam Sheffer, and Csaba D. Tóth. Bounds on the maximum multiplicity of some common geometric graphs. SIAM Journal on Discrete Mathematics, 27(2):802-826, 2013.
Jacob E. Goodman. Proof of a conjecture of Burr, Grünbaum, and Sloane. Discrete Math., 32(1):27-35, 1980. URL: http://dx.doi.org/10.1016/0012-365X(80)90096-5.
http://dx.doi.org/10.1016/0012-365X(80)90096-5
Jacob E. Goodman and Richard Pollack. Proof of Grünbaum’s conjecture on the stretchability of certain arrangements of pseudolines. J. Comb. Theory, Ser. A, 29(3):385-390, 1980. URL: http://dx.doi.org/10.1016/0097-3165(80)90038-2.
http://dx.doi.org/10.1016/0097-3165(80)90038-2
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. Comb. Theory, Ser. A, 37(3):257-293, 1984.
Michael Hoffmann, André Schulz, Micha Sharir, Adam Sheffer, Csaba D. Tóth, and Emo Welzl. Thirty Essays on Geometric Graph Theory, chapter Counting Plane Graphs: Flippability and Its Applications, pages 303-325. Springer New York, New York, NY, 2013.
Hannes Krasser. Order Types of Point Sets in the Plane. PhD thesis, Institute for Theoretical Computer Science, Graz University of Technology, October 2003.
Paul McCabe and Raimund Seidel. New lower bounds for the number of straight-edge triangulations of a planar point set. In Proc. 20th European Workshop on Computational Geometry (EWCG 2004), pages 175-176, 2004.
Alexander Pilz and Emo Welzl. Order on order types. In Proc. 31st Int. Symposium on Computational Geometry (SoCG 2015), volume 34 of LIPIcs, pages 285-299, 2015.
Saurabh Ray and Raimund Seidel. A simple and less slow method for counting triangulations and for related problems. In Proc. 20th European Workshop on Computational Geometry (EWCG 2004), pages 177-180, 2004.
Andreas Razen, Jack Snoeyink, and Emo Welzl. Number of crossing-free geometric graphs vs. triangulations. Electr. Notes Discrete Math., 31:195-200, 2008.
Francisco Santos and Raimund Seidel. A better upper bound on the number of triangulations of a planar point set. J. Comb. Theory, Ser. A, 102(1):186-193, 2003.
Micha Sharir and Adam Sheffer. Counting triangulations of planar point sets. Electr. J. Comb., 18(1), 2011.
Micha Sharir and Adam Sheffer. Counting plane graphs: Cross-graph charging schemes. Combinatorics, Probability & Computing, 22(6):935-954, 2013.
Micha Sharir, Adam Sheffer, and Emo Welzl. On degrees in random triangulations of point sets. J. Comb. Theory, Ser. A, 118(7):1979-1999, 2011.
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Recognizing Weakly Simple Polygons
We present an O(n log n)-time algorithm that determines whether a given planar n-gon is weakly simple. This improves upon an O(n^2 log n)-time algorithm by [Chang, Erickson, and Xu, SODA, 2015]. Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question.
weakly simple polygon
crossing
8:1-8:16
Regular Paper
Hugo A.
Akitaya
Hugo A. Akitaya
Greg
Aloupis
Greg Aloupis
Jeff
Erickson
Jeff Erickson
Csaba
Tóth
Csaba Tóth
10.4230/LIPIcs.SoCG.2016.8
Hugo A. Akitaya, Greg Aloupis, Jeff Erickson, and Csaba D. Tóth. Recognizing weakly simple polygons. Preprint, 2016. URL: http://arxiv.org/abs/1603.07401.
http://arxiv.org/abs/1603.07401
Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Martin L. Demaine, Joseph S.B. Mitchell, Saurabh Sethia, and Steven S. Skiena. When can you fold a map? Computational Geometry, 29(1):23-46, 2004.
Marshall Bern and Barry Hayes. The complexity of flat origami. In Proc. 7th ACM-SIAM Sympos. on Discrete Algorithms, pages 175-183, 1996.
Hsien-Chih Chang, Jeff Erickson, and Chao Xu. Detecting weakly simple polygons. In Proc. 26th ACM-SIAM Sympos. on Discrete Algorithms, pages 1655-1670, 2015.
Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete Comput. Geom., 6(3):485-524, 1991.
Pier Francesco Cortese, Giuseppe Di Battista, Maurizio Patrignani, and Maurizio Pizzonia. On embedding a cycle in a plane graph. Discrete Math., 309(7):1856-1869, 2009.
Ares Ribó Mor. Realization and counting problems for planar structures. PhD thesis, Freie Universität Berlin, Department of Mathematics and Computer Science, 2006.
Michael Ian Shamos and Dan Hoey. Geometric intersection problems. In Proc. 17th IEEE Sympos. Foundations of Computer Science, pages 208-215, 1976.
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Tight Lower Bounds for Data-Dependent Locality-Sensitive Hashing
We prove a tight lower bound for the exponent rho for data-dependent Locality-Sensitive Hashing schemes, recently used to
design efficient solutions for the c-approximate nearest neighbor search. In particular, our lower bound matches the bound of rho<= 1/(2c-1)+o(1) for the l_1 space, obtained via the recent algorithm from [Andoni-Razenshteyn, STOC'15].
In recent years it emerged that data-dependent hashing is strictly superior to the classical Locality-Sensitive Hashing, when
the hash function is data-independent. In the latter setting, the best exponent has been already known: for the l_1 space, the tight bound is rho=1/c, with the upper bound from [Indyk-Motwani,STOC'98] and the matching lower bound from [O'Donnell-Wu-Zhou,ITCS'11].
We prove that, even if the hashing is data-dependent, it must hold that rho>=1/(2c-1)-o(1). To prove the result, we need to
formalize the exact notion of data-dependent hashing that also captures the complexity of the hash functions (in addition to their collision properties). Without restricting such complexity, we would allow for obviously infeasible solutions such as the Voronoi diagram of a dataset. To preclude such solutions, we require our hash functions to be succinct. This condition is satisfied by all the known algorithmic results.
similarity search
high-dimensional geometry
LSH
data structures
lower bounds
9:1-9:11
Regular Paper
Alexandr
Andoni
Alexandr Andoni
Ilya
Razensteyn
Ilya Razensteyn
10.4230/LIPIcs.SoCG.2016.9
Alexandr Andoni. Nearest Neighbor Search: the Old, the New, and the Impossible. PhD thesis, Massachusetts Institute of Technology, 2009.
Alexandr Andoni and Piotr Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 459-468, 2006.
Alexandr Andoni and Piotr Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Communications of the ACM, 51(1):117-122, 2008.
Alexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn, and Ludwig Schmidt. Practical and optimal LSH for angular distance. In Proceedings of the 29th Annual Conference on Neural Information Processing Systems (NIPS'15), 2015.
Alexandr Andoni, Piotr Indyk, Huy L. Nguyen, and Ilya Razenshteyn. Beyond locality-sensitive hashing. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'14), pages 1018-1028, 2014.
Alexandr Andoni and Ilya Razenshteyn. Optimal data-dependent hashing for approximate near neighbors. In Proceedings of the ACM Symposium on the Theory of Computing (STOC'15), 2015.
Anja Becker, Léo Ducas, Nicolas Gama, and Thijs Laarhoven. New directions in nearest neighbor searching with applications to lattice sieving. In Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA'16), 2016.
Sanjoy Dasgupta and Anupam Gupta. An elementary proof of a theorem of Johnson and Lindenstrauss. Random Structures and Algorithms, 22(1):60-65, 2003.
Mayur Datar, Nicole Immorlica, Piotr Indyk, and Vahab S. Mirrokni. Locality-sensitive hashing scheme based on p-stable distributions. In Proceedings of the 20th ACM Symposium on Computational Geometry (SoCG'04), pages 253-262, 2004.
Moshe Dubiner. Bucketing coding and information theory for the statistical highdimensional nearest-neighbor problem. IEEE Transactions on Information Theory, 56(8):4166-4179, 2010.
Uriel Feige and Gideon Schechtman. On the optimality of the random hyperplane rounding technique for MAX CUT. Random Structures and Algorithms, 20(3):403-440, 2002.
Sariel Har-Peled, Piotr Indyk, and Rajeev Motwani. Approximate nearest neighbor: towards removing the curse of dimensionality. Theory of Computing, 8(1):321-350, 2012.
Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In Proceedings of the 30th ACM Symposium on the Theory of Computing (STOC'98), pages 604-613, 1998.
William B. Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Conference in modern analysis and probability (New Haven, Connecticut, 1982), volume 26 of Contemporary Mathematics, pages 189-206. 1984.
Subhash Khot and Nisheeth K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into l₁. J. ACM, 62(1):8:1-8:39, 2015.
Eyal Kushilevitz, Rafail Ostrovky, and Yuval Rabani. Efficient search for approximate nearest neighbor in high dimensional spaces. SIAM Journal on Computing, 30(2):457-474, 2000.
Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995.
Rajeev Motwani, Assaf Naor, and Rina Panigrahy. Lower bounds on locality sensitive hashing. SIAM Journal on Discrete Mathematics, 21(4):930-935, 2007.
Ryan O'Donnell, Yi Wu, and Yuan Zhou. Optimal lower bounds for locality sensitive hashing (except when q is tiny). In Proceedings of Innovations in Computer Science (ICS'11), pages 275-283, 2011.
Rina Panigrahy, Kunal Talwar, and Udi Wieder. A geometric approach to lower bounds for approximate near-neighbor search and partial match. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS'08), pages 414-423, 2008.
Rina Panigrahy, Kunal Talwar, and Udi Wieder. Lower bounds on near neighbor search via metric expansion. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS'10), pages 805-814, 2010.
Hannan Samet. Foundations of Multidimensional and Metric Data Structures. Elsevier, 2006.
Kengo Terasawa and Yuzuru Tanaka. Spherical LSH for approximate nearest neighbor search on unit hypersphere. In Algorithms and Data Structures, 10th International Workshop, WADS 2007, Halifax, Canada, August 15-17, 2007, Proceedings, pages 27-38, 2007.
Jingdong Wang, Heng Tao Shen, Jingkuan Song, and Jianqiu Ji. Hashing for similarity search: A survey. CoRR, abs/1408.2927, 2014.
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The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions
We show that the union of translates of a convex body in three dimensional space can have a cubic number holes in the worst case, where a hole in a set is a connected component of its compliment. This refutes a 20-year-old conjecture. As a consequence, we also obtain improved lower bounds on the complexity of motion planning problems and of Voronoi diagrams with convex distance functions.
Union complexity
Convex sets
Motion planning
10:1-10:16
Regular Paper
Boris
Aronov
Boris Aronov
Otfried
Cheong
Otfried Cheong
Michael Gene
Dobbins
Michael Gene Dobbins
Xavier
Goaoc
Xavier Goaoc
10.4230/LIPIcs.SoCG.2016.10
Pankaj K. Agarwal, Sariel Har-Peled, Haim Kaplan, and Micha Sharir. Union of random Minkowski sums and network vulnerability analysis. Discrete Comput. Geom., 52(3):551-582, 2014.
Pankaj K. Agarwal, János Pach, and Micha Sharir. State of the union (of geometric objects). In Proc. Joint Summer Research Conf. on Discrete and Computational Geometry: 20 Years Later, volume 452 of Contemp. Math., pages 9-48. AMS, 2008.
Boris Aronov, Otfried Cheong, Michael G. Dobbins, and Xavier Goaoc. The number of holes in the union of translates of a convex set in three dimensions. To appear in Discrete Comput. Geom., 2015. URL: http://arxiv.org/abs/1502.01779.
http://arxiv.org/abs/1502.01779
Boris Aronov and Micha Sharir. On translational motion planning of a convex polyhedron in 3-space. SIAM J. Comput., 26(6):1785-1803, 1997.
Franz Aurenhammer. Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Comput. Surv., 23:345-405, 1991.
Karol Borsuk. On the imbedding of systems of compacta in simplicial complexes. Fundamenta Mathematicae, 35:217-234, 1948.
Kenneth L. Clarkson and Kasturi Varadarajan. Improved approximation algorithms for geometric set cover. Discrete Comput. Geom., 37(1):43-58, 2007.
Alon Efrat and Micha Sharir. On the complexity of the union of fat convex objects in the plane. Discrete Comput. Geom., 23(2):171-189, 2000.
Steven Fortune. Voronoi diagrams and Delaunay triangulations. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 23, pages 513-528. CRC Press LLC, Boca Raton, FL, 2 edition, 2004.
Allen Hatcher. Algebraic topology. Cambridge University Press, Cambridge, UK, 2002.
Christian Icking, Rolf Klein, Ngoc-Minh Lé, and Lihong Ma. Convex distance functions in 3-space are different. Fund. Inform., 22:331-352, 1995. URL: http://dx.doi.org/10.3233/FI-1995-2242.
http://dx.doi.org/10.3233/FI-1995-2242
Klara Kedem, Ron Livne, János Pach, and Micha Sharir. On the union of Jordan regions and collision-free translational motion amidst polygonal ostacles. Discrete Comput. Geom., 1(1):59-71, 1986.
Mikhail D. Kovalev. Svoistvo vypuklykh mnozhestv i ego prilozhenie (A property of convex sets and its application). Mat. Zametki, 44:89-99, 1988. In Russian.
Joseph S. B. Mitchell and Joseph O'Rourke. Computational geometry column 42. Int. J. Comput. Geom. Ap., 11(05):573-582, 2001.
James R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Menlo Park, CA, 1984.
János Pach and Gábor Tardos. On the boundary complexity of the union of fat triangles. SIAM J. Comput., 31(6):1745-1760, 2002.
Micha Sharir. Algorithmic motion planning. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 47, pages 1037-1064. CRC Press LLC, Boca Raton, FL, 2 edition, 2004.
Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York, NY, USA, 2010.
Boaz Tagansky. The Complexity of Substructures in Arrangments of Surfaces. PhD thesis, Tel Aviv University, 1996.
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On the Combinatorial Complexity of Approximating Polytopes
Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body K of diameter $diam(K)$ is given in Euclidean d-dimensional space, where $d$ is a constant. Given an error parameter eps > 0, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from K is at most eps diam(K). By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that O(1/eps^{(d-1)/2}) facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is O-tilde(1/eps^{(d-1)/2}), where O-tilde conceals a polylogarithmic factor in 1/eps. This is an improvement upon the best known bound, which is roughly O(1/eps^{d-2}).
Our result is based on a novel combination of both new and old ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of Barany and Larman's economical cap covering, which may be of independent interest. Finally, we use a deterministic variation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.
Polytope approximation
Convex polytopes
Macbeath regions
11:1-11:15
Regular Paper
Sunil
Arya
Sunil Arya
Guilherme D.
da Fonseca
Guilherme D. da Fonseca
David M.
Mount
David M. Mount
10.4230/LIPIcs.SoCG.2016.11
P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan. Approximating extent measures of points. J. Assoc. Comput. Mach., 51:606-635, 2004.
G. E. Andrews. A lower bound for the volumes of strictly convex bodies with many boundary points. Trans. Amer. Math. Soc., 106:270-279, 1963.
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, 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, 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
I. Bárány. Intrinsic volumes and f-vectors of random polytopes. Math. Ann., 285:671-699, 1989.
I. Bárány. The technique of M-regions and cap-coverings: A survey. Rend. Circ. Mat. Palermo, 65:21-38, 2000.
I. Bárány. Extremal problems for convex lattice polytopes: A survey. Contemp. Math., 453:87-103, 2008.
I. Bárány and D. G. Larman. Convex bodies, economic cap coverings, random polytopes. Mathematika, 35:274-291, 1988.
K. Böröczky, Jr. Approximation of general smooth convex bodies. Adv. Math., 153:325-341, 2000.
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.
E. M. Bronstein. Approximation of convex sets by polytopes. J. Math. Sci., 153(6):727-762, 2008.
K. L. Clarkson. Building triangulations using ε-nets. In Proc. 38th Annu. ACM Sympos. Theory Comput., pages 326-335, 2006.
O. Devillers, M. Glisse, and X. Goaoc. Complexity analysis of random geometric structures made simpler. In Proc. 29th Annu. Sympos. Comput. Geom., pages 167-176, 2013.
R. M. Dudley. Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory, 10(3):227-236, 1974.
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.
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S. Har-Peled. Geometric approximation algorithms. Number 173 in Mathematical surveys and monographs. American Mathematical Society, 2011.
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P. McMullen. The maximum numbers of faces of a convex polytope. Mathematika, 17:179-184, 1970.
R. Schneider. Polyhedral approximation of smooth convex bodies. J. Math. Anal. Appl., 128:470-474, 1987.
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Efficient Algorithms to Decide Tightness
Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but more efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces.
In this article, we present a new polynomial time procedure to decide tightness for triangulations of 3-manifolds - a problem which previously was thought to be hard. In addition, for the more difficult problem of deciding tightness of 4-dimensional combinatorial manifolds, we describe an algorithm that is fixed parameter tractable in the treewidth of the 1-skeletons of the vertex links. Finally, we show that simpler treewidth parameters are not viable: for all non-trivial inputs, we show that the treewidths of both the 1-skeleton and the dual graph must grow too quickly for a standard treewidth-based algorithm to remain tractable.
discrete geometry and topology
polynomial time algorithms
fixed parameter tractability
tight triangulations
simplicial complexes
12:1-12:15
Regular Paper
Bhaskar
Bagchi
Bhaskar Bagchi
Basudeb
Datta
Basudeb Datta
Benjamin A.
Burton
Benjamin A. Burton
Nitin
Singh
Nitin Singh
Jonathan
Spreer
Jonathan Spreer
10.4230/LIPIcs.SoCG.2016.12
Aleksandr D. Alexandrov. On a class of closed surfaces. Recueil Math. (Moscow), 4:69-72, 1938.
Bhaskar Bagchi. The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds. European J. Combin., 51:69-83, 2016.
Bhaskar Bagchi and Basudeb Datta. On stellated spheres and a tightness criterion for combinatorial manifolds. European J. Combin., 36:294-313, 2014.
Bhaskar Bagchi, Basudeb Datta, and Jonathan Spreer. Tight triangulations of closed 3-manifolds. European J. Combin., 54:103-120, 2016.
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Benjamin A. Burton, Basudeb Datta, Nitin Singh, and Jonathan Spreer. A construction principle for tight and minimal triangulations of manifolds, 2015. 27 pages, 2 figures. URL: http://arxiv.org/abs/1511.04500.
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Benjamin A. Burton and Rodney G. Downey. Courcelle’s theorem for triangulations. arXiv:1403.2926 [math.GT], 24 pages , 7 figures, 2014.
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Anchored Rectangle and Square Packings
For points p_1,...,p_n in the unit square [0,1]^2, an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r_1,...,r_n in [0,1]^2 such that point p_i is a corner of the rectangle r_i (that is, r_i is anchored at p_i) for i=1,...,n. We show that for every set of n points in [0,1]^2, there is an anchored rectangle packing of area at least 7/12-O(1/n), and for every n, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27.
The above constructive lower bounds immediately yield constant-factor approximations, of 7/12 -epsilon for rectangles and 5/32 for squares, for computing anchored packings of maximum area in O(n log n) time. We prove that a simple greedy strategy achieves a 9/47-approximation for anchored square packings, and 1/3 for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n^{O(1/epsilon)} and exp(poly(log (n/epsilon))) time, respectively.
Rectangle packing
anchored rectangle
greedy algorithm
charging scheme
approximation algorithm.
13:1-13:16
Regular Paper
Kevin
Balas
Kevin Balas
Adrian
Dumitrescu
Adrian Dumitrescu
Csaba
Tóth
Csaba Tóth
10.4230/LIPIcs.SoCG.2016.13
Creative Commons Attribution 3.0 Unported license
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On Variants of k-means Clustering
Clustering problems often arise in fields like data mining and machine learning. Clustering usually refers to the task of partitioning a collection of objects into groups with similar elements, with respect to a similarity (or dissimilarity) measure. Among the clustering problems, k-means clustering in particular has received much attention from researchers. Despite the fact that k-means is a well studied problem, its status in the plane is still open. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound achievable in polynomial time is 9+epsilon.
In this paper, we consider the following variant of k-means. Given a set C of points in R^d and a real f > 0, find a finite set F of points in R^d that minimizes the quantity f*|F|+sum_{p in C} min_{q in F} {||p-q||}^2. For any fixed dimension d, we design a PTAS for this problem that is based on local search. We also give a "bi-criterion" local search algorithm for k-means which uses (1+epsilon)k centers and yields a solution whose cost is at most (1+epsilon) times the cost of an optimal k-means solution. The algorithm runs in polynomial time for any fixed dimension.
The contribution of this paper is two-fold. On the one hand, we are able to handle the square of distances in an elegant manner, obtaining a near-optimal approximation bound. This leads us towards a better understanding of the k-means problem. On the other hand, our analysis of local search might also be useful for other geometric problems. This is important considering that little is known about the local search method for geometric approximation.
k-means
Facility location
Local search
Geometric approximation
14:1-14:15
Regular Paper
Sayan
Bandyapadhyay
Sayan Bandyapadhyay
Kasturi
Varadarajan
Kasturi Varadarajan
10.4230/LIPIcs.SoCG.2016.14
Daniel Aloise, Amit Deshpande, Pierre Hansen, and Preyas Popat. Np-hardness of euclidean sum-of-squares clustering. Machine Learning, 75(2):245-248, 2009. URL: http://dx.doi.org/10.1007/s10994-009-5103-0.
http://dx.doi.org/10.1007/s10994-009-5103-0
Sanjeev Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM, 45(5):753-782, 1998. URL: http://dx.doi.org/10.1145/290179.290180.
http://dx.doi.org/10.1145/290179.290180
Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation schemes for euclidean k-medians and related problems. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC'98, pages 106-113, New York, NY, USA, 1998. ACM. URL: http://dx.doi.org/10.1145/276698.276718.
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Incremental Voronoi diagrams
We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in R^3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(n^(1/2)). A matching Omega(n^(1/2)) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log(n)) upper bound of Aronov et al. [Aronov et al., in proc. of LATIN, 2006] for farthest-point Voronoi diagrams in the special case where points are inserted in clockwise order along their convex hull.
We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of S U (p) from the diagram of S, in time O(K polylog n) worst case, which is O(n^(1/2) polylog n) amortized by the aforementioned combinatorial result.
The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.
Voronoi diagrams
dynamic data structures
Delaunay triangulation
15:1-15:16
Regular Paper
Sarah R.
Allen
Sarah R. Allen
Luis
Barba
Luis Barba
John
Iacono
John Iacono
Stefan
Langerman
Stefan Langerman
10.4230/LIPIcs.SoCG.2016.15
Greg Aloupis, Luis Barba, and Stefan Langerman. Circle separability queries in logarithmic time. In Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG'12, pages 121-125, August 2012.
Boris Aronov, Prosenjit Bose, Erik D Demaine, Joachim Gudmundsson, John Iacono, Stefan Langerman, and Michiel Smid. Data structures for halfplane proximity queries and incremental Voronoi diagrams. In LATIN 2006: Theoretical Informatics, pages 80-92. Springer, 2006.
Luis Barba. Disk constrained 1-center queries. In Proceedings of the 24th Canadian Conference on Computational Geometry, CCCG'12, pages 15-19, August 2012.
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Timothy M Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. Journal of the ACM (JACM), 57(3):16, 2010.
Yi-Jen Chiang and Roberto Tamassia. Dynamic algorithms in computational geometry. Proceedings of the IEEE, 80(9):1412-1434, 1992.
Mark De Berg, Marc Van Kreveld, Mark Overmars, and Otfried Cheong Schwarzkopf. Computational geometry. Springer, 2000.
Herbert Edelsbrunner and Raimund Seidel. Voronoi diagrams and arrangements. Discrete &Computational Geometry, 1(1):25-44, 1986.
Rolf Klein. Concrete and Abstract Voronoi Diagrams, volume 400 of Lecture Notes in Computer Science. Springer, 1989.
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Seth Pettie. Applications of forbidden 0–1 matrices to search tree and path compression-based data structures. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1457-1467, 2010.
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Dimension Reduction Techniques for l_p (1<p<2), with Applications
For Euclidean space (l_2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [Conf. in modern analysis and probability, AMS 1984], with a host of known applications. Here, we consider the problem of dimension reduction for all l_p spaces 1<p<2. Although strong lower bounds are known for dimension reduction in l_1, Ostrovsky and Rabani [JACM 2002] successfully circumvented these by presenting an l_1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to l_1 and do not naturally extend to other norms.
In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1<p<2, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for l_1 can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering.
Dimension reduction
embeddings
16:1-16:15
Regular Paper
Yair
Bartal
Yair Bartal
Lee-Ad
Gottlieb
Lee-Ad Gottlieb
10.4230/LIPIcs.SoCG.2016.16
I. Abraham, Y. Bartal, and O. Neiman. Embedding metric spaces in their intrinsic dimension. In SODA'08, pages 363-372. SIAM, 2008.
D. Achlioptas. Database-friendly random projections. In PODS'01, pages 274-281, New York, NY, USA, 2001. ACM.
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N. Ailon and B. Chazelle. Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. In STOC'06, pages 557-563. ACM, 2006.
N. Alon. Problems and results in extremal combinatorics, part I. Disc. Math, 273, 2003.
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A. Andoni, M. Charikar, O. Neiman, and H.L. Nguyen. Near linear lower bounds for dimension reduction in 𝓁₁. In FOCS'11. IEEE Computer Society, 2011.
Alexandr Andoni. Nearest Neighbor Search: the Old, the New, and the Impossible. PhD thesis, MIT, 2009.
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S. Har-Peled, P. Indyk, and R. Motwani. Approximate nearest neighbor: Towards removing the curse of dimensionality. Theory of Computing, 8(14):321-350, 2012.
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https://creativecommons.org/licenses/by/3.0/legalcode
Testing Convexity of Figures Under the Uniform Distribution
We consider the following basic geometric problem: Given epsilon in (0,1/2), a 2-dimensional figure that consists of a black object and a white background is epsilon-far from convex if it differs in at least an epsilon fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is epsilon-far from convex are needed to detect a violation of convexity with probability at least 2/3? This question arises in the context of designing property testers for convexity. Specifically, a (1-sided error) tester for convexity gets samples from the figure, labeled by their color; it always accepts if the black object is convex; it rejects with probability at least 2/3 if the figure is epsilon-far from convex.
We show that Theta(epsilon^{-4/3}) uniform samples are necessary and sufficient for detecting a violation of convexity in an epsilon-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our testing algorithm runs in time O(epsilon^{-4/3}) and thus beats the Omega(epsilon^{-3/2}) sample lower bound for learning convex figures under the uniform distribution from the work of Schmeltz (Data Structures and Efficient Algorithms,1992). It shows that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.
Convex sets
2D geometry
randomized algorithms
property testing
17:1-17:15
Regular Paper
Piotr
Berman
Piotr Berman
Meiram
Murzabulatov
Meiram Murzabulatov
Sofya
Raskhodnikova
Sofya Raskhodnikova
10.4230/LIPIcs.SoCG.2016.17
A. M. Andrew. Another efficient algorithm for convex hulls in two dimensions. Inf. Process. Lett., 9(5):216-219, 1979. URL: http://dx.doi.org/10.1016/0020-0190(79)90072-3.
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Imre Barany. Extremal problems for convex lattice polytopes: a survey. Contemporary Mathematics, 2000.
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Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. The role of adaptivity in image testing. Manuscript, 2015.
Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L_p-testing. In STOC, pages 164-173, 2014. URL: http://dx.doi.org/10.1145/2591796.2591887.
http://dx.doi.org/10.1145/2591796.2591887
Eric Blais, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, 2014, pages 309-320, 2014.
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Sofya Raskhodnikova and Grigory Yaroslavtsev. Learning pseudo-Boolean k-DNF and submodular functions. In SODA, pages 1356-1368, 2013.
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Separating a Voronoi Diagram via Local Search
Given a set P of n points in R^d , we show how to insert a set Z of O(n^(1-1/d)) additional points, such that P can be broken into two sets P1 and P2 , of roughly equal size, such that in the Voronoi diagram V(P u Z), the cells of P1 do not touch the cells of P2; that is, Z separates P1 from P2 in the Voronoi diagram (and also in the dual Delaunay triangulation). In addition, given such a partition (P1,P2) of P , we present an approximation algorithm to compute a minimum size separator realizing this partition. We also present a simple local search algorithm that is a PTAS for approximating the optimal Voronoi partition.
Separators
Local search
Approximation
Voronoi diagrams
Delaunay triangulation
Meshing
Geometric hitting set
18:1-18:16
Regular Paper
Vijay V. S. P.
Bhattiprolu
Vijay V. S. P. Bhattiprolu
Sariel
Har-Peled
Sariel Har-Peled
10.4230/LIPIcs.SoCG.2016.18
N. Alon, P. Seymour, and R. Thomas. A separator theorem for nonplanar graphs. In J. Amer. Math. Soc., volume 3, pages 801-808, 1990. URL: http://dx.doi.org/10.1090/S0894-0347-1990-1065053-0.
http://dx.doi.org/10.1090/S0894-0347-1990-1065053-0
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http://dx.doi.org/10.1137/S0097539702416402
F. Aurenhammer, R. Klein, and D.-T. Lee. Voronoi Diagrams and Delaunay Triangulations. World Scientific, 2013. URL: http://dx.doi.org/10.1142/8685.
http://dx.doi.org/10.1142/8685
S. Bandyapadhyay and K. R. Varadarajan. On variants of k-means clustering. CoRR, abs/1512.02985, 2015. Also in SoCG 16. URL: http://arxiv.org/abs/1512.02985.
http://arxiv.org/abs/1512.02985
S. Basu, R. Pollack, and M. F. Roy. Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics. Springer, 2006. URL: http://dx.doi.org/10.1007/3-540-33099-2.
http://dx.doi.org/10.1007/3-540-33099-2
M. de Berg, O. Cheong, M. van Kreveld, and M. H. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, Santa Clara, CA, USA, 3rd edition, 2008. URL: http://www.cs.uu.nl/geobook/.
http://www.cs.uu.nl/geobook/
V. V. S. P. Bhattiprolu and S. Har-Peled. Separating a voronoi diagram via local search. CoRR, abs/1401.0174, 2014. URL: http://arxiv.org/abs/1401.0174.
http://arxiv.org/abs/1401.0174
J. Böttcher, K. P. Pruessmann, A. Taraz, and A. Würfl. Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs. Eur. J. Comb., 31(5):1217-1227, July 2010. URL: http://dx.doi.org/10.1016/j.ejc.2009.10.010.
http://dx.doi.org/10.1016/j.ejc.2009.10.010
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http://dx.doi.org/10.1016/S0196-6774(02)00294-8
T. M. Chan and S. Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete Comput. Geom., 48:373-392, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9417-5.
http://dx.doi.org/10.1007/s00454-012-9417-5
Z. Dvorak and S. Norin. Strongly sublinear separators and polynomial expansion. ArXiv e-prints, April 2015. URL: http://arxiv.org/abs/1504.04821,
http://arxiv.org/abs/1504.04821
A. Efrat and S. Har-Peled. Guarding galleries and terrains. Inform. Process. Lett., 100(6):238-245, 2006. URL: http://dx.doi.org/10.1016/j.ipl.2006.05.014.
http://dx.doi.org/10.1016/j.ipl.2006.05.014
D. Eisenstat and P. N. Klein. Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs. In Proc. 45th Annu. ACM Sympos. Theory Comput. (STOC), pages 735-744, 2013. URL: http://dx.doi.org/10.1145/2488608.2488702.
http://dx.doi.org/10.1145/2488608.2488702
T. Erlebach, K. Jansen, and E. Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput., 34(6):1302-1323, 2005. URL: http://dx.doi.org/10.1137/S0097539702402676.
http://dx.doi.org/10.1137/S0097539702402676
J. Fakcharoenphol and S. Rao. Planar graphs, negative weight edges, shortest paths, and near linear time. J. Comput. Sys. Sci., 72(5):868-889, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2005.05.007.
http://dx.doi.org/10.1016/j.jcss.2005.05.007
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http://dx.doi.org/10.1016/0196-6774(84)90019-1
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http://arxiv.org/abs/0809.2554
S. Har-Peled. Geometric Approximation Algorithms, volume 173 of Mathematical Surveys and Monographs. Amer. Math. Soc., Boston, MA, USA, 2011. URL: http://sarielhp.org/book/, URL: http://dx.doi.org/10.1090/surv/173.
http://dx.doi.org/10.1090/surv/173
S. Har-Peled. A simple proof of the existence of a planar separator. ArXiv e-prints, 2011. URL: http://arxiv.org/abs/1105.0103,
http://arxiv.org/abs/1105.0103
S. Har-Peled and M. Lee. Weighted geometric set cover problems revisited. J. Comput. Geom., 3(1):65-85, 2012. URL: http://sarielhp.org/papers/08/expand_cover/.
http://sarielhp.org/papers/08/expand_cover/
S. Har-Peled and K. Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. In Proc. 23nd Annu. Euro. Sympos. Alg.\CNFESA, volume 9294 of Lect. Notes in Comp. Sci., pages 717-728, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_60.
http://dx.doi.org/10.1007/978-3-662-48350-3_60
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http://dx.doi.org/10.1137/060649562
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N. H. Mustafa, R. Raman, and S. Ray. Settling the APX-hardness status for geometric set cover. In Proc. 55th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 541-550, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.64.
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N. H. Mustafa and S. Ray. Improved results on geometric hitting set problems. Discrete Comput. Geom., 44(4):883-895, 2010. URL: http://dx.doi.org/10.1007/s00454-010-9285-9.
http://dx.doi.org/10.1007/s00454-010-9285-9
J. Nešetřil and P. Ossona de Mendez. Grad and classes with bounded expansion I. decompositions. Eur. J. Comb., 29(3):760-776, 2008.
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Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On Visibility Representations of Non-Planar Graphs
A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption that an embedding in the plane of the input graph is fixed and we are looking for an RVR that reflects this embedding. We show that in this case the problem can be solved in polynomial time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden configurations, which extends the set known for straight-line drawings of 1-plane graphs. These forbidden configurations can be tested for in linear time, and so in linear time we can test whether a 1-plane graph has an RVR and either compute such a representation or report a negative witness. Finally, we discuss some extensions of our study to the case when the embedding is not fixed but the RVR can have at most one crossing per edge.
Visibility Representations
1-Planarity
Recognition Algorithm
Forbidden Configuration
19:1-19:16
Regular Paper
Therese
Biedl
Therese Biedl
Giuseppe
Liotta
Giuseppe Liotta
Fabrizio
Montecchiani
Fabrizio Montecchiani
10.4230/LIPIcs.SoCG.2016.19
Md. Jawaherul Alam, Franz J. Brandenburg, and Stephen G. Kobourov. Straight-line grid drawings of 3-connected 1-planar graphs. In GD 2013, volume 8242 of LNCS, pages 83-94. Springer, 2013.
Therese Biedl, Giuseppe Liotta, and Fabrizio Montecchiani. On visibility representations of non-planar graphs. CoRR, abs/1511.08592, 2015. URL: http://arxiv.org/abs/1511.08592.
http://arxiv.org/abs/1511.08592
Therese Biedl, Anna Lubiw, Mark Petrick, and Michael J. Spriggs. Morphing orthogonal planar graph drawings. ACM Trans. Algorithms, 9(4):29, 2013. URL: http://dx.doi.org/10.1145/2500118.
http://dx.doi.org/10.1145/2500118
Prosenjit Bose, Alice M. Dean, Joan P. Hutchinson, and Thomas C. Shermer. On rectangle visibility graphs. In GD 1996, volume 1190 of LNCS, pages 25-44. Springer, 1997.
Franz J. Brandenburg. 1-Visibility representations of 1-planar graphs. J. Graph Algorithms Appl., 18(3):421-438, 2014.
Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput., 14(1):210-223, 1985.
Sabine Cornelsen and Andreas Karrenbauer. Accelerated bend minimization. J. Graph Algorithms Appl., 16(3):635-650, 2012.
Alice M. Dean, William S. Evans, Ellen Gethner, Joshua D. Laison, Mohammad Ali Safari, and William T. Trotter. Bar k-visibility graphs. J. Graph Algorithms Appl., 11(1):45-59, 2007.
Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, 1999.
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William S. Evans, Michael Kaufmann, William Lenhart, Tamara Mchedlidze, and Stephen K. Wismath. Bar 1-visibility graphs vs. other nearly planar graphs. J. Graph Algorithms Appl., 18(5):721-739, 2014.
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Creative Commons Attribution 3.0 Unported license
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Delaunay Triangulations on Orientable Surfaces of Low Genus
Earlier work on Delaunay triangulation of point sets on the 2D flat torus, which is locally isometric to the Euclidean plane, was based on lifting the point set to a locally isometric 9-sheeted covering space of the torus. Under mild conditions the Delaunay triangulation of the lifted point set, consisting of 9 copies of the input set, projects to the Delaunay triangulation of the input set.
We improve and generalize this work. First we present a new construction based on an 8-sheeted covering space, which shows that eight copies suffice for the standard flat torus. Then we generalize this construction to the context of compact orientable surfaces of higher genus, which are locally isometric to the hyperbolic plane. We investigate more thoroughly the Bolza surface, homeomorphic to a sphere with two handles, both because it is the hyperbolic surface with lowest genus, and because triangulations on the Bolza surface have applications in various fields such as neuromathematics and cosmological models. While the general properties (existence results of appropriate covering spaces) show similarities with the results for the flat case, explicit constructions and their proofs are much more complex, even in the case of the apparently simple Bolza surface. One of the main reasons is the fact that two hyperbolic translations do not commute in general.
To the best of our knowledge, the results in this paper are the first ones of this kind. The interest of our contribution lies not only in the results, but most of all in the construction of covering spaces itself and the study of their properties.
covering spaces
hyperbolic surfaces
finitely presented groups
Fuchsian groups
systole
20:1-20:17
Regular Paper
Mikhail
Bogdanov
Mikhail Bogdanov
Monique
Teillaud
Monique Teillaud
Gert
Vegter
Gert Vegter
10.4230/LIPIcs.SoCG.2016.20
M. Artin. Algebra. Prentice-Hall, 2002.
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R. vd Weygaert, G. Vegter, H. Edelsbrunner, B.J.T. Jones, P. Pranav, C. Park, W. A. Hellwing, B. Eldering, N. Kruithof, E.G.P. Bos, J. Hidding, J. Feldbrugge, E. ten Have, M. v Engelen, M. Caroli, and M. Teillaud. Alpha, Betti and the megaparsec universe: on the homology and topology of the cosmic web. In Trans. on Comp. Science XIV, volume 6970 of LNCS, pages 60-101. Springer-Verlag, 2011. URL: http://dx.doi.org/10.1007/978-3-642-25249-5_3.
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An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams
Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before.
Order-k Voronoi Diagrams
Abstract Voronoi Diagrams
Randomized Geometric Algorithms
21:1-21:15
Regular Paper
Cecilia
Bohler
Cecilia Bohler
Rolf
Klein
Rolf Klein
Chih-Hung
Liu
Chih-Hung Liu
10.4230/LIPIcs.SoCG.2016.21
Pankaj K. Agarwal, Mark de Berg, Jirí Matousek, and Otfried Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27(3):654-667, 1998.
Franz Aurenhammer and Otfried Schwarzkopf. A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams. In Proceedings of the Seventh Annual Symposium on Computational Geometry (SoCG), pages 142-151, 1991.
Cecilia Bohler, Panagiotis Cheilaris, Rolf Klein, Chih-Hung Liu, Evanthia Papadopoulou, and Maksym Zavershynskyi. On the complexity of higher order abstract Voronoi diagrams. Computational Geometry, 48(8):539-551, 2015.
Cecilia Bohler and Rolf Klein. Abstract Voronoi diagrams with disconnected regions. Int. J. Comput. Geometry Appl., 24(4):347-372, 2014.
Cecilia Bohler, Chih-Hung Liu, Evanthia Papadopoulou, and Maksym Zavershynskyi. A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams. In Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC), pages 27-37, 2014.
Jean-Daniel Boissonnat, Olivier Devillers, and Monique Teillaud. A semidynamic construction of higher-order Voronoi diagrams and its randomized analysis. Algorithmica, 9(4):329-356, 1993.
Timothy M. Chan. Random sampling, halfspace range reporting, and construction of (less= k)-levels in three dimensions. SIAM J. Comput., 30(2):561-575, 2000.
Timothy M. Chan and Konstantinos Tsakalidis. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. In Proceeding of the 31st International Symposium on Computational Geometry (SoCG), pages 719-732, 2015.
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Bernard Chazelle and Herbert Edelsbrunner. An improved algorithm for constructing k th-order Voronoi diagrams. IEEE Trans. Computers, 36(11):1349-1354, 1987.
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Kenneth L. Clarkson. New applications of random sampling in computational geometry. Discrete & Computational Geometry, 2:195-222, 1987.
Kenneth L. Clarkson and Peter W. Shor. Application of random sampling in computational geometry, II. Discrete & Computational Geometry, 4:387-421, 1989.
Andreas Gemsa, D. T. Lee, Chih-Hung Liu, and Dorothea Wagner. Higher order city Voronoi diagrams. In Proceedings of the 13th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 59-70, 2012.
David Haussler and Emo Welzl. epsilon-nets and simplex range queries. Discrete & Computational Geometry, 2:127-151, 1987.
Rolf Klein. Concrete and Abstract Voronoi Diagrams, volume 400 of Lecture Notes in Computer Science. Springer, 1989.
Rolf Klein, Elmar Langetepe, and Zahra Nilforoushan. Abstract Voronoi diagrams revisited. Comput. Geom., 42(9):885-902, 2009.
Rolf Klein, Kurt Mehlhorn, and Stefan Meiser. Randomized incremental construction of abstract Voronoi diagrams. Comput. Geom., 3:157-184, 1993.
D. T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Computers, 31(6):478-487, 1982.
Chih-Hung Liu and D. T. Lee. Higher-order geodesic Voronoi diagrams in a polygonal domain with holes. In Proceedings of the Twenty-Fourth Annual Symposium on Discrete Algorithms (SODA), pages 1633-1645, 2013.
Kurt Mehlhorn, Stefan Meiser, and Ronald Rasch. Furthest site abstract Voronoi diagrams. Int. J. Comput. Geometry Appl., 11(6):583-616, 2001.
Ketan Mulmuley. Computational geometry - an introduction through randomized algorithms. Prentice Hall, 1994.
Evanthia Papadopoulou and Marksim Zavershynskyi. On higher order Voronoi diagrams of line segments. Algorithmica, 2014. Published on-line.
Edgar A. Ramos. On range reporting, ray shooting and k-level construction. In Proceedings of the Fifteenth Annual Symposium on Computational Geometry (SoCG), pages 390-399, 1999.
Robert Endre Tarjan and Christopher J. Van Wyk. An O(n log log n)-time algorithm for triangulating a simple polygon. SIAM J. Comput., 17(1):143-178, 1988.
Creative Commons Attribution 3.0 Unported license
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All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs
For an undirected n-vertex graph G with non-negative edge-weights, we consider the following type of query: given two vertices s and t in G, what is the weight of a minimum st-cut in G? We solve this problem in preprocessing time O(n log^3 n) for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space O(n) that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is 2^{O(g^2)}.
minimum cuts
surface-embedded graphs
Gomory-Hu tree
22:1-22:16
Regular Paper
Glencora
Borradaile
Glencora Borradaile
David
Eppstein
David Eppstein
Amir
Nayyeri
Amir Nayyeri
Christian
Wulff-Nilsen
Christian Wulff-Nilsen
10.4230/LIPIcs.SoCG.2016.22
G. Borradaile, E. Chambers, K. Fox, and A. Nayyeri. Computing minimum homology basis and minimum cycle basis in surface embedded graphs. In submission., 2015.
G. Borradaile, P.N. Klein, S. Mozes, Y. Nussbaum, and C. Wulff-Nilsen. Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. In Proc. 52nd Symp. Found. of Computer Science (FOCS 2011), pages 170-179, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.73.
http://dx.doi.org/10.1109/FOCS.2011.73
G. Borradaile, P. Sankowski, and C. Wulff-Nilsen. Min st-cut oracle for planar graphs with near-linear preprocessing time. In Proc. 51st IEEE Symp. Foundations of Computer Science (FOCS 2010), pages 601-610, 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.63.
http://dx.doi.org/10.1109/FOCS.2010.63
G. Borradaile, P. Sankowski, and C. Wulff-Nilsen. Min st-cut oracle for planar graphs with near-linear preprocessing time. ACM Trans. Algorithms, 2014. To appear.
S. Cabello, É. Colin de Verdière, and F. Lazarus. Finding shortest non-trivial cycles in directed graphs on surfaces. In Proc. of the 26th Annual Symposium on Computational Geometry, SoCG'10, pages 156-165, New York, NY, USA, 2010. ACM. URL: http://dx.doi.org/10.1145/1810959.1810988.
http://dx.doi.org/10.1145/1810959.1810988
E. Chambers, J. Erickson, and A. Nayyeri. Homology flows, cohomology cuts. In Proc. 41st ACM Symposium on Theory of Computing (STOC'09), pages 273-282, 2009. URL: http://dx.doi.org/10.1145/1536414.1536453.
http://dx.doi.org/10.1145/1536414.1536453
E. Chambers, J. Erickson, and A. Nayyeri. Minimum cuts and shortest homologous cycles. In Proc. 25th Symp. Computational Geometry (SoCG'09), pages 377-385, 2009. URL: http://dx.doi.org/10.1145/1542362.1542426.
http://dx.doi.org/10.1145/1542362.1542426
Erin W. Chambers, Jeff Erickson, and Amir Nayyeri. Homology flows, cohomology cuts. SIAM J. Comput., 41:1605-1634, 2009. URL: http://dx.doi.org/10.1137/090766863.
http://dx.doi.org/10.1137/090766863
E. Demaine, G. Landau, and O. Weimann. On Cartesian trees and range minimum queries. Algorithmica, 68(3):610-625, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9683-x.
http://dx.doi.org/10.1007/s00453-012-9683-x
J. Erickson. Shortest non-trivial cycles in directed surface graphs. In Proc. of the 27th Annual Symp. on Computational Geometry, SoCG'11, pages 236-243, New York, NY, USA, 2011. ACM. URL: http://dx.doi.org/10.1145/1998196.1998231.
http://dx.doi.org/10.1145/1998196.1998231
J. Erickson, K. Fox, and A. Nayyeri. Global minimum cuts in surface embedded graphs. In Proc. 23rd Annual ACM-SIAM Symp. on Discrete Algorithms, pages 1309-1318, 2012.
J. Erickson and A. Nayyeri. Minimum cuts and shortest non-separating cycles via homology covers. In Proc. 22nd ACM-SIAM Symp. Discrete Algorithms (SODA 2011), pages 1166-1176, 2011.
J. Erickson and A. Sidiropoulos. A near-optimal approximation algorithm for asymmetric tsp on embedded graphs. In Proc. 30th Annual Symposium on Computational Geometry, SOCG'14, pages 130:130-130:135. ACM, 2014. URL: http://dx.doi.org/10.1145/2582112.2582136.
http://dx.doi.org/10.1145/2582112.2582136
J. Fakcharoenphol and S. Rao. Planar graphs, negative weight edges, shortest paths, and near linear time. J. Comput. Syst. Sci., 72(5):868-889, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2005.05.007.
http://dx.doi.org/10.1016/j.jcss.2005.05.007
R. Gomory and T. Hu. Multi-terminal network flows. Journal of SIAM, 9(4):551-570, 1961. URL: http://dx.doi.org/10.1137/0109047.
http://dx.doi.org/10.1137/0109047
D. Gusfield. Very simple methods for all pairs network flow analysis. SIAM J. Comput., 19(1):143-155, 1990. URL: http://dx.doi.org/10.1137/0219009.
http://dx.doi.org/10.1137/0219009
D. Hartvigsen and R. Mardon. The all-pairs min cut problem and the minimum cycle basis problem on planar graphs. SIAM J. on Discrete Math., 7(3):403-418, 1994. URL: http://dx.doi.org/10.1137/S0895480190177042.
http://dx.doi.org/10.1137/S0895480190177042
A. Hatcher. Algebraic Topology. Cambridge University Press, 2002.
J. Łacki, Y. Nussbaum, P. Sankowski, and C. Wulff-Nilsen. Single source - all sinks max flows in planar digraphs. In Proc. 53rd Symp. Found. of Computer Science (FOCS 2012), pages 599-608, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.66.
http://dx.doi.org/10.1109/FOCS.2012.66
G. Miller. Finding small simple cycle separators for 2-connected planar graphs. J. Comput. Syst. Sci., 32(3):265-279, 1986. URL: http://dx.doi.org/10.1016/0022-0000(86)90030-9.
http://dx.doi.org/10.1016/0022-0000(86)90030-9
K. Mulmuley, V. Vazirani, and U. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):345-354, 1987.
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https://creativecommons.org/licenses/by/3.0/legalcode
Minimum Cycle and Homology Bases of Surface Embedded Graphs
We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the 1-dimensional (Z_2)-homology classes) of an undirected graph embedded on an orientable surface of genus g. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 1-skeleton of any graph is exactly its minimum cycle basis.
For the minimum cycle basis problem, we give a deterministic O(n^omega + 2^2g n^2)-time algorithm. The best known existing algorithms for surface embedded graphs are those for general sparse graphs: an O(n^omega) time Monte Carlo algorithm [Amaldi et. al., ESA'09] and a deterministic O(n^3) time algorithm [Mehlhorn and Michail, TALG'09]. For the minimum homology basis problem, we give an O(g^3 n log n)-time algorithm, improving on existing algorithms for many values of g and n.
Cycle basis
Homology basis
Topological graph theory
23:1-23:15
Regular Paper
Glencora
Borradaile
Glencora Borradaile
Erin Wolf
Chambers
Erin Wolf Chambers
Kyle
Fox
Kyle Fox
Amir
Nayyeri
Amir Nayyeri
10.4230/LIPIcs.SoCG.2016.23
L. Aleksandrov and H. Djidjev. Linear algorithms for partitioning embedded graphs of bounded genus. SIAM J. of Disc. Math., 9(1):129-150, 1996.
Edoardo Amaldi, Claudio Iuliano, Tomasz Jurkiewicz, Kurt Mehlhorn, and Romeo Rizzi. Breaking the O(m² n) barrier for minimum cycle bases. In Proc. 17th Ann. Euro. Symp. Algo., pages 301-312, 2009.
Franziska Berger, Peter Gritzmann, and Sven de Vries. Minimum cycle bases for network graphs. Algorithmica, 40(1):51-62, 2004.
G. Borradaile, P. Sankowski, and C. Wulff-Nilsen. Min st-cut oracle for planar graphs with near-linear preprocessing time. ACM Trans. Algo., 11(3):16, 2015.
Glencora Borradaile, David Eppstein, Amir Nayyeri, and Christian Wulff-Nilsen. All-pairs minimum cuts in near-linear time for surface-embedded graphs. In Proc. 32nd Ann. Int. Symp. Comput. Geom., 2016.
Oleksiy Busaryev, Sergio Cabello, Chao Chen, Tamal K. Dey, and Yusu Wang. Annotating simplicies with a homology basis and its applications. In Proc. 13th Scandinavian Workshop on Algo. Theory, pages 189-200, 2012.
Sergio Cabello, Erin W. Chambers, and Jeff Erickson. Multiple-source shortest paths in embedded graphs. SIAM J. Comput., 42(4):1542-1571, 2013.
A. C. Cassell, J. C. de Henderson, and K. Ramachandran. Cycle bases of minimal measure for the structural analysis of skeletal structures by the flexibility method. Proc. R. Soc. Lond. Ser. A, 350:61-70, 1976.
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Tamal K. Dey, Jian Sun, and Yusu Wang. Approximating loops in a shortest homology basis from point data. In Proc. 26th Ann. Symp. Comput. Geom., pages 166-175, 2010.
David Eppstein. Dynamic generators of topologically embedded graphs. In Proc. 14th Ann. ACM-SIAM Symp. Disc. Algo., pages 599-608, 2003.
Jeff Erickson. Shortest non-trivial cycles in directed surface graphs. In Proc. 27th Ann. Symp. Comput. Geom., pages 236-243, 2011.
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Jeff Erickson and Kim Whittlesey. Greedy optimal homotopy and homology generators. In Proc. 16th Ann. ACM-SIAM Symp. on Disc. Algo., pages 1038-1046, 2005.
Kyle Fox. Shortest non-trivial cycles in directed and undirected surface graphs. In Proc. 24th Ann. ACM-SIAM Symp. Disc. Algo., pages 352-364, 2013.
Alexander Golynski and Joseph D. Horton. A polynomial time algorithm to find the minimum cycle basis of a regular matroid. In Proc. 8th Scandinavian Workshop on Algo. Theory, pages 200-209, 2002.
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Telikepalli Kavitha, Kurt Mehlhorn, Dimitrios Michail, and Katarzyna E. Paluch. An Õ(m²n) algorithm for minimum cycle basis of graphs. Algorithmica, 52(3):333-349, 2008.
G. Kirchhoff. Ueber die auflösung der gleichungen, auf welche man bei der untersuchung der linearen vertheilung galvanischer ströme geführt wird. Poggendorf Ann. Physik, 72:497-508, 1847. English transl. in Trans. Inst. Radio Engrs. CT-5 (1958), pp. 4-7.
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Finding Non-Orientable Surfaces in 3-Manifolds
We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds.
We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.
3-manifold
low-dimensional topology
embedding
non-orientability
normal surfaces
24:1-24:15
Regular Paper
Benjamin A.
Burton
Benjamin A. Burton
Arnaud
de Mesmay
Arnaud de Mesmay
Uli
Wagner
Uli Wagner
10.4230/LIPIcs.SoCG.2016.24
Ian Agol, Joel Hass, and William Thurston. The computational complexity of knot genus and spanning area. Transactions of the American Mathematical Society, 358:3821-3850, 2006.
Glen E. Bredon and John W. Wood. Non-orientable surfaces in orientable 3-manifolds. Invent. Math., 7:83-110, 1969.
Benjamin A. Burton. A new approach to crushing 3-manifold triangulations. Discrete &Computational Geometry, 52(1):116-139, 2014.
Benjamin A. Burton, Arnaud de Mesmay, and Uli Wagner. Finding non-orientable surfaces in 3-manifolds. arXiv:1602.07907, 2016.
Benjamin A. Burton and Melih Ozlen. Computing the crosscap number of a knot using integer programming and normal surfaces. ACM Trans. Math. Softw., 39(1):4:1-4:18, November 2012.
Werner End. Non-orientable surfaces in 3-manifolds. Archiv der Mathematik, 59(2):173-185, 1992.
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Joel Hass and Greg Kuperberg. New results on the complexity of recognizing the 3-sphere. In Oberwolfach Reports, volume 9, pages 1425-1426, 2012.
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Miwa Iwakura and Chuichiro Hayashi. Non-orientable fundamental surfaces in lens spaces. Topology and its Applications, 156(10):1753-1766, 2009.
William Jaco and Ulrich Oertel. An algorithm to decide if a 3-manifold is a Haken manifold. Topology, 23(2):195-209, 1984.
William Jaco and J. Hyam Rubinstein. 0-efficient triangulations of 3-manifolds. Journal of Differential Geometry, 65:61-168, 2003.
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K Johannson. Homotopy equivalence of 3-manifolds with boundary. Lecture Notes in Math, 761, 1979.
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Greg Kuperberg. Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization. arXiv:1508.06720, 2015.
Adam Levine, Daniel Ruberman, and Sašo Strle. Nonorientable surfaces in homology cobordisms. Geometry &Topology, 19(1):439-494, 2015.
Tao Li. Heegaard surfaces and measured laminations, I: the Waldhausen conjecture. Inventiones mathematicae, 167(1):135-177, 2007.
Tao Li. An algorithm to determine the Heegaard genus of a 3-manifold. Geometry &Topology, 15(2):1029-1106, 2011.
Jiří Matousšek, Martin Tancer, and Uli Wagner. Hardness of embedding simplicial complexes in R^d. Journal of the European Mathematical Society, 13(2):259-295, 2011.
Jiří Matoušek, Eric Sedgwick, Martin Tancer, and Uli Wagner. Embeddability in the 3-sphere is decidable. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 78:78-78:84, New York, NY, USA, 2014. ACM.
Sergei V. Matveev. Algorithmic topology and classification of 3-manifolds, volume 9 of Algorithms and Computation in Mathematics. Springer-Verlag, 2003.
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Richard Rannard. Incompressible surfaces in Seifert fibered spaces. Topology and its Applications, 72(1):19-30, 1996.
Joachim Hyam Rubinstein. On 3-manifolds that have finite fundamental group and contain Klein bottles. Transactions of the American Mathematical Society, 251:129-137, 1979.
Joachim Hyam Rubinstein. Nonorientable surfaces in some non-Haken 3-manifolds. Transactions of the American Mathematical Society, 270(2):503-524, 1982.
Joachim Hyam Rubinstein. An algorithm to recognize the 3-sphere. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 601-611, Basel, 1995. Birkhäuser.
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Structure and Stability of the 1-Dimensional Mapper
Given a continuous function f:X->R and a cover I of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover f^{-1}(I). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework relating the structure of the Mapper to that of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover I goes to zero.
Mapper
Reeb Graph
Extended Persistence
Topological Data Analysis
25:1-25:16
Regular Paper
Mathieu
Carrière
Mathieu Carrière
Steve
Oudot
Steve Oudot
10.4230/LIPIcs.SoCG.2016.25
M. Alagappan. From 5 to 13: Redefining the Positions in Basketball. MIT Sloan Sports Analytics Conference, 2012.
A. Babu. Zigzag Coarsenings, Mapper Stability and Gene-network Analyses. 2013.
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 Proc. 30th Sympos. Comput. Geom., pages 464-473, 2014.
U. Bauer, E. Munch, and Y. Wang. Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs. In Proc. 31st Sympos. Comput. Geom., 2015.
S. Biasotti, D. Giorgi, M. Spagnuolo, and B. Falcidieno. Reeb Graphs for Shape Analysis and Applications. Theor. Comput. Sci., 392(1-3):5-22, 2008.
G. Carlsson, V. de Silva, and D. Morozov. Zigzag Persistent Homology and Real-valued Functions. In Proc. 25th Sympos. Comput. Geom., pages 247-256, 2009.
H. Carr and D. Duke. Joint Contour Nets. IEEE Trans. Vis. Comput. Graph., 20(8):1100-1113, 2014.
M. Carrière and S. Oudot. Structure and Stability of the 1-Dimensional Mapper. arXiv 1511.05823, 2015.
M. Carrière, S. Oudot, and M. Ovsjanikov. Stable Topological Signatures for Points on 3D Shapes. In Proc. 13th Sympos. Geom. Proc., 2015.
A. Chattopadhyay, H. Carr, D. Duke, Z. Geng, and O. Saeki. Multivariate Topology Simplification. arXiv 1509.04465, 2015.
F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The Structure and Stability of Persistence Modules. arXiv 1207.3674, 2012.
F. Chazal, L. Guibas, S. Oudot, and P. Skraba. Analysis of scalar fields over point cloud data. In Proc. 20th Sympos. Discr. Algo., pages 1021-1030, 2009.
F. Chazal and J. Sun. Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph. In Proc. 30th Sympos. Comput. Geom., 2014.
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Extending persistence using Poincaré and Lefschetz duality. Found. Comput. Math., 9(1):79-103, 2009.
É. Colin de Verdière, G. Ginot, and X. Goaoc. Multinerves and helly numbers of acyclic families. In Proc. 28th Sympos. Comput. Geom., pages 209-218, 2012.
V. de Silva, E. Munch, and A. Patel. Categorified Reeb Graphs. arXiv 1501.04147, 2015.
T. Dey, F. Mémoli, and Y. Wang. Mutiscale Mapper: A Framework for Topological Summarization of Data and Maps. arXiv 1504.03763, 2015.
T. Dey and Y. Wang. Reeb graphs: Approximation and persistence. Discr. Comput. Geom., 49(1):46-73, 2013.
E. Munch and B. Wang. Convergence between Categorical Representations of Reeb Space and Mapper. In Proc. 32nd Sympos. Comput. Geom., 2016.
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. Proc. National Acad. Sci., 108(17):7265-7270, 2011.
G. Singh, F. Mémoli, and G. Carlsson. Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition. In Sympos. PB Graphics, 2007.
R. B. Stovner. On the Mapper Algorithm. 2012.
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Max-Sum Diversity Via Convex Programming
Diversity maximization is an important concept in information retrieval, computational geometry and operations research. Usually, it is a variant of the following problem: Given a ground set, constraints, and a function f that measures diversity of a subset, the task is to select a feasible subset S such that f(S) is maximized. The sum-dispersion function f(S) which is the sum of the pairwise distances in S, is in this context a prominent diversification measure. The corresponding diversity maximization is the "max-sum" or "sum-sum" diversification. Many recent results deal with the design of constant-factor approximation algorithms of diversification problems involving sum-dispersion function under a matroid constraint.
In this paper, we present a PTAS for the max-sum diversity problem under a matroid constraint for distances d(.,.) of negative type. Distances of negative type are, for example, metric distances stemming from the l_2 and l_1 norms, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search.
Our algorithm is based on techniques developed in geometric algorithms like metric embeddings and convex optimization. We show that one can compute a fractional solution of the usually non-convex relaxation of the problem which yields an upper bound on the optimum integer solution. Starting from this fractional solution, we employ a deterministic rounding approach which only incurs a small loss in terms of objective, thus leading to a PTAS. This technique can be applied to other previously studied variants of the max-sum dispersion function, including combinations of diversity with linear-score maximization, improving over the previous constant-factor approximation algorithms.
Geometric Dispersion
Embeddings
Approximation Algorithms
Convex Programming
Matroids
26:1-26:14
Regular Paper
Alfonso
Cevallos
Alfonso Cevallos
Friedrich
Eisenbrand
Friedrich Eisenbrand
Rico
Zenklusen
Rico Zenklusen
10.4230/LIPIcs.SoCG.2016.26
Z. Abbassi, V. S. Mirrokni, and M. Thakur. Diversity maximization under matroid constraints. In Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 32-40. ACM, 2013.
A. A. Ageev and M. I. Sviridenko. Pipage rounding: a new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization, 8(23):307-328, 2004.
N. Alon, S. Arora, R. Manokaran, D. Moshkovitz, and O. Weinstein. Inapproximability of densest κ-subgraph from average case hardness. Unpublished manuscript, 2011.
S. Bhattacharya, S. Gollapudi, and K. Munagala. Consideration set generation in commerce search. In Proceedings of the 20th international conference on World wide web, pages 317-326. ACM, 2011.
B. Birnbaum and K. J. Goldman. An improved analysis for a greedy remote-clique algorithm using factor-revealing LPs. Algorithmica, 55(1):42-59, 2009.
L. M. Blumenthal. Theory and Applications of Distance Geometry, volume 347. Oxford, 1953.
A. Borodin, H. C. Lee, and Y. Ye. Max-sum diversification, monotone submodular functions and dynamic updates. In Proceedings of the 31st Symposium on Principles of Database Systems, pages 155-166. ACM, 2012.
G. Calinescu, C. Chekuri, M. Pál, and J. Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011.
M. S. Charikar. Similarity estimation techniques from rounding algorithms. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 380-388. ACM, 2002.
C. Chekuri, J. Vondrák, and R. Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science, pages 575-584, 2010.
M. M. Deza and M. Laurent. Geometry of Cuts and Metrics. Springer-Verlag, Berlin, 1997.
M. M. Deza and H. Maehara. Metric transforms and euclidean embeddings. Transactions of the American Mathematical Society, 317(2):661-671, 1990.
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F. R. Giles. Submodular Functions, Graphs and Integer Polyhedra. PhD thesis, University of Waterloo, 1975.
S. Gollapudi and A. Sharma. An axiomatic approach for result diversification. In Proceedings of the 18th International Conference on World Wide Web, pages 381-390. ACM, 2009.
J. C. Gower and P. Legendre. Metric and euclidean properties of dissimilarity coefficients. Journal of Classification, 3(1):5-48, 1986.
M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 1988.
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L. G. Khachiyan. A polynomial algorithm in linear programming. Doklady Akademii Nauk SSSR, 244:1093-1097, 1979.
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Q. Lv, M. Charikar, and K. Li. Image similarity search with compact data structures. In Proceedings of the 13th ACM International Conference on Information and Knowledge Management, pages 208-217. ACM, 2004.
K Makarychev, W Schudy, and M Sviridenko. Concentration inequalities for nonlinear matroid intersection. Random Structures &Algorithms, 46(3):541-571, 2015.
C. D. Manning, P. Raghavan, and H. Schütze. Introduction to Information Retrieval, volume 1. Cambridge university press Cambridge, 2008.
J. Matoušek. Lecture notes on metric embeddings. https://kam.mff.cuni.cz/~matousek/ba-a4.pdf, 2013.
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Dynamic Streaming Algorithms for Epsilon-Kernels
Introduced by Agarwal, Har-Peled, and Varadarajan [J. ACM, 2004], an epsilon-kernel of a point set is a coreset that can be used to approximate the width, minimum enclosing cylinder, minimum bounding box, and solve various related geometric optimization problems. Such coresets form one of the most important tools in the design of linear-time approximation algorithms in computational geometry, as well as efficient insertion-only streaming algorithms and dynamic (non-streaming) data structures. In this paper, we continue the theme and explore dynamic streaming algorithms (in the so-called turnstile model).
Andoni and Nguyen [SODA 2012] described a dynamic streaming algorithm for maintaining a (1+epsilon)-approximation of the width using O(polylog U) space and update time for a point set in [U]^d for any constant dimension d and any constant epsilon>0. Their sketch, based on a "polynomial method", does not explicitly maintain an epsilon-kernel. We extend their method to maintain an epsilon-kernel, and at the same time reduce some of logarithmic factors. As an application, we obtain the first randomized dynamic streaming algorithm for the width problem (and related geometric optimization problems) that supports k outliers, using poly(k, log U) space and time.
coresets
streaming algorithms
dynamic algorithms
polynomial method
randomization
outliers
27:1-27:11
Regular Paper
Timothy M.
Chan
Timothy M. Chan
10.4230/LIPIcs.SoCG.2016.27
Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. J. ACM, 51(4):606-635, 2004.
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, Sariel Har-Peled, and Hai Yu. Robust shape fitting via peeling and grating coresets. Discrete & Computational Geometry, 39(1-3):38-58, 2008. URL: http://dx.doi.org/10.1007/s00454-007-9013-2.
http://dx.doi.org/10.1007/s00454-007-9013-2
Pankaj K. Agarwal, Jeff M. Phillips, and Hai Yu. Stability of ε-kernels. In Proceedings of the 18th Annual European Symposium on Algorithms, Part I, pages 487-499, 2010. URL: http://dx.doi.org/10.1007/978-3-642-15775-2_42.
http://dx.doi.org/10.1007/978-3-642-15775-2_42
Alexandr Andoni and Huy L. Nguyen. Width of points in the streaming model. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 447-452, 2012. ACM Trans. Algorithms, to appear.
Sunil Arya and Timothy M. Chan. Better ε-dependencies for offline approximate nearest neighbor search, Euclidean minimum spanning trees, and ε-kernels. In Proceedings of the 30th Annual Symposium on Computational Geometry, pages 416-425, 2014. URL: http://dx.doi.org/10.1145/2582112.2582161.
http://dx.doi.org/10.1145/2582112.2582161
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.
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http://dx.doi.org/10.1145/235809.235813
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http://dx.doi.org/10.1007/s00453-010-9392-2
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Two Approaches to Building Time-Windowed Geometric Data Structures
Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.'s O(n log^2 n) and O(n log n loglog n) solutions respectively.
Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.
time window
geometric data structures
range searching
dynamic convex hull
28:1-28:15
Regular Paper
Timothy M.
Chan
Timothy M. Chan
Simon
Pratt
Simon Pratt
10.4230/LIPIcs.SoCG.2016.28
Pankaj K. Agarwal and Jirı Matoušek. Dynamic half-space range reporting and its applications. Algorithmica, 13(4):325-345, 1995.
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http://dx.doi.org/10.1016/0925-7721(91)90001-U
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http://dx.doi.org/10.1109/SFCS.1994.365724
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Drago Bokal, Sergio Cabello, and David Eppstein. Finding all maximal subsequences with hereditary properties. In Proceedings of the 31st International Symposium on Computational Geometry (SoCG), pages 240-254, 2015.
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Timothy M. Chan. Dynamic planar convex hull operations in near-logarithmaic amortized time. Journal of the ACM, 48(1):1-12, 2001. URL: http://dx.doi.org/10.1145/363647.363652.
http://dx.doi.org/10.1145/363647.363652
Timothy M. Chan. A fully dynamic algorithm for planar width. In Proceedings of the 17th Annual Symposium on Computational Geometry (SoCG), pages 172-176, 2001.
Timothy M. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. J. ACM, 57(3):16:1-16:15, 2010. URL: http://dx.doi.org/10.1145/1706591.1706596.
http://dx.doi.org/10.1145/1706591.1706596
Timothy M. Chan. Persistent predecessor search and orthogonal point location on the word ram. ACM Transactions on Algorithms, 9(3):22, 2013.
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.
Timothy M. Chan and Simon Pratt. Time-windowed closest pair. In Proceedings of the 27th Canadian Conference on Computational Geometry (CCCG), 2015.
Timothy M. Chan and Konstantinos Tsakalidis. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. In Proceedings of the 31st International Symposium on Computational Geometry (SoCG), pages 719-732, 2015.
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Untangling Planar Curves
Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n^2), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. This lower bound also implies that Omega(n^{3/2}) degree-1 reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to a simple closed curve.
computational topology
homotopy
planar graphs
Delta-Y transformations
defect
Reidemeister moves
tangles
29:1-29:16
Regular Paper
Hsien-Chih
Chang
Hsien-Chih Chang
Jeff
Erickson
Jeff Erickson
10.4230/LIPIcs.SoCG.2016.29
Francesca Aicardi. Tree-like curves. In Vladimir I. Arnold, editor, Singularities and Bifurcations, volume 21 of Advances in Soviet Mathematics, pages 1-31. Amer. Math. Soc., 1994.
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Dan Archdeacon, Charles J. Colbourn, Isidoro Gitler, and J. Scott Provan. Four-terminal reducibility and projective-planar wye-delta-wye-reducible graphs. J. Graph Theory, 33(2):83-93, 2000.
Vladimir I. Arnold. Plane curves, their invariants, perestroikas and classifications. In Vladimir I. Arnold, editor, Singularities and Bifurcations, volume 21 of Adv. Soviet Math., pages 33-91. Amer. Math. Soc., 1994.
Vladimir I. Arnold. Topological Invariants of Plane Curves and Caustics, volume 5 of University Lecture Series. Amer. Math. Soc., 1994.
Hsien-Chih Chang and Jeff Erickson. Electrical reduction, homtoopy moves, and defect. Preprint, October 2015. URL: http://arxiv.org/abs/1510.00571.
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Chaim Even-Zohar, Joel Hass, Nati Linial, and Tahl Nowik. Invariants of random knots and links. Preprint, November 2014. URL: http://arxiv.org/abs/1411.3308.
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Inserting Multiple Edges into a Planar Graph
Let G be a connected planar (but not yet embedded) graph and F a set of additional edges not in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. An optimal solution to this problem is known to approximate the crossing number of the graph G+F.
Finding an exact solution to MEI is NP-hard for general F, but linear time solvable for the special case of |F|=1 [Gutwenger et al, SODA 2001/Algorithmica] and polynomial time solvable when all of F are incident to a new vertex [Chimani et al, SODA 2009]. The complexity for general F but with constant k=|F| was open, but algorithms both with relative and absolute approximation guarantees have been presented [Chuzhoy et al, SODA 2011], [Chimani-Hlineny, ICALP 2011]. We show that the problem is fixed parameter tractable (FPT) in k for biconnected G, or if the cut vertices of G have bounded degrees. We give the first exact algorithm for this problem; it requires only O(|V(G)|) time for any constant k.
crossing number
edge insertion
parameterized complexity
path homotopy
funnel algorithm
30:1-30:15
Regular Paper
Markus
Chimani
Markus Chimani
Petr
Hlinený
Petr Hlinený
10.4230/LIPIcs.SoCG.2016.30
C. Batini, M. Talamo, and R. Tamassia. Computer aided layout of entity relationship diagrams. Journal of Systems and Software, 4:163-173, 1984.
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M. Chimani. Computing Crossing Numbers. PhD thesis, TU Dortmund, Germany, 2008. URL: http://hdl.handle.net/2003/25955.
http://hdl.handle.net/2003/25955
M. Chimani and C. Gutwenger. Advances in the planarization method: effective multiple edge insertions. J. Graph Algorithms Appl., 16(3):729-757, 2012.
M. Chimani, C. Gutwenger, P. Mutzel, and C. Wolf. Inserting a vertex into a planar graph. In Proc. SODA'09, pages 375-383, 2009.
M. Chimani and P. Hliněný. A tighter insertion-based approximation of the crossing number. In Proc. ICALP'11, volume 6755 of LNCS, pages 122-134. Springer, 2011.
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I. Gitler, P. Hliněný, J. Leanos, and G. Salazar. The crossing number of a projective graph is quadratic in the face-width. Electronic Journal of Combinatorics, 15(1):#R46, 2008.
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C. Gutwenger and P. Mutzel. A linear time implementation of SPQR trees. In Proc. GD'00, volume 1984 of LNCS, pages 77-90. Springer, 2001.
C. Gutwenger, P. Mutzel, and R. Weiskircher. Inserting an edge into a planar graph. Algorithmica, 41(4):289-308, 2005.
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P. Hliněný and G. Salazar. Approximating the crossing number of toroidal graphs. In Proc. ISAAC'07, volume 4835 of LNCS, pages 148-159. Springer, 2007.
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M. Schaefer. The graph crossing number and its variants: A survey. Electronic Journal of Combinatorics, #DS21, May 15, 2014.
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Polynomial-Sized Topological Approximations Using the Permutahedron
Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in R^d, we obtain a O(d)-approximation with at most n2^{O(d log k)} simplices of dimension k or lower. In conjunction with dimension reduction techniques, our approach yields a O(polylog (n))-approximation of size n^{O(1)} for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry.
Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every (1+epsilon)-approximation of the Cech filtration has to contain n^{Omega(log log n)} features, provided that epsilon < 1/(log^{1+c}n) for c in (0,1).
Persistent Homology
Topological Data Analysis
Simplicial Approximation
Permutahedron
Approximation Algorithms
31:1-31:16
Regular Paper
Aruni
Choudhary
Aruni Choudhary
Michael
Kerber
Michael Kerber
Sharath
Raghvendra
Sharath Raghvendra
10.4230/LIPIcs.SoCG.2016.31
J. Baek and A.Adams. Some useful properties of the Permutohedral Lattice for Gaussian filtering. Technical report, Stanford University, 2009. URL: http://graphics.stanford.edu/papers/permutohedral/permutohedral_techreport.pdf.
http://graphics.stanford.edu/papers/permutohedral/permutohedral_techreport.pdf
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M. Botnan and G. Spreemann. Approximating Persistent Homology in Euclidean space through collapses. Applied Algebra in Engineering, Communication and Computing, 26(1-2):73-101, 2015. URL: http://dx.doi.org/10.1007/s00200-014-0247-y.
http://dx.doi.org/10.1007/s00200-014-0247-y
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http://dx.doi.org/10.1007/s00454-014-9573-x
F. Chazal, D. Cohen-Steiner, M. Glisse, L.J. Guibas, and S.Y. Oudot. Proximity of Persistence Modules and their Diagrams. In ACM Symposium on Computational Geometry (SoCG), pages 237-246, 2009. URL: http://dx.doi.org/10.1145/1542362.1542407.
http://dx.doi.org/10.1145/1542362.1542407
A. Choudhary, M. Kerber, and S. Raghvendra. Polynomial-sized Topological Approximations using the Permutahedron. CoRR, abs/1601.02732, 2016. URL: http://arxiv.org/abs/1601.02732.
http://arxiv.org/abs/1601.02732
A. Choudhary, M. Kerber, and R. Sharathkumar. Approximate Čech complexes in low and high dimensions. URL: http://people.mpi-inf.mpg.de/~achoudha/Files/AppCech.pdf.
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T.K. Dey, F. Fan, and Y. Wang. Computing topological persistence for simplicial maps. In ACM Symposium on Computational Geometry (SoCG), pages 345-354, 2014. URL: http://dx.doi.org/10.1145/2582112.2582165.
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M. Kerber and S. Raghvendra. Approximation and streaming algorithms for projective clustering via random projections. In Canadian Conference on Computational Geometry (CCCG), pages 179-185, 2015.
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D. Sheehy. The persistent homology of distance functions under random projection. In ACM Symposium on Computational Geometry (SoCG), 2014.
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Faster Algorithms for Computing Plurality Points
Let V be a set of n points in R^d, which we call voters, where d is a fixed constant. A point p in R^d is preferred over another point p' in R^d by a voter v in V if dist(v,p) < dist(v,p'). A point p is called a plurality point if it is preferred by at least as many voters as any other point p'.
We present an algorithm that decides in O(n log n) time whether V admits a plurality point in the L_2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute the smallest subset W of V such that V - W admits a plurality point, and to compute a so-called minimum-radius plurality ball.
Finally, we consider the problem in the personalized L_1 norm, where each point v in V has a preference vector <w_1(v), ...,w_d(v)> and the distance from v to any point p in R^d is given by sum_{i=1}^d w_i(v) cdot |x_i(v)-x_i(p)|. For this case we can compute in O(n^(d-1)) time the set of all plurality points of V. When all preference vectors are equal, the running time improves to O(n).
computational geometry
computational social choice
voting theory
plurality points
Condorcet points
32:1-32:15
Regular Paper
Mark
de Berg
Mark de Berg
Joachim
Gudmundsson
Joachim Gudmundsson
Mehran
Mehr
Mehran Mehr
10.4230/LIPIcs.SoCG.2016.32
H. K. Ahn, S. W. Cheng, O. Cheong, M. Golin, and R. van Oostrum. Competitive facility location: the Voronoi game. Theor. Comput. Sci., 310(1-3):457-467, 2004.
T. Chan. An optimal randomized algorithm for maximum tukey depth. In Proc. 15th ACM-SIAM Symp. Discr. Alg. (SODA), pages 430-436, 2004.
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D. Kress and E. Pesch. Sequential competitive location on networks. Europ. J. Op. Res., 217(3):483-499, 2012.
W. Y. Lin, Y. W. Wu, H. L. Wang, and K. M. Chao. Forming plurality at minimum cost. In Proc. 9th Int. Workshop Alg. Comput. (WALCOM), LNCS 8973, pages 77-88, 2015.
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Qualitative Symbolic Perturbation
In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in epsilon for the inputs of a predicate. Instead of a single perturbation, we propose to use a sequence of (simpler) perturbations. Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.
Robustness issues
Symbolic perturbations
Apollonius diagram
33:1-33:17
Regular Paper
Olivier
Devillers
Olivier Devillers
Menelaos
Karavelas
Menelaos Karavelas
Monique
Teillaud
Monique Teillaud
10.4230/LIPIcs.SoCG.2016.33
P. Alliez, O. Devillers, and J. Snoeyink. Removing degeneracies by perturbing the problem or the world. Reliable Computing, 6:61-79, 2000. URL: http://hal.inria.fr/inria-00338566/.
http://hal.inria.fr/inria-00338566/
H. Brönnimann, O. Devillers, V. Dujmović, H. Everett, M. Glisse, X. Goaoc, S. Lazard, H.-S. Na, and S. Whitesides. Lines and free line segments tangent to arbitrary three-dimensional convex polyhedra. SIAM Journal on Computing, 37:522-551, 2007. URL: http://hal.inria.fr/inria-00103916.
http://hal.inria.fr/inria-00103916
C. Burnikel, K. Mehlhorn, and S. Schirra. On degeneracy in geometric computations. In 5th ACM-SIAM Sympos. Discrete Algorithms, pages 16-23, 1994. URL: http://dl.acm.org/citation.cfm?id=314474.
http://dl.acm.org/citation.cfm?id=314474
O. Devillers, M. Glisse, and S. Lazard. Predicates for line transversals to lines and line segments in three-dimensional space. In Proc. 24th Annual Symposium on Computational Geometry, pages 174-181, 2008. URL: http://hal.inria.fr/inria-00336256/.
http://hal.inria.fr/inria-00336256/
O. Devillers, M. Karavelas, and M. Teillaud. Qualitative symbolic perturbation: two applications of a new geometry-based perturbation framework. Research Report 8153, INRIA, 2015. version 4. URL: http://hal.inria.fr/hal-00758631/.
http://hal.inria.fr/hal-00758631/
O. Devillers and M. 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.
http://dx.doi.org/10.1016/j.comgeo.2010.09.010
H. Edelsbrunner and E. 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://dl.acm.org/citation.cfm?id=77639.
http://dl.acm.org/citation.cfm?id=77639
I. Emiris and J. Canny. A general approach to removing degeneracies. SIAM J. Comput., 24:650-664, 1995. URL: http://epubs.siam.org/sicomp/resource/1/smjcat/v24/i3/p650_s1.
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I. Emiris and M. Karavelas. The predicates of the Apollonius diagram: algorithmic analysis and implementation. Computational Geometry: Theory and Applications, 33(1-2):18-57, January 2006. URL: http://dx.doi.org/10.1016/j.comgeo.2004.02.006.
http://dx.doi.org/10.1016/j.comgeo.2004.02.006
G. Irving and F. Green. A deterministic pseudorandom perturbation scheme for arbitrary polynomial predicates. Technical Report 1308.1986v1, arXiv, 2013. URL: http://arxiv.org/abs/1308.1986.
http://arxiv.org/abs/1308.1986
K. Mehlhorn, R. Osbild, and M. Sagraloff. A general approach to the analysis of controlled perturbation algorithms. Comput. Geom. Theory Appl., 44:507-528, 2011. URL: http://dx.doi.org/10.1016/j.comgeo.2011.06.001.
http://dx.doi.org/10.1016/j.comgeo.2011.06.001
R. Seidel. The nature and meaning of perturbations in geometric computing. Discrete Comput. Geom., 19:1-17, 1998.
R. Seidel. Perturbations in geometric computing, 2013. Workshop on Geometric Computing, Heraklion. URL: http://www.acmac.uoc.gr/GC2013/files/Seidel-slides.pdf.
http://www.acmac.uoc.gr/GC2013/files/Seidel-slides.pdf
C. K. Yap. A geometric consistency theorem for a symbolic perturbation scheme. J. Comput. Syst. Sci., 40(1):2-18, 1990. URL: http://www.sciencedirect.com/science/article/pii/002200009090016E.
http://www.sciencedirect.com/science/article/pii/002200009090016E
C. K. Yap. Symbolic treatment of geometric degeneracies. J. Symbolic Comput., 10:349-370, 1990. URL: http://www.sciencedirect.com/science/article/pii/S0747717108800697.
http://www.sciencedirect.com/science/article/pii/S0747717108800697
C. K. Yap and T. Dubé. The exact computation paradigm. In Computing in Euclidean Geometry, volume 4 of Lecture Notes Series on Computing, pages 452-492. World Scientific, 1995. URL: http://www.cs.nyu.edu/~exact/doc/paradigm.ps.gz.
http://www.cs.nyu.edu/~exact/doc/paradigm.ps.gz
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https://creativecommons.org/licenses/by/3.0/legalcode
Finding Global Optimum for Truth Discovery: Entropy Based Geometric Variance
Truth Discovery is an important problem arising in data analytics related fields such as data mining, database, and big data. It concerns about finding the most trustworthy information from a dataset acquired from a number of unreliable sources. Due to its importance, the problem has been extensively studied in recent years and a number techniques have already been proposed. However, all of them are of heuristic nature and do not have any quality guarantee. In this paper, we formulate the problem as a high dimensional geometric optimization problem, called Entropy based Geometric Variance. Relying on a number of novel geometric techniques (such as Log-Partition and Modified Simplex Lemma), we further discover new insights to this problem. We show, for the first time, that the truth discovery problem can be solved with guaranteed quality of solution. Particularly, we show that it is possible to achieve a (1+eps)-approximation within nearly linear time under some reasonable assumptions. We expect that our algorithm will be useful for other data related applications.
geometric optimization
data mining
high dimension
entropy
34:1-34:16
Regular Paper
Hu
Ding
Hu Ding
Jing
Gao
Jing Gao
Jinhui
Xu
Jinhui Xu
10.4230/LIPIcs.SoCG.2016.34
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On Expansion and Topological Overlap
We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map X -> R^d there exists a point p in R^d whose preimage intersects a positive fraction mu > 0 of the d-cells of X. More generally, the conclusion holds if R^d is replaced by any d-dimensional piecewise-linear (PL) manifold M, with a constant \mu that depends only on d and on the expansion properties of X, but not on M.
Combinatorial Topology
Selection Lemmas
Higher-Dimensional Expanders
35:1-35:10
Regular Paper
Dominic
Dotterrer
Dominic Dotterrer
Tali
Kaufman
Tali Kaufman
Uli
Wagner
Uli Wagner
10.4230/LIPIcs.SoCG.2016.35
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On the Number of Maximum Empty Boxes Amidst n Points
We revisit the following problem (along with its higher dimensional variant): Given a set S of n points inside an axis-parallel rectangle U in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S.
1. We prove that the number of maximum-area empty rectangles amidst n points in the plane is O(n log n 2^alpha(n)), where alpha(n) is the extremely slowly growing inverse of Ackermann's function. The previous best bound, O(n^2), is due to Naamad, Lee, and Hsu (1984).
2. For any d at least 3, we prove that the number of maximum-volume empty boxes amidst n points in R^d is always O(n^d) and sometimes Omega(n^floor(d/2)).
This is the first superlinear lower bound derived for this problem.
3. We discuss some algorithmic aspects regarding the search for a maximum empty box in R^3. In particular, we present an algorithm that finds a (1-epsilon)-approximation of the maximum empty box amidst n points in O(epsilon^{-2} n^{5/3} log^2{n}) time.
Maximum empty box
Davenport-Schinzel sequence
approximation algorithm
data mining.
36:1-36:13
Regular Paper
Adrian
Dumitrescu
Adrian Dumitrescu
Minghui
Jiang
Minghui Jiang
10.4230/LIPIcs.SoCG.2016.36
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Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of vertices there is a path which is monotonically increasing in some direction, and it is called a strongly monotone drawing if the direction of monotonicity is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. The drawings of 3-connected planar graphs are based on primal-dual circle packings. Our drawings of outerplanar graphs depend on a new algorithm that constructs strongly monotone drawings of trees which are also convex. For irreducible trees, these drawings are strictly convex.
graph drawing
planar graphs
strongly monotone
strictly convex
primal-dual circle packing
37:1-37:15
Regular Paper
Stefan
Felsner
Stefan Felsner
Alexander
Igamberdiev
Alexander Igamberdiev
Philipp
Kindermann
Philipp Kindermann
Boris
Klemz
Boris Klemz
Tamara
Mchedlidze
Tamara Mchedlidze
Manfred
Scheucher
Manfred Scheucher
10.4230/LIPIcs.SoCG.2016.37
Soroush Alamdari, Timothy M. Chan, Elyot Grant, Anna Lubiw, and Vinayak Pathak. Self-approaching graphs. In Walter Didimo and Maurizio Patrignani, editors, Proc. 20th Int. Symp. Graph Drawing (GD'12), volume 7704 of Lecture Notes Comput. Sci., pages 260-271. Springer, 2013. 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
Patrizio Angelini, Enrico Colasante, Giuseppe Di Battista, Fabrizio Frati, and Maurizio Patrignani. Monotone drawings of graphs. J. Graph Algorithms Appl., 16(1):5-35, 2012. URL: http://dx.doi.org/10.7155/jgaa.00249.
http://dx.doi.org/10.7155/jgaa.00249
Patrizio Angelini, Walter Didimo, Stephen Kobourov, Tamara Mchedlidze, Vincenzo Roselli, Antonios Symvonis, and Stephen Wismath. Monotone drawings of graphs with fixed embedding. Algorithmica, 71:1-25, 2013. URL: http://dx.doi.org/10.1007/s00453-013-9790-3.
http://dx.doi.org/10.1007/s00453-013-9790-3
Esther M. Arkin, Robert Connelly, and Joseph S. B. Mitchell. On monotone paths among obstacles with applications to planning assemblies. In Proc. 5th Ann. ACM Symp. Comput. Geom. (SoCG'89), pages 334-343. ACM, 1989. URL: http://dx.doi.org/10.1145/73833.73870.
http://dx.doi.org/10.1145/73833.73870
Graham R. Brightwell and Edward R. Scheinerman. Representations of planar graphs. SIAM J. Discrete Math., 6(2):214-229, 1993. URL: http://dx.doi.org/10.1137/0406017.
http://dx.doi.org/10.1137/0406017
Hooman R. Dehkordi, Fabrizio Frati, and Joachim Gudmundsson. Increasing-chord graphs on point sets. In Christian Duncan and Antonios Symvonis, editors, Proc. 22nd Int. Symp. Graph Drawing (GD'14), volume 8871 of Lecture Notes Comput. Sci., 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
Raghavan Dhandapani. Greedy drawings of triangulations. Discrete Comput. Geom., 43(2):375-392, 2010. URL: http://dx.doi.org/10.1007/s00454-009-9235-6.
http://dx.doi.org/10.1007/s00454-009-9235-6
Stefan Felsner. Convex drawings of planar graphs and the order dimension of 3-polytopes. Order, 18(1):19-37, 2001. URL: http://dx.doi.org/10.1023/A:1010604726900.
http://dx.doi.org/10.1023/A:1010604726900
Dayu He and Xin He. Nearly optimal monotone drawing of trees. Theoretical Computer Science, 2016. To appear. URL: http://dx.doi.org/10.1016/j.tcs.2016.01.009.
http://dx.doi.org/10.1016/j.tcs.2016.01.009
Xin He and Dayu He. Compact monotone drawing of trees. In Dachuan Xu, Donglei Du, and Dingzhu Du, editors, Proc. 21st Int. Conf. Comput. Combin. (COCOON'15), volume 9198 of Lecture Notes Comput. Sci., pages 457-468. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21398-9_36.
http://dx.doi.org/10.1007/978-3-319-21398-9_36
Xin He and Dayu He. Monotone drawings of 3-connected plane graphs. In Nikhil Bansal and Irene Finocchi, editors, Proc. 23rd Ann. Europ. Symp. Algorithms (ESA'15), volume 9294 of Lecture Notes Comput. Sci., pages 729-741. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_61.
http://dx.doi.org/10.1007/978-3-662-48350-3_61
Md. Iqbal Hossain and Md. Saidur Rahman. Monotone grid drawings of planar graphs. In Jianer Chen, John E. Hopcroft, and Jianxin Wang, editors, Proc. 8th Int. Workshop Front. Algorithmics (FAW'14), volume 8497 of Lecture Notes Comput. Sci., pages 105-116. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-08016-1_10.
http://dx.doi.org/10.1007/978-3-319-08016-1_10
Weidong Huang, Peter Eades, and Seok-Hee Hong. A graph reading behavior: Geodesic-path tendency. In Peter Eades, Thomas Ertl, and Han-Wei Shen, editors, Proc. 2nd IEEE Pacific Visualization Symposium (PacificVis'09), pages 137-144. IEEE Computer Society, 2009. URL: http://dx.doi.org/10.1109/PACIFICVIS.2009.4906848.
http://dx.doi.org/10.1109/PACIFICVIS.2009.4906848
Christian Icking, Rolf Klein, and Elmar Langetepe. Self-approaching curves. Math. Proc. Camb. Philos. Soc., 125:441-453, 1995. URL: http://dx.doi.org/10.1017/S0305004198003016.
http://dx.doi.org/10.1017/S0305004198003016
Philipp Kindermann, André Schulz, Joachim Spoerhase, and Alexander Wolff. On monotone drawings of trees. In Christian Duncan and Antonis Symvonis, editors, Proc. 22nd Int. Symp. Graph Drawing (GD'14), volume 8871 of Lecture Notes Comput. Sci., pages 488-500. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-45803-7_41.
http://dx.doi.org/10.1007/978-3-662-45803-7_41
Bongshin Lee, Catherine Plaisant, Cynthia Sims Parr, Jean-Daniel Fekete, and Nathalie Henry. Task taxonomy for graph visualization. In Enrico Bertini, Catherine Plaisant, and Giuseppe Santucci, editors, Proc. AVI Workshop Beyond Time Errors: Novel Eval. Methods Inform. Vis. (BELIC'06), pages 1-5. ACM, 2006. URL: http://dx.doi.org/10.1145/1168149.1168168.
http://dx.doi.org/10.1145/1168149.1168168
Tom Leighton and Ankur Moitra. Some results on greedy embeddings in metric spaces. Discrete Comput. Geom., 44(3):686-705, 2010. URL: http://dx.doi.org/10.1007/s00454-009-9227-6.
http://dx.doi.org/10.1007/s00454-009-9227-6
Martin Nöllenburg and Roman Prutkin. Euclidean greedy drawings of trees. In Hans L. Bodlaender and Giuseppe F. Italiano, editors, Proc. 21st Europ. Symp. Algorithms (ESA'13), volume 8125 of Lecture Notes Comput. Sci., pages 767-778. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40450-4_65.
http://dx.doi.org/10.1007/978-3-642-40450-4_65
Martin Nöllenburg, Roman Prutkin, and Ignaz Rutter. On self-approaching and increasing-chord drawings of 3-connected planar graphs. J. Comput. Geom., 7(1):47-69, 2016. URL: http://jocg.org/v7n1p3.
http://jocg.org/v7n1p3
Ananth Rao, Sylvia Ratnasamy, Christos H. Papadimitriou, Scott Shenker, and Ion Stoica. Geographic routing without location information. In David B. Johnson, Anthony D. Joseph, and Nitin H. Vaidya, editors, Proc. 9th Ann. Int. Conf. Mob. Comput. Netw. (MOBICOM'03), pages 96-108. ACM, 2003. URL: http://dx.doi.org/10.1145/938985.938996.
http://dx.doi.org/10.1145/938985.938996
Kenneth Stephenson. Introduction to circle packing: the theory of discrete analytic functions. Cambridge Univ. Press, 2005. URL: http://www.cambridge.org/9780521823562.
http://www.cambridge.org/9780521823562
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Hyperplane Separability and Convexity of Probabilistic Point Sets
We describe an O(n^d) time algorithm for computing the exact probability that two d-dimensional probabilistic point sets are linearly separable, for any fixed d >= 2. A probabilistic point in d-space is the usual point, but with an associated (independent) probability of existence. We also show that the d-dimensional separability problem is equivalent to a (d+1)-dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of n probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership by a factor of n [Agarwal et al., ESA, 2014]. In addition, our algorithms can handle "input degeneracies" in which more than k+1 points may lie on a k-dimensional subspace, thus resolving an open problem in [Agarwal et al., ESA, 2014]. Finally, we prove lower bounds for the separability problem via a reduction from the k-SUM problem, which shows in particular that our O(n^2) algorithms for 2-dimensional separability and 3-dimensional convex hull membership are nearly optimal.
probabilistic separability
uncertain data
3-SUM hardness
topological sweep
hyperplane separation
multi-dimensional data
38:1-38:16
Regular Paper
Martin
Fink
Martin Fink
John
Hershberger
John Hershberger
Nirman
Kumar
Nirman Kumar
Subhash
Suri
Subhash Suri
10.4230/LIPIcs.SoCG.2016.38
A. Abdullah, S. Daruki, and J. M. Phillips. Range counting coresets for uncertain data. In Proc. 29th SCG, pages 223-232. ACM, 2013.
P. Afshani, P. K. Agarwal, L. Arge, K. G. Larsen, and J. M. Phillips. (Approximate) uncertain skylines. Theory of Comput. Syst., 52:342-366, 2013.
P. K. Agarwal, B. Aronov, S. Har-Peled, J. M. Phillips, K. Yi, and W. Zhang. Nearest neighbor searching under uncertainty II. In Proc. 32nd ACM PODS, pages 115-126, 2013.
P. K. Agarwal, S.-W. Cheng, and K. Yi. Range searching on uncertain data. ACM Trans. on Algorithms, 8(4):43:1-43:17, 2012.
P. K. Agarwal, A. Efrat, S. Sankararaman, and W. Zhang. Nearest-neighbor searching under uncertainty. In Proc. 31st ACM PODS, pages 225-236. ACM, 2012.
P. K. Agarwal, S. Har-Peled, S. Suri, H. Yıldız, and W. Zhang. Convex hulls under uncertainty. In Proc. 22nd ESA, pages 37-48, 2014.
C. C. Aggarwal. Managing and Mining Uncertain Data. Springer, 2009.
C. C. Aggarwal and P. S. Yu. A survey of uncertain data algorithms and applications. IEEE TKDE., 21(5):609-623, 2009.
N. Ailon and B. Chazelle. Lower bounds for linear degeneracy testing. J. ACM, 52(2):157-171, 2005.
C. Böhm, F. Fiedler, A. Oswald, C. Plant, and B. Wackersreuther. Probabilistic skyline queries. In Proc. CIKM, pages 651-660, 2009.
K. L. Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small. J. ACM, 42(2):488-499, 1995.
M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications. Springer, 3rd edition, 2008.
M. de Berg, A. D. Mehrabi, and F. Sheikhi. Separability of imprecise points. In Proc. 14th SWAT, pages 146-157. Springer, 2014.
H. Edelsbrunner and L. J. Guibas. Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38(1):165-194, 1989.
H. Edelsbrunner and E. P. Mücke. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Trans. on Graphics, 9(1):66-104, 1990.
J. Erickson. Lower bounds for linear satisfiability problems. Chicago J. Theoret. Comp. Sci., 1999(8), August 1999.
A. Gajentaan and M. H. Overmars. On a class of O(n²) problems in computational geometry. CGTA, 5(3):165-185, 1995.
A. Gronlund and S. Pettie. Threesomes, degenerates, and love triangles. In Proc. 55th FOCS, pages 621-630, 2014.
A. Jørgensen, M. Löffler, and J. M. Phillips. Geometric computations on indecisive and uncertain points. CoRR, abs/1205.0273, 2012.
P. Kamousi, T. M. Chan, and S. Suri. Stochastic minimum spanning trees in Euclidean spaces. In Proc. 27th SCG, pages 65-74, 2011.
P. Kamousi, T. M. Chan, and S. Suri. Closest pair and the post office problem for stochastic points. CGTA, 47(2):214-223, 2014.
T. Kopelowitz, S. Pettie, and E. Porat. Higher lower bounds from the 3SUM conjecture. In SODA, 2016.
H.-P. Kriegel, P. Kunath, and M. Renz. Probabilistic nearest-neighbor query on uncertain objects. In Advances in Databases: Concepts, Systems and Applications, volume 4443, pages 337-348. Springer, 2007.
Y. Li, J. Xue, A. Agrawal, and R. Janardan. On the arrangement of stochastic lines in R². Unpublished manuscript (personal communication), 2015.
N. Megiddo. Linear programming in linear time when the dimension is fixed. J. ACM, 31(1):114-127, 1984.
C. F. Olson. Probabilistic indexing for object recognition. IEEE PAMI, 17(5):518-522, 1995.
J. S. Provan and M. O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput., 12(4):777-788, 1983.
S. Suri and K. Verbeek. On the most likely Voronoi diagram and nearest neighbor searching. In Proc. 25th ISAAC, pages 338-350, 2014.
S. Suri, K. Verbeek, and H. Yıldız. On the most likely convex hull of uncertain points. In Proc. 21st ESA, pages 791-802, 2013.
S. P. Vadhan. The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput., 31:398-427, 1997.
L. G. Valiant. The complexity of enumeration and reliability problems. SIAM J. Comput., 8(3):410-421, 1979.
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Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems
A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time.
Rectilinear graphs
Steiner arborescence
parameterized algorithms
39:1-39:15
Regular Paper
Fedor
Fomin
Fedor Fomin
Sudeshna
Kolay
Sudeshna Kolay
Daniel
Lokshtanov
Daniel Lokshtanov
Fahad
Panolan
Fahad Panolan
Saket
Saurabh
Saket Saurabh
10.4230/LIPIcs.SoCG.2016.39
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets Möbius: fast subset convolution. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), pages 67-74, New York, 2007. ACM.
Marcus Brazil and Martin Zachariasen. Optimal Interconnection Trees in the Plane: Theory, Algorithms and Applications. Springer, 2015.
Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer-Verlag, Berlin, 2015.
Linda L. Deneen, Gary M. Shute, and Clark D. Thomborson. A probably fast, provably optimal algorithm for rectilinear Steiner trees. Random Structures Algorithms, 5(4):535-557, 1994.
Stuart E. Dreyfus and Robert A. Wagner. The Steiner problem in graphs. Networks, 1(3):195-207, 1971. URL: http://dx.doi.org/10.1002/net.3230010302.
http://dx.doi.org/10.1002/net.3230010302
Bernhard Fuchs, Walter Kern, Daniel Mölle, Stefan Richter, Peter Rossmanith, and Xinhui Wang. Dynamic programming for minimum Steiner trees. Theory of Computing Systems, 41(3):493-500, 2007. URL: http://dx.doi.org/10.1007/s00224-007-1324-4.
http://dx.doi.org/10.1007/s00224-007-1324-4
Joseph L. Ganley. Computing optimal rectilinear Steiner trees: a survey and experimental evaluation. Discrete Appl. Math., 90(1-3):161-171, 1999.
Joseph L. Ganley and James P. Cohoon. Improved computation of optimal rectilinear Steiner minimal trees. Internat. J. Comput. Geom. Appl., 7(5):457-472, 1997. URL: http://dx.doi.org/10.1142/S0218195997000272.
http://dx.doi.org/10.1142/S0218195997000272
M. R. Garey and D. S. Johnson. The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math., 32(4):826-834, 1977.
Qian-Ping Gu and Hisao Tamaki. Improved bounds on the planar branchwidth with respect to the largest grid minor size. Algorithmica, 64(3):416-453, 2012. URL: http://dx.doi.org/10.1007/s00453-012-9627-5.
http://dx.doi.org/10.1007/s00453-012-9627-5
M. Hanan. On Steiner’s problem with rectilinear distance. SIAM J. Appl. Math., 14:255-265, 1966.
Frank K Hwang. On Steiner minimal trees with rectilinear distance. SIAM J. Appl. Math., 30(1):104-114, 1976.
Frank K. Hwang, Dana S. Richards, and Pawel Winter. The Steiner tree problem, volume 53 of Annals of Discrete Mathematics. North-Holland Publishing Co., Amsterdam, 1992.
Philip N. Klein and Dániel Marx. A subexponential parameterized algorithm for Subset TSP on planar graphs. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1812-1830. SIAM, 2014.
L. Nastansky, S. M. Selkow, and N. F. Stewart. Cost-minimal trees in directed acyclic graphs. Z. Operations Res. Ser. A-B, 18:A59-A67, 1974.
Jesper Nederlof. Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica, 65(4):868-884, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9630-x.
http://dx.doi.org/10.1007/s00453-012-9630-x
Marcin Pilipczuk, Michał Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Subexponential-time parameterized algorithm for Steiner tree on planar graphs. In Proc. of the 30th International Symp. on Theoretical Aspects of Computer Science (STACS), volume 20 of Leibniz International Proceedings in Informatics (LIPIcs), pages 353-364. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2013.
Marcin Pilipczuk, Michal Pilipczuk, Piotr Sankowski, and Erik Jan van Leeuwen. Network sparsification for Steiner problems on planar and bounded-genus graphs. In Proc. 55th Annual Symp. on Foundations of Computer Science (FOCS), pages 276-285. IEEE, 2014.
Hans Jürgen Prömel and Angelika Steger. The Steiner Tree Problem. Advanced Lectures in Mathematics. Friedr. Vieweg &Sohn, Braunschweig, 2002.
Weiping Shi and Chen Su. The Rectilinear Steiner Arborescence Problem Is NP-Complete. SIAM J. Comput., 35(3):729-740, 2005. URL: http://dx.doi.org/10.1137/S0097539704371353.
http://dx.doi.org/10.1137/S0097539704371353
Clark D Thomborson, Linda L Deneen, and Gary M Shute. Computing a rectilinear steiner minimal tree in n^O (√n) time. In Parallel Algorithms and Architectures, pages 176-183. Springer, 1987.
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Random Sampling with Removal
Random sampling is a classical tool in constrained optimization. Under favorable conditions, the optimal solution subject to a small subset of randomly chosen constraints violates only a small subset of the remaining constraints. Here we study the following variant that we call random sampling with removal: suppose that after sampling the subset, we remove a fixed number of constraints from the sample, according to an arbitrary rule. Is it still true that the optimal solution of the reduced sample violates only a small subset of the constraints?
The question naturally comes up in situations where the solution subject to the sampled constraints is used as an approximate solution to the original problem. In this case, it makes sense to improve cost and volatility of the sample solution by removing some of the constraints that appear most restricting. At the same time, the approximation quality (measured in terms of violated constraints) should remain high.
We study random sampling with removal in a generalized, completely abstract setting where we assign to each subset R of the constraints an arbitrary set V(R) of constraints disjoint from R; in applications, V(R) corresponds to the constraints violated by the optimal solution subject to only the constraints in R. Furthermore, our results are parametrized by the dimension d, i.e., we assume that every set R has a subset B of size at most d with the same set of violated constraints. This is the first time this generalized setting is studied.
In this setting, we prove matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of k elements. For a large range of values of k, the new upper bounds improve the previously best bounds for LP-type problems, which moreover had only been known in special cases. We show that this bound on special LP-type problems, can be derived in the much more general setting of violator spaces, and with very elementary proofs.
LP-type problem
violator space
random sampling
sampling with removal
40:1-40:16
Regular Paper
Bernd
Gärtner
Bernd Gärtner
Johannes
Lengler
Johannes Lengler
May
Szedlák
May Szedlák
10.4230/LIPIcs.SoCG.2016.40
Y. Brise and B. Gärtner. Clarkson’s algorithm for violator spaces. Computational Geometry, 44(2):70-81, 2011. Special issue of selected papers from the 21st Annual Canadian Conference on Computational Geometry. URL: http://dx.doi.org/10.1016/j.comgeo.2010.09.003.
http://dx.doi.org/10.1016/j.comgeo.2010.09.003
M. C. Campi and S. Garatti. The exact feasibility of randomized solutions of uncertain convex programs. SIAM J. Optim., 19:1211-1230, 2008.
M. C. Campi and S. Garatti. A sampling-and-discarding approach to chance-constrained optimization: feasibility and optimality. J. Optim. Theory Appl., 148:257-280, 2011.
T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. The MIT Press, Cambridge, MA., 1990.
B. Gärtner. Sampling with removal in LP-type problems. Journal of Computational Geometry, 6(2):93-112, 2015.
B. Gärtner, J. Matoušek, L. Rüst, and P. Škovroň. Violator spaces: Structure and algorithms. Discrete Appl. Math., 156(11):2124-2141, June 2008. URL: http://dx.doi.org/10.1016/j.dam.2007.08.048.
http://dx.doi.org/10.1016/j.dam.2007.08.048
B. Gärtner, W. D. Morris, Jr., and L. Rüst. Unique sink orientations of grids. In Proc. 11th Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 3509 of Lecture Notes in Computer Science, pages 210-224. Springer-Verlag, 2005.
B. Gärtner and E. Welzl. A simple sampling lemma: Analysis and applications in geometric optimization. Discrete &Computational Geometry, 25(4):569-590, 2001.
J. Matoušek, M. Sharir, and E. Welzl. A subexponential bound for linear programming. Algorithmica, 16:498-516, 1996.
J. Matoušek. Removing degeneracy in LP-type problems revisited. Discrete &Computational Geometry, 42(4):517-526, 2009. URL: http://dx.doi.org/10.1007/s00454-008-9085-7.
http://dx.doi.org/10.1007/s00454-008-9085-7
M. Sharir and E. Welzl. A combinatorial bound for linear programming and related problems. In Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science, STACS'92, pages 569-579, London, UK, UK, 1992. Springer-Verlag. URL: http://dx.doi.org/10.1007/3-540-55210-3_213.
http://dx.doi.org/10.1007/3-540-55210-3_213
T. Szabó and E. Welzl. Unique sink orientations of cubes. In Proc. 42nd IEEE Symposium on Foundations of Computer Science (FOCS), pages 547-555, 2000.
E. Welzl. Smallest enclosing disks (balls and ellipsoids). In Results and New Trends in Computer Science, pages 359-370. Springer-Verlag, 1991.
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The Planar Tree Packing Theorem
Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garcia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.
graph drawing
simultaneous embedding
planar graph
graph packin
41:1-41:15
Regular Paper
Markus
Geyer
Markus Geyer
Michael
Hoffmann
Michael Hoffmann
Michael
Kaufmann
Michael Kaufmann
Vincent
Kusters
Vincent Kusters
Csaba
Tóth
Csaba Tóth
10.4230/LIPIcs.SoCG.2016.41
J. Akiyama and V. Chvátal. Packing paths perfectly. Discrete Math., 85(3):247-255, 1990.
P. Braß, E. Cenek, C. A. Duncan, A. Efrat, C. Erten, D. Ismailescu, S. G. Kobourov, A. Lubiw, and J. S. B. Mitchell. On simultaneous planar graph embeddings. Comput. Geom., 36(2):117-130, 2007.
Y. Caro and R. Yuster. Packing graphs: The packing problem solved. Electr. J. Combin., 4(1), 1997.
D. Eppstein. Arboricity and bipartite subgraph listing algorithms. Inform. Process. Lett., 51(4):207-211, 1994.
P. Erdős. Extremal problems in graph theory. In M. Fiedler, editor, Theory of Graphs and its Applications, pages 29-36. Academic Press, 1965.
A. Frank and Z. Szigeti. A note on packing paths in planar graphs. Math. Program., 70(2):201-209, 1995.
F. Frati. Planar packing of diameter-four trees. In Proc. 21st Canad. Conf. Comput. Geom., pages 95-98, 2009.
F. Frati, M. Geyer, and M. Kaufmann. Planar packings of trees and spider trees. Inform. Process. Lett., 109(6):301-307, 2009.
A. García, C. Hernando, F. Hurtado, M. Noy, and J. Tejel. Packing trees into planar graphs. J. Graph Theory, pages 172-181, 2002.
M. Geyer, M. Hoffmann, M. Kaufmann, V. Kusters, and Cs. D. Tóth. Planar packing of binary trees. In Proc. 13th Algorithms and Data Struct. Sympos., volume 8037 of Lecture Notes Comput. Sci., pages 353-364. Springer, 2013.
D. Gonçalves. Edge partition of planar graphs into two outerplanar graphs. In Proc. 37th Annu. ACM Sympos. Theory Comput., pages 504-512. ACM Press, 2005.
A. Gyárfás and J. Lehel. Packing trees of different order into K_n. In Combinatorics, volume 18 of Colloq. Math. Soc. János Bolyai, pages 463-469. North Holland, 1978.
S. M. Hedetniemi, S. T. Hedetniemi, and P. J. Slater. A note on packing two trees into K_N. Ars Combin., 11:149-153, 1981.
M. Maheo, J.-F. Saclé, and M. Woźniak. Edge-disjoint placement of three trees. European J. Combin., 17(6):543-563, 1996.
P. Mutzel, T. Odenthal, and M. Scharbrodt. The thickness of graphs: A survey. Graphs and Combinatorics, 14(1):59-73, 1998.
C. St. J. A. Nash-Williams. Edge-disjoint spanning trees of finite graphs. Journal of The London Mathematical Society, s1-36:445-450, 1961.
Y. Oda and K. Ota. Tight planar packings of two trees. In European Workshop on Computational Geometry, pages 215-216, 2006.
W. Schnyder. Planar graphs and poset dimension. Order, 5:323-343, 1989.
S.K. Teo and H.P. Yap. Packing two graphs of order n having total size at most 2n-2. Graphs and Combinatorics, 6(2):197-205, 1990.
W. T. Tutte. On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society, s1-36(1):221-230, 1961.
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Crossing Number is Hard for Kernelization
The graph crossing number problem, cr(G)<=k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixed-parameter tractable for the parameter k [Grohe, STOC 2001]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G)<=k can be in polynomial time reduced to an equivalent instance of size polynomial in k (and independent of |G|). We answer this question in the negative. Along the proof we show that the tile crossing number problem of twisted planar tiles is NP-hard, which has been an open problem for some time, too, and then employ the complexity technique of cross-composition. Our result holds already for the special case of graphs obtained from planar graphs by adding one edge.
crossing number; tile crossing number; parameterized complexity; polynomial kernel; cross-composition
42:1-42:10
Regular Paper
Petr
Hlinený
Petr Hlinený
Marek
Dernár
Marek Dernár
10.4230/LIPIcs.SoCG.2016.42
Michael J. Bannister, David Eppstein, and Joseph A. Simons. Fixed parameter tractability of crossing minimization of almost-trees. In Graph Drawing - GD 2013, volume 8242 of Lecture Notes in Computer Science, pages 340-351. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-319-03841-4_30.
http://dx.doi.org/10.1007/978-3-319-03841-4_30
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014. URL: http://dx.doi.org/10.1137/120880240.
http://dx.doi.org/10.1137/120880240
Drago Bokal, Mojca Bracic, Marek Derňár, and Petr Hliněný. On degree properties of crossing-critical families of graphs. In Graph Drawing and Network Visualization - GD 2015, volume 9411 of Lecture Notes in Computer Science, pages 75-86. Springer, 2015.
Drago Bokal, Bogdan Oporowski, R. Bruce Richter, and Gelasio Salazar. Characterizing 2-crossing-critical graphs. Manuscript, 171 pages. http://arxiv.org/abs/1312.3712, 2013.
Sergio Cabello. Hardness of approximation for crossing number. Discrete & Computational Geometry, 49(2):348-358, 2013. URL: http://dx.doi.org/10.1007/s00454-012-9440-6.
http://dx.doi.org/10.1007/s00454-012-9440-6
Sergio Cabello and Bojan Mohar. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput., 42(5):1803-1829, 2013. URL: http://dx.doi.org/10.1137/120872310.
http://dx.doi.org/10.1137/120872310
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
http://dx.doi.org/10.1007/978-1-4471-5559-1
J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006.
Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.007.
http://dx.doi.org/10.1016/j.jcss.2010.06.007
M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM J. Alg. Discr. Meth., 4:312-316, 1983.
Martin Grohe. Computing crossing numbers in quadratic time. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 231-236. ACM, 2001. URL: http://dx.doi.org/10.1145/380752.380805.
http://dx.doi.org/10.1145/380752.380805
Petr Hliněný. Crossing number is hard for cubic graphs. J. Comb. Theory, Ser. B, 96(4):455-471, 2006. URL: http://dx.doi.org/10.1016/j.jctb.2005.09.009.
http://dx.doi.org/10.1016/j.jctb.2005.09.009
Ken-ichi Kawarabayashi and Bruce A. Reed. Computing crossing number in linear time. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 382-390. ACM, 2007. URL: http://dx.doi.org/10.1145/1250790.1250848.
http://dx.doi.org/10.1145/1250790.1250848
Martin Kochol. Construction of crossing-critical graphs. Discrete Mathematics, 66(3):311-313, 1987. URL: http://dx.doi.org/10.1016/0012-365X(87)90108-7.
http://dx.doi.org/10.1016/0012-365X(87)90108-7
B. Mohar and C. Thomassen. Graphs on Surfaces. Johns Hopkins University Press, Baltimore, 2001.
Michael J. Pelsmajer, Marcus Schaefer, and Daniel Stefankovic. Crossing numbers of graphs with rotation systems. Algorithmica, 60(3):679-702, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9343-y.
http://dx.doi.org/10.1007/s00453-009-9343-y
Benny Pinontoan and R. Bruce Richter. Crossing numbers of sequences of graphs II: planar tiles. Journal of Graph Theory, 42(4):332-341, 2003. URL: http://dx.doi.org/10.1002/jgt.10097.
http://dx.doi.org/10.1002/jgt.10097
Benny Pinontoan and R. Bruce Richter. Crossing numbers of sequences of graphs I: General tiles. The Australasian Journal of Combinatorics, 30:197-206, 2004.
R. Bruce Richter and Carsten Thomassen. Minimal graphs with crossing number at least k. J. Comb. Theory, Ser. B, 58(2):217-224, 1993. URL: http://dx.doi.org/10.1006/jctb.1993.1038.
http://dx.doi.org/10.1006/jctb.1993.1038
Marcus Schaefer. The graph crossing number and its variants: A survey. Electronic Journal of Combinatorics, #DS21, May 15, 2014.
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Shortest Path Embeddings of Graphs on Surfaces
The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property.
Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.
Graph embedding
surface
shortest path
crossing number
hyperbolic geometry
43:1-43:16
Regular Paper
Alfredo
Hubard
Alfredo Hubard
Vojtech
Kaluža
Vojtech Kaluža
Arnaud
de Mesmay
Arnaud de Mesmay
Martin
Tancer
Martin Tancer
10.4230/LIPIcs.SoCG.2016.43
Dan Archdeacon and C. Paul Bonnington. Two maps on one surface. Journal of Graph Theory, 36(4):198-216, 2001.
Robert Brooks and Eran Makover. Random construction of Riemann surfaces. J. Differential Geom., 68(1):121-157, 2004.
Éric Colin de Verdière and Jeff Erickson. Tightening nonsimple paths and cycles on surfaces. SIAM Journal on Computing, 39(8):3784-3813, 2010.
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Jiří Matoušek, Eric Sedgwick, Martin Tancer, and Uli Wagner. Embeddability in the 3-sphere is decidable. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 78-84. ACM, 2014.
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Simultaneous Nearest Neighbor Search
Motivated by applications in computer vision and databases, we introduce and study the Simultaneous Nearest Neighbor Search (SNN) problem. Given a set of data points, the goal of SNN is to design a data structure that, given a collection of queries, finds a collection of close points that are compatible with each other. Formally, we are given k query points Q=q_1,...,q_k, and a compatibility graph G with vertices in Q, and the goal is to return data points p_1,...,p_k that minimize (i) the weighted sum of the distances from q_i to p_i and (ii) the weighted sum, over all edges (i,j) in the compatibility graph G, of the distances between p_i and p_j. The problem has several applications in computer vision and databases, where one wants to return a set of *consistent* answers to multiple related queries. Furthermore, it generalizes several well-studied computational problems, including Nearest Neighbor Search, Aggregate Nearest Neighbor Search and the 0-extension problem.
In this paper we propose and analyze the following general two-step method for designing efficient data structures for SNN. In the first step, for each query point q_i we find its (approximate) nearest neighbor point p'_i; this can be done efficiently using existing approximate nearest neighbor structures. In the second step, we solve an off-line optimization problem over sets q_1,...,q_k and p'_1,...,p'_k; this can be done efficiently given that k is much smaller than n. Even though p'_1,...,p'_k might not constitute the optimal answers to queries q_1,...,q_k, we show that, for the unweighted case, the resulting algorithm satisfies a O(log k/log log k)-approximation guarantee. Furthermore, we show that the approximation factor can be in fact reduced to a constant for compatibility graphs frequently occurring in practice, e.g., 2D grids, 3D grids or planar graphs.
Finally, we validate our theoretical results by preliminary experiments. In particular, we show that the empirical approximation factor provided by the above approach is very close to 1.
Approximate Nearest Neighbor
Metric Labeling
0-extension
Simultaneous Nearest Neighbor
Group Nearest Neighbor
44:1-44:15
Regular Paper
Piotr
Indyk
Piotr Indyk
Robert
Kleinberg
Robert Kleinberg
Sepideh
Mahabadi
Sepideh Mahabadi
Yang
Yuan
Yang Yuan
10.4230/LIPIcs.SoCG.2016.44
Pankaj K Agarwal, Alon Efrat, and Wuzhou Zhang. Nearest-neighbor searching under uncertainty. In Proceedings of the 32nd symposium on Principles of database systems. ACM, 2012.
Alexandr Andoni, Piotr Indyk, Huy L Nguyen, and Ilya Razenshteyn. Beyond locality-sensitive hashing. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1018-1028. SIAM, 2014.
Aaron Archer, Jittat Fakcharoenphol, Chris Harrelson, Robert Krauthgamer, Kunal Talwar, and Éva Tardos. Approximate classification via earthmover metrics. In Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pages 1079-1087. Society for Industrial and Applied Mathematics, 2004.
Sunil Arya, David M Mount, Nathan S Netanyahu, Ruth Silverman, and Angela Y Wu. An optimal algorithm for approximate nearest neighbor searching fixed dimensions. Journal of the ACM (JACM), 45(6):891-923, 1998.
Connelly Barnes, Eli Shechtman, Adam Finkelstein, and Dan Goldman. Patchmatch: A randomized correspondence algorithm for structural image editing. ACM Transactions on Graphics-TOG, 28(3):24, 2009.
Jon Louis Bentley. Multidimensional binary search trees used for associative searching. Communications of the ACM, 18(9):509-517, 1975.
Yuri Boykov and Vladimir Kolmogorov. An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 26(9):1124-1137, 2004.
Yuri Boykov, Olga Veksler, and Ramin Zabih. Fast approximate energy minimization via graph cuts. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 23(11):1222-1239, 2001.
Gruia Calinescu, Howard Karloff, and Yuval Rabani. Approximation algorithms for the 0-extension problem. SIAM Journal on Computing, 34(2):358-372, 2005.
Jittat Fakcharoenphol, Chris Harrelson, Satish Rao, and Kunal Talwar. An improved approximation algorithm for the 0-extension problem. In Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, pages 257-265. Society for Industrial and Applied Mathematics, 2003.
Pedro Felzenszwalb, William Freeman, Piotr Indyk, Robert Kleinberg, and Ramin Zabih. Bigdata: F: Dka: Collaborative research: Structured nearest neighbor search in high dimensions, 2015. URL: http://cs.brown.edu/~pff/SNN/.
http://cs.brown.edu/~pff/SNN/
Pedro F Felzenszwalb and Daniel P Huttenlocher. Efficient belief propagation for early vision. International journal of computer vision, 70(1):41-54, 2006.
Pedro F Felzenszwalb, Gyula Pap, Eva Tardos, and Ramin Zabih. Globally optimal pixel labeling algorithms for tree metrics. In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, pages 3153-3160. IEEE, 2010.
William T Freeman, Thouis R Jones, and Egon C Pasztor. Example-based super-resolution. Computer Graphics and Applications, IEEE, 22(2):56-65, 2002.
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.
Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 604-613. ACM, 1998.
Howard Karloff, Subhash Khot, Aranyak Mehta, and Yuval Rabani. On earthmover distance, metric labeling, and 0-extension. SIAM Journal on Computing, 39(2):371-387, 2009.
Alexander V Karzanov. Minimum 0-extensions of graph metrics. European Journal of Combinatorics, 19(1):71-101, 1998.
Jon Kleinberg and Eva Tardos. Approximation algorithms for classification problems with pairwise relationships: Metric labeling and markov random fields. Journal of the ACM (JACM), 49(5):616-639, 2002.
Tsvi Kopelowitz and Robert Krauthgamer. Faster clustering via preprocessing. arXiv preprint arXiv:1208.5247, 2012.
Robert Krauthgamer and James R Lee. Navigating nets: simple algorithms for proximity search. In Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pages 798-807. Society for Industrial and Applied Mathematics, 2004.
Eyal Kushilevitz, Rafail Ostrovsky, and Yuval Rabani. Efficient search for approximate nearest neighbor in high dimensional spaces. SIAM Journal on Computing, 30(2):457-474, 2000.
James R Lee and Assaf Naor. Metric decomposition, smooth measures, and clustering. Preprint, 2004.
Feifei Li, Bin Yao, and Piyush Kumar. Group enclosing queries. Knowledge and Data Engineering, IEEE Transactions on, 23(10):1526-1540, 2011.
Yang Li, Feifei Li, Ke Yi, Bin Yao, and Min Wang. Flexible aggregate similarity search. In Proceedings of the 2011 ACM SIGMOD international conference on management of data, pages 1009-1020. ACM, 2011.
David R Martin, Charless C Fowlkes, and Jitendra Malik. Learning to detect natural image boundaries using local brightness, color, and texture cues. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 26(5):530-549, 2004.
Man Lung Yiu, Nikos Mamoulis, and Dimitris Papadias. Aggregate nearest neighbor queries in road networks. Knowledge and Data Engineering, IEEE Transactions on, 17(6):820-833, 2005.
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Degree Four Plane Spanners: Simpler and Better
Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p, q is realized as the line segment [pq], and is assigned a weight equal to the Euclidean distance |pq|. In this paper, we show how to construct in O(nlg{n}) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance.
Nonnumerical Algorithms and Problems,Computational Geometry and Object Modeling
45:1-45:15
Regular Paper
Iyad
Kanj
Iyad Kanj
Ljubomir
Perkovic
Ljubomir Perkovic
Duru
Türkoglu
Duru Türkoglu
10.4230/LIPIcs.SoCG.2016.45
N. Bonichon, C. Gavoille, N. Hanusse, and D. Ilcinkas. Connections between theta-graphs, delaunay triangulations, and orthogonal surfaces. In Proceedings of the 36th International Workshop on Graph Theoretic Concepts in Computer Science, volume 6410 of Lecture Notes in Computer Science, pages 266-278, 2010.
N. Bonichon, C. Gavoille, N. Hanusse, and L. Perković. Plane spanners of maximum degree six. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP), volume 6198 of Lecture Notes in Computer Science, pages 19-30. Springer, 2010.
N. Bonichon, C. Gavoille, N. Hanusse, and L. Perković. The stretch factor of L₁- and L_∞-Delaunay triangulations. In Proceedings of the 20th Annual European Symposium on Algorithms (ESA), volume 7501 of Lecture Notes in Computer Science, pages 205-216. Springer, 2012.
N. Bonichon, I. Kanj, L. Perkovic, and G. Xia. There are plane spanners of degree 4 and moderate stretch factor. Discrete & Computational Geometry, 53(3):514-546, 2015.
P. Bose, P. Carmi, and L. Chaitman-Yerushalmi. On bounded degree plane strong geometric spanners. J. Discrete Algorithms, 15:16-31, 2012.
P. Bose, P. Carmi, S. Collette, and M. Smid. On the stretch factor of convex delaunay graphs. Journal of Computational Geometry, 1(1):41-56, 2010.
P. Bose, J. Gudmundsson, and M. Smid. Constructing plane spanners of bounded degree and low weight. Algorithmica, 42(3-4):249-264, 2005.
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.
P. Bose, M. Smid, and D. Xu. Delaunay and diamond triangulations contain spanners of bounded degree. International Journal of Computational Geometry and Applications, 19(2):119-140, 2009.
L. P. Chew. There is a planar graph almost as good as the complete graph. In Proceedings of the Second Annual Symposium on Computational Geometry (SoCG), pages 169-177, 1986.
L. P. Chew. There are planar graphs almost as good as the complete graph. Journal of Computer and System Sciences, 39(2):205-219, 1989.
G. Das and P.J. Heffernan. Constructing degree-3 spanners with other sparseness properties. Int. J. Found. Comput. Sci., 7(2):121-136, 1996.
D. Dobkin, S. Friedman, and K. Supowit. Delaunay graphs are almost as good as complete graphs. Discrete &Computational Geometry, 5(4):399-407, December 1990. URL: http://dx.doi.org/10.1007/BF02187801.
http://dx.doi.org/10.1007/BF02187801
I. Kanj and L. Perković. On geometric spanners of Euclidean and unit disk graphs. In In Proceedings of the 25^th Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume hal-00231084, pages 409-420. HAL, 2008.
I. Kanj, L. Perković, and D. Türkoğlu. Degree Four Plane Spanners: Simpler and Better. CoRR, abs/1603.03818, 2016. URL: http://arxiv.org/abs/1603.03818.
http://arxiv.org/abs/1603.03818
J. M. Keil and C. A. Gutwin. Classes of graphs which approximate the complete Euclidean graph. Discrete &Computational Geometry, 7(1):13-28, 1992.
X.-Y. Li and Y. Wang. Efficient construction of low weight bounded degree planar spanner. International Journal of Computational Geometry and Applications, 14(1-2):69-84, 2004.
J. Salowe. Euclidean spanner graphs with degree four. Discrete Applied Mathematics, 54(1):55-66, 1994.
Y. Wang and Xiang-Yang L. Localized construction of bounded degree and planar spanner for wireless ad hoc networks. Mobile Networks and Applications, 11(2):161-175, 2006.
G. Xia. The stretch factor of the Delaunay triangulation is less than 1.998. SIAM J. Comput., 42(4):1620-1659, 2013. URL: http://dx.doi.org/10.1137/110832458.
http://dx.doi.org/10.1137/110832458
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A Lower Bound on Opaque Sets
It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle.
barriers; Cauchy-Crofton formula; lower bound; opaque sets
46:1-46:10
Regular Paper
Akitoshi
Kawamura
Akitoshi Kawamura
Sonoko
Moriyama
Sonoko Moriyama
Yota
Otachi
Yota Otachi
János
Pach
János Pach
10.4230/LIPIcs.SoCG.2016.46
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Fixed Points of the Restricted Delaunay Triangulation Operator
The restricted Delaunay triangulation can be conceived as an operator that takes as input a k-manifold (typically smooth) embedded in R^d and a set of points sampled with sufficient density on that manifold, and produces as output a k-dimensional triangulation of the manifold, the input points serving as its vertices. What happens if we feed that triangulation back into the operator, replacing the original manifold, while retaining the same set of input points? If k = 2 and the sample points are sufficiently dense, we obtain another triangulation of the manifold. Iterating this process, we soon reach an iteration for which the input and output triangulations are the same. We call this triangulation a fixed point of the restricted Delaunay triangulation operator.
With this observation, and a new test for distinguishing "critical points" near the manifold from those near its medial axis, we develop a provably good surface reconstruction algorithm for R^3 with unusually modest sampling requirements. We develop a similar algorithm for constructing a simplicial complex that models a 2-manifold embedded in a high-dimensional space R^d, also with modest sampling requirements (especially compared to algorithms that depend on sliver exudation). The latter algorithm builds a non-manifold representation similar to the flow complex, but made solely of Delaunay simplices. The algorithm avoids the curse of dimensionality: its running time is polynomial, not exponential, in d.
restricted Delaunay triangulation
fixed point
manifold reconstruction
surface reconstruction
computational geometry
47:1-47:15
Regular Paper
Marc
Khoury
Marc Khoury
Jonathan Richard
Shewchuk
Jonathan Richard Shewchuk
10.4230/LIPIcs.SoCG.2016.47
Nina Amenta and Marshall Bern. Surface Reconstruction by Voronoi Filtering. Discrete &Computational Geometry, 22(4):481-504, June 1999.
Nina Amenta, Sunghee Choi, Tamal Krishna Dey, and Naveen Leekha. A Simple Algorithm for Homeomorphic Surface Reconstruction. International Journal of Computational Geometry and Applications, 12(1-2):125-141, 2002.
Nina Amenta, Sunghee Choi, and Günter Rote. Incremental Constructions con BRIO. In Proceedings of the Nineteenth Annual Symposium on Computational Geometry, pages 211-219, San Diego, California, June 2003. Association for Computing Machinery.
Marshall Bern and David Eppstein. Mesh Generation and Optimal Triangulation. In Computing in Euclidean Geometry, volume 1 of Lecture Notes Series on Computing, pages 23-90. World Scientific, Singapore, 1992.
Jean-Daniel Boissonnat and Arijit Ghosh. Manifold Reconstruction Using Tangential Delaunayomplexes. Discrete &Computational Geometry, 51(1):221-267, January 2014.
Jean-Daniel Boissonnat and Steve Oudot. Provably Good Sampling and Meshing of Surfaces. Graphical Models, 67(5):405-451, September 2005.
Jean-Daniel Boissonnat and Steve Oudot. Provably Good Sampling and Meshing of Lipschitzurfaces. In Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, pages 337-346, Sedona, Arizona, June 2006.
Siu-Wing Cheng, Tamal Krishna Dey, Herbert Edelsbrunner, Michael A. Facello, and Shang-Hua Teng. Sliver Exudation. Journal of the ACM, 47(5):883-904, September 2000.
Siu-Wing Cheng, Tamal Krishna Dey, and Edgar A. Ramos. Manifold Reconstruction from Point Samples. In Proceedings of the Sixteenth Annual Symposium on Discrete Algorithms, pages 1018-1027, Vancouver, British Columbia, Canada, January 2005. ACM-SIAM.
Siu-Wing Cheng, Tamal Krishna Dey, and Jonathan Richard Shewchuk. Delaunay Mesh Generation. CRC Press, Boca Raton, Florida, December 2012.
Kenneth L. Clarkson and Peter W. Shor. Applications of Random Sampling in Computational Geometry, II. Discrete &Computational Geometry, 4(1):387-421, December 1989.
Tamal K. Dey, Joachim Giesen, Edgar A. Ramos, and Bardia Sadri. Critical Points of Distance to an ε-Sampling of a Surface and Flow-Complex-Based Surface Reconstruction. International Journal of Computational Geometry and Applications, 18(1-2):29-62, 2008.
Tamal Krishna Dey. Curve and Surface Reconstruction: Algorithms with Mathematical Analysis. Cambridge University Press, New York, 2007.
Herbert Edelsbrunner. Surface Reconstruction by Wrapping Finite Sets in Space. In Boris Aronov, Saugata Basu, János Pach, and Micha Sharir, editors, Discrete and Computational Geometry: The Goodman-Pollack Festschrift, pages 379-404. Springer-Verlag, Berlin, 2003.
Herbert Edelsbrunner and Nimish R. Shah. Triangulating Topological Spaces. International Journal of Computational Geometry and Applications, 7(4):365-378, August 1997.
Julia Flötotto. A Coordinate System Associated to a Point Cloud Issued from a Manifold: Definition, Properties and Applications. PhD thesis, Université Nice Sophia Antipolis, 2003.
Joachim Giesen and Matthias John. The Flow Complex: A Data Structure for Geometric Modeling. Computational Geometry: Theory and Applications, 39:178-190, 2008.
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Congruence Testing of Point Sets in 4-Space
We give a deterministic O(n log n)-time algorithm to decide if two n-point sets in 4-dimensional Euclidean space are the same up to rotations and translations. It has been conjectured that O(n log n) algorithms should exist for any fixed dimension. The best algorithms in d-space so far are a deterministic algorithm by Brass and Knauer [Int. J. Comput. Geom. Appl., 2000] and a randomized Monte Carlo algorithm by Akutsu [Comp. Geom., 1998]. They take time O(n^2 log n) and O(n^(3/2) log n) respectively in 4-space. Our algorithm exploits many geometric structures and properties of 4-dimensional space.
Congruence Testing Algorithm
Symmetry
Computational Geometry
48:1-48:16
Regular Paper
Heuna
Kim
Heuna Kim
Günter
Rote
Günter Rote
10.4230/LIPIcs.SoCG.2016.48
Tatsuya Akutsu. On determining the congruence of point sets in d dimensions. Computational Geometry: Theory and Applications, 4(9):247-256, 1998.
Helmut Alt, Kurt Mehlhorn, Hubert Wagener, and Emo Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete &Computational Geometry, 3(1):237-256, 1988.
M. J. Atallah. On symmetry detection. IEEE Trans. Computers, 100(7):663-666, 1985.
M. D. Atkinson. An optimal algorithm for geometrical congruence. Journal of Algorithms, 8(2):159-172, 1987.
Jon Louis Bentley and Michael Ian Shamos. Divide-and-conquer in multidimensional space. In Proc. 8th Ann. ACM Symp. Theory of Computing (STOC), pages 220-230. ACM, 1976.
Peter Brass and Christian Knauer. Testing the congruence of d-dimensional point sets. International Journal of Computational Geometry and Applications, 12(1-2):115-124, 2002.
Harold S. M. Coxeter. Regular Polytopes. Dover Publications, 3rd edition, 1973.
C. Dieckmann. Approximate Symmetries of Point Patterns. PhD thesis, FU Berlin, 2012.
N. P. Dolbilin and D. H. Huson. Periodic Delone tilings. Per. Math. Hung., 34:57-64, 1997.
Sebastian Iwanowski. Testing approximate symmetry in the plane is NP-hard. Theoretical Computer Science, 80(2):227-262, 1991.
H. Kim and G. Rote. Congruence testing of point sets in 4 dimensions. arXiv: URL: http://arxiv.org/abs/1603.07269.
http://arxiv.org/abs/1603.07269
Glenn Manacher. An application of pattern matching to a problem in geometrical complexity. Information Processing Letters, 5(1):6-7, 1976.
Henry P. Manning. Geometry of Four Dimensions. Macmillan, 1914.
Kōkichi Sugihara. An n log n algorithm for determining the congruity of polyhedra. Journal of Computer and System Sciences, 29(1):36-47, 1984.
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On the Complexity of Minimum-Link Path Problems
We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the min-link path's vertices or edges can be restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2D, and provide first results in dimensions 3 and higher for several versions of the problem.
Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [Ghosh et al. 2012] despite a large body of work on the topic. We also resolve the open problem from [Mitchell et al. 1992] mentioned in the handbook [Goodman and O'Rourke, 2004] (see Chapter 27.5, Open problem 3) and The Open Problems Project [Demaine et al. TOPP] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.
minimum-linkpath
diffuse reflection
terrain
bit complexity
NP-hardness
49:1-49:16
Regular Paper
Irina
Kostitsyna
Irina Kostitsyna
Maarten
Löffler
Maarten Löffler
Valentin
Polishchuk
Valentin Polishchuk
Frank
Staals
Frank Staals
10.4230/LIPIcs.SoCG.2016.49
John Adegeest, Mark H. Overmars, and Jack Snoeyink. Minimum-link c-oriented paths: Single-source queries. Int. J. of Computational Geometry and Applications, 4(1):39-51, 1994.
Pankaj K Agarwal and Micha Sharir. Arrangements and their applications. Handbook of computational geometry, pages 49-119, 2000.
Boris Aronov, Alan R. Davis, Tamal K. Dey, Sudebkumar Prasant Pal, and D. Chithra Prasad. Visibility with multiple reflections. Discrete & Computational Geometry, 20(1):61-78, 1998. URL: http://dx.doi.org/10.1007/PL00009378.
http://dx.doi.org/10.1007/PL00009378
Boris Aronov, Alan R. Davis, Tamal K. Dey, Sudebkumar Prasant Pal, and D. Chithra Prasad. Visibility with one reflection. Discrete & Computational Geometry, 19(4):553-574, 1998. URL: http://dx.doi.org/10.1007/PL00009368.
http://dx.doi.org/10.1007/PL00009368
Boris Aronov, Alan R. Davis, John Iacono, and Albert Siu Cheong Yu. The complexity of diffuse reflections in a simple polygon. In Proc. 7th Latin American Symposium on Theoretical Informatics, pages 93-104, 2006.
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M. de Berg, M. J. van Kreveld, B. J. Nilsson, and M. H. Overmars. Shortest path queries in rectilinear worlds. Int. J. of Computational Geometry and Applications, 3(2):287-309, 1992.
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A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino
A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a O(n*log^2(n))-time algorithm for deciding if a polyomino with n edges can tile the plane isohedrally. This improves on the O(n^{18})-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provençal, Fédou, and the second author.
Plane tiling
polyomino
boundary word
isohedral
50:1-50:15
Regular Paper
Stefan
Langerman
Stefan Langerman
Andrew
Winslow
Andrew Winslow
10.4230/LIPIcs.SoCG.2016.50
N. Alon, R. Yuster, and U. Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997.
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A. Winslow. An optimal algorithm for tiling the plane with a translated polyomino. In 26th International Symposium on Algorithms and Computation (ISAAC), 2015.
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Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range
Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections.
We focus on conditions for the existence of almost r-embeddings, i.e., maps f: K -> R^d such that the intersection of f(sigma_1), ..., f(sigma_r) is empty whenever sigma_1,...,sigma_r are pairwise disjoint simplices of K.
Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions: If r d > (r+1) dim K + 2 then there exists an almost r-embedding K-> R^d if and only if there exists an equivariant map of the r-fold deleted product of K to the sphere S^(d(r-1)-1).
This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 2), and settles an open question raised there.
Topological Combinatorics
Tverberg-Type Problems
Simplicial Complexes
Piecewise-Linear Topology
Haefliger-Weber Theorem
51:1-51:12
Regular Paper
Isaac
Mabillard
Isaac Mabillard
Uli
Wagner
Uli Wagner
10.4230/LIPIcs.SoCG.2016.51
S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner. Eliminating higher-multiplicity intersections, III. Codimension 2. Preprint, http://arxiv.org/abs/1511.03501, 2015.
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Isaac Mabillard and Uli Wagner. Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range. Preprint http://arxiv.org/abs/1601.00876, 2016.
http://arxiv.org/abs/1601.00876
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Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Peeling and Nibbling the Cactus: Subexponential-Time Algorithms for Counting Triangulations and Related Problems
Given a set of n points S in the plane, a triangulation T of S is a maximal set of non-crossing segments with endpoints in S. We present an algorithm that computes the number of triangulations on a given set of n points in time n^{ (11+ o(1)) sqrt{n} }, significantly improving the previous best running time of O(2^n n^2) by Alvarez and Seidel [SoCG 2013]. Our main tool is identifying separators of size O(sqrt{n}) of a triangulation in a canonical way. The definition of the separators are based on the decomposition of the triangulation into nested layers ("cactus graphs"). Based on the above algorithm, we develop a simple and formal framework to count other non-crossing straight-line graphs in n^{O(sqrt{n})} time. We demonstrate the usefulness of the framework by applying it to counting non-crossing Hamilton cycles, spanning trees, perfect matchings, 3-colorable triangulations, connected graphs, cycle decompositions, quadrangulations, 3-regular graphs, and more.
computational geometry
triangulations
exponential-time algorithms
52:1-52:16
Regular Paper
Dániel
Marx
Dániel Marx
Tillmann
Miltzow
Tillmann Miltzow
10.4230/LIPIcs.SoCG.2016.52
Oswin Aichholzer. The path of a triangulation. In SoCG'99, pages 14-23. ACM, 1999.
Victor Alvarez, Karl Bringmann, Radu Curticapean, and Saurabh Ray. Counting triangulations and other crossing-free structures via onion layers. D&C, 53(4):675-690, 2015. URL: http://dx.doi.org/10.1007/s00454-015-9672-3.
http://dx.doi.org/10.1007/s00454-015-9672-3
Victor Alvarez, Karl Bringmann, and Saurabh Ray. A simple sweep line algorithm for counting triangulations and pseudo-triangulations. CoRR, abs/1312.3188, 2013. URL: http://arxiv.org/abs/1312.3188.
http://arxiv.org/abs/1312.3188
Victor Alvarez, Karl Bringmann, Saurabh Ray, and Raimund Seidel. Counting triangulations and other crossing-free structures approximately. Comput. Geom., 48(5):386-397, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2014.12.006.
http://dx.doi.org/10.1016/j.comgeo.2014.12.006
Victor Alvarez and Raimund Seidel. A simple aggregative algorithm for counting triangulations of planar point sets and related problems. In SoCG'13, pages 1-8, 2013. URL: http://dx.doi.org/10.1145/2462356.2462392.
http://dx.doi.org/10.1145/2462356.2462392
Efthymios Anagnostou and Derek Corneil. Polynomial-time instances of the minimum weight triangulation problem. Computational Geometry, 3(5):247-259, 1993.
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http://dx.doi.org/10.1007/BF02574703
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, pages 1782-1801, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.129.
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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 Graph Drawing, 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, pages 251-262, 2010. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2010.2459.
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http://dx.doi.org/10.1007/978-3-642-23719-5_31
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Convergence between Categorical Representations of Reeb Space and Mapper
The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets. First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships. A related construction called mapper, and a special case of the mapper construction called the Joint Contour Net have been shown to be effective in visual analytics. Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees. An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit.
In this paper, we are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations. Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution.
Topological data analysis
Reeb space
mapper
category theory
53:1-53:16
Regular Paper
Elizabeth
Munch
Elizabeth Munch
Bei
Wang
Bei Wang
10.4230/LIPIcs.SoCG.2016.53
Peter Bubenik, Vin de Silva, and Jonathan Scott. Metrics for generalized persistence modules. Foundations of Computational Mathematics, 15(6):1501-1531, 2015.
Hamish Carr and David Duke. Joint contour nets. IEEE Transactions on Visualization and Computer Graphics, 20(8):1100-1113, 2014.
Hamish Carr, Jack Snoeyink, and Michiel van de Panne. Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree. Computational Geometry, 43:42-58, 2010.
Mathieu Carriére and Steve Oudot. Structure and stability of the 1-dimensional mapper. Symposium on Computational Geometry (to appear); arXiv:1511.05823, 2016.
Frédéric Chazal and Jian Sun. Gromov-Hausdorff approximation of filament structure using Reeb-type graph. Proceedings 13th Annual Symposium on Computational Geometry, pages 491-500, 2014.
Justin Curry. Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania, 2014.
Vin de Silva, Elizabeth Munch, and Amit Patel. Categorification of Reeb graphs. Discrete and Computational Geometry (to appear); arXiv:1501.04147, 2016.
Tamal K. Dey, Facundo Mémoli, and Yusu Wang. Mutiscale mapper: A framework for topological summarization of data and maps. arXiv:1504.03763, 2015.
Herbert Edelsbrunner and John Harer. Jacobi sets of multiple Morse functions. In F. Cucker, R. DeVore, P. Olver, and E. Süli, editors, Foundations of Computational Mathematics, Minneapolis 2002, pages 37-57. Cambridge University Press, 2002.
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William Harvey, Yusu Wang, and Rephael Wenger. A randomized O(mlogm) algorithm for computing Reeb graphs of arbitrary simplicial complexes. ACM Symposium on Computational Geometry, pages 267-276, 2010.
Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.
Dimitry Kozlov. Combinatorial Algebraic Topology. Springer, 2008.
P. Y. 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, 2013.
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Elizabeth Munch and Bei Wang. Convergence between categorical representations of Reeb space and mapper. arXiv:1307.7760, 2016.
Monica Nicolaua, Arnold J. Levineb, and Gunnar Carlsson. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings National Academy of Sciences of the United States of America, 108(17):7265-7270, 2011.
Salman Parsa. A deterministic O(mlogm) time algorithm for the Reeb graph. Proceedings 29th Annual Symposium on Computational Geometry, pages 269-276, 2012.
Amit Patel. Reeb Spaces and the Robustness of Preimages. PhD thesis, Duke University, 2010.
Michael A. Penna. On the geometry of combinatorial manifolds. Pacific Journal of Mathematics, 77(2):499-522, 1978.
G. Reeb. Sur les points singuliers d'une forme de pfaff completement intergrable ou d'une fonction numerique [on the singular points of a complete integral pfaff form or of a numerical function]. Comptes Rendus Acad.Science Paris, 222:847-849, 1946.
Gurjeet Singh, Facundo Mémoli, and Gunnar Carlsson. Topological methods for the analysis of high dimensional data sets and 3D object recognition. In Eurographics Symposium on Point-Based Graphics, 2007.
Roar Bakken Stovner. On the mapper algorithm: A study of a new topological method for data analysis. Master’s thesis, Norwegian University of Science and Technology, 2012.
Jon Woolf. The fundamental category of a stratified space. arXiv:0811.2580, 2013.
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New Lower Bounds for epsilon-Nets
Following groundbreaking work by Haussler and Welzl (1987), the use of small epsilon-nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest epsilon-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Konemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in R^4 by a family of half-spaces such that the size of any epsilon-net for them is at least (1/(9*epsilon)) log (1/epsilon) (Pach and Tardos).
The present paper completes both of these avenues of research. We (i) give a lower bound, matching the result of Chan et al., and (ii) generalize the construction of Pach and Tardos to half-spaces in R^d, for any d >= 4, to show that the general upper bound of Haussler and Welzl for the size of the smallest epsilon-nets is tight.
epsilon-nets; lower bounds; geometric set systems; shallow-cell complexity; half-spaces
54:1-54:16
Regular Paper
Andrey
Kupavskii
Andrey Kupavskii
Nabil
Mustafa
Nabil Mustafa
János
Pach
János Pach
10.4230/LIPIcs.SoCG.2016.54
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.
N. Alon. A non-linear lower bound for planar epsilon-nets. Discrete & Computational Geometry, 47(2):235-244, 2012.
B. Aronov, E. Ezra, and M. Sharir. Small-size ε-nets for axis-parallel rectangles and boxes. SIAM J. Comput., 39(7):3248-3282, 2010.
P. Ashok, U. Azmi, and S. Govindarajan. Small strong epsilon nets. Comput. Geom., 47(9):899-909, 2014.
N. Bus, S. Garg, N. H. Mustafa, and S. Ray. Tighter estimates for epsilon-nets for disks. Comput. Geom., 53:27-35, 2016.
T. M. Chan, E. Grant, J. Könemann, and M. Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of Symposium on Discrete Algorithms (SODA), 2012.
B. Chazelle and J. Friedman. A deterministic view of random sampling and its use in geometry. Combinatorica, 10(3):229-249, 1990.
K. Clarkson and K. Varadarajan. Improved approximation algorithms for geometric set cover. Discrete &Computational Geometry, 37:43-58, 2007.
K. L. Clarkson and P. W. Shor. Application of random sampling in computational geometry, II. Discrete & Computational Geometry, 4:387-421, 1989.
D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete &Computational Geometry, 2:127-151, 1987.
J. Komlós, J. Pach, and G. J. Woeginger. Almost tight bounds for epsilon-nets. Discrete & Computational Geometry, 7:163-173, 1992.
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J. Matoušek. Lectures in Discrete Geometry. Springer-Verlag, New York, NY, 2002.
J. Matoušek, R. Seidel, and E. Welzl. How to net a lot with little: Small epsilon-nets for disks and halfspaces. In Proceedings of Symposium on Computational Geometry, pages 16-22, 1990.
N. H. Mustafa, K. Dutta, and A. Ghosh. A simple proof of optimal epsilon-nets. Submitted, 2016.
N. H. Mustafa and S. Ray. Near-optimal generalisations of a theorem of Macbeath. In Proceedings of the 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, 2016, to appear.
J. Pach and P. K. Agarwal. Combinatorial Geometry. John Wiley &Sons, New York, NY, 1995.
J. Pach and G. Tardos. Tight lower bounds for the size of epsilon-nets. Journal of the AMS, 26:645-658, 2013.
E. Pyrga and S. Ray. New existence proofs for epsilon-nets. In Proceedings of the Symposium on Computational Geometry (SoCG), pages 199-207, 2008.
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Micha Sharir. On k-sets in arrangement of curves and surfaces. Discrete & Computational Geometry, 6:593-613, 1991.
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K. Varadarajan. Epsilon nets and union complexity. In Proceedings of the Symposium on Computational Geometry (SoCG), pages 11-16, 2009.
K. Varadarajan. Weighted geometric set cover via quasi uniform sampling. In Proceedings of the Symposium on Theory of Computing (STOC), pages 641-648, New York, USA, 2010. ACM.
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On Computing the Fréchet Distance Between Surfaces
We describe two (1+epsilon)-approximation algorithms for computing the Fréchet distance between two homeomorphic piecewise linear surfaces R and S of genus zero and total complexity n, with Frechet distance delta.
(1) A 2^{O((n + ( (Area(R)+Area(S))/(epsilon.delta)^2 )^2 )} time algorithm if R and S are composed of fat triangles (triangles with angles larger than a constant).
(2) An O(D/(epsilon.delta)^2) n + 2^{O(D^4/(epsilon^4.delta^2))} time algorithm if R and S are polyhedral terrains over [0,1]^2 with slope at most D.
Although, the Fréchet distance between curves has been studied extensively, very little is known for surfaces. Our results are the first algorithms (both for surfaces and terrains) that are guaranteed to terminate in finite time. Our latter result, in particular, implies a linear time algorithm for terrains of constant maximum slope and constant Frechet distance.
Surfaces
Terrains
Frechet distance
Parametrized complexity
normal coordinates
55:1-55:15
Regular Paper
Amir
Nayyeri
Amir Nayyeri
Hanzhong
Xu
Hanzhong Xu
10.4230/LIPIcs.SoCG.2016.55
Helmut Alt and Maike Buchin. Semi-computability of the Fréchet distance between surfaces. In Proceedings of the 21st European Workshop on Computational Geometery, pages 45-48, 2005.
Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geometry Appl., 5:75-91, 1995. URL: http://dx.doi.org/10.1142/S0218195995000064.
http://dx.doi.org/10.1142/S0218195995000064
Boris Aronov, Sariel Har-Peled, Christian Knauer, Yusu Wang, and Carola Wenk. Fréchet distance for curves, revisited. In Proceedings of the 14th Conference on Annual European Symposium, pages 52-63, 2006. URL: http://dx.doi.org/10.1007/11841036_8.
http://dx.doi.org/10.1007/11841036_8
Rinat Ben Avraham, Omrit Filtser, Haim Kaplan, Matthew J. Katz, and Micha Sharir. The discrete fréchet distance with shortcuts via approximate distance counting and selection. In Proceedings of the 30th Annual Symposium on Computational Geometry, pages 377-386, 2014. URL: http://dx.doi.org/10.1145/2582112.2582155.
http://dx.doi.org/10.1145/2582112.2582155
Sotiris Brakatsoulas, Dieter Pfoser, Randall Salas, and Carola Wenk. On map-matching vehicle tracking data. In Proceedings of the 31st Conference on Very Large Data Bases, pages 853-864, 2005.
Kevin Buchin, Maike Buchin, and Joachim Gudmundsson. Detecting single file movement. In Proceedings of the 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pages 1-10, 2008.
Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Maarten Löffler, and Jun Luo. Detecting commuting patterns by clustering subtrajectories. In Proceedings of the 19th International Symposium on Algorithms and Computation, pages 644-655, 2008.
Kevin Buchin, Maike Buchin, and André Schulz. Fréchet distance of surfaces: Some simple hard cases. In Proceedings of the 18th Conference on Annual European Symposium, pages 63-74, 2010.
Kevin Buchin, Maike Buchin, and Carola Wenk. Computing the Fréchet distance between simple polygons. Comp. Geom. Theo. Appl., 41(1-2):2-20, October 2008. URL: http://dx.doi.org/10.1016/j.comgeo.2007.08.003.
http://dx.doi.org/10.1016/j.comgeo.2007.08.003
Atlas F. Cook IV, Anne Driemel, Jessica Sherette, and Carola Wenk. Computing the frechet distance between folded polygons. Computational Geometry, 50:1-16, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2015.08.002.
http://dx.doi.org/10.1016/j.comgeo.2015.08.002
Anne Driemel and Sariel Har-Peled. Jaywalking your dog: Computing the fréchet distance with shortcuts. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 318-337, 2012.
Anne Driemel, Sariel Har-Peled, and Carola Wenk. Approximating the Fréchet distance for realistic curves in near linear time. Discrete & Computational Geometry, 48(1):94-127, 2012. URL: http://dx.doi.org/10.1007/s00454-012-9402-z.
http://dx.doi.org/10.1007/s00454-012-9402-z
Thomas Eiter and Heikki Mannila. Computing discrete Fréchet distance. Tech. Report CD-TR 94/64, Christian Doppler Lab. Expert Sys., TU Vienna, Austria, 1994.
Jeff Erickson and Amir Nayyeri. Shortest non-crossing walks in the plane. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 297-308, 2011.
Jeff Erickson and Amir Nayyeri. Tracing compressed curves in triangulated surfaces. In Proceedings of the 28th Annual Symposium on Computational Geometry, pages 131-140, 2012.
Michael S. Floater and Kai Hormann. Surface parameterization: a tutorial and survey. In Advances in Multiresolution for Geometric Modelling, Mathematics and Visualization, pages 157-186. Springer Berlin Heidelberg, 2005. URL: http://dx.doi.org/10.1007/3-540-26808-1_9.
http://dx.doi.org/10.1007/3-540-26808-1_9
Michael Godau. On the Complexity of Measuring the Similarity Between Geometric Objects in Higher Dimensions. PhD thesis, Freie Universität Berlin, 1998.
Wolfgang Haken. Theorie der Normalflächen: Ein Isotopiekriterium für den Kreisknoten. Acta Mathematica, 105:245-375, 1961.
Sariel Har-Peled and Benjamin Raichel. The fréchet distance revisited and extended. ACM Transactions on Algorithms, 10(1):3, 2014. URL: http://dx.doi.org/10.1145/2532646.
http://dx.doi.org/10.1145/2532646
Man-Soon Kim, Sang-Wook Kim, and Miyoung Shin. Optimization of subsequence matching under time warping in time-series databases. In Proceedings of ACM Symposium on Applied Computing, pages 581-586, 2005.
Ariane Mascret, Thomas Devogele, Iwan Le Berre, and Alain Hénaff. Coastline matching process based on the discrete Fréchet distance. In Proceedings of the12th International symposium on Spatial Data Handling, pages 383-400, 2006.
Amir Nayyeri and Anastasios Sidiropoulos. Computing the fréchet distance between polygons with holes. In The proceedings of The 42nd International Colloquium on Automata, Languages, and Programming, pages 997-1009, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_81.
http://dx.doi.org/10.1007/978-3-662-47672-7_81
Marcus Schaefer, Eric Sedgwick, and Daniel Stefankovic. Algorithms for normal curves and surfaces. In Proceedings if the 8th Annual International Computing and Combinatorics Conference, pages 370-380, 2002.
Marcus Schaefer and Daniel Štefankovič. Decidability of string graphs. J. Comput. Syst. Sci., 68(2):319-334, 2004.
Joan Serrà, Emillia Gómez, Perfecto Herrera, and Xavier Serra. Chroma binary similarity and local alignment applied to cover song identification. IEEE Transactions on Audio, Speech & Language Processing, 16(6):1138-1151, 2008.
Daniel Stefankovic. Algorithms for Simple Curves on Surfaces, String Graphs, and Crossing Numbers. PhD thesis, University of Chicago, Chicago, IL, USA, 2005. AAI3168396.
Oliver van Kaick, Hao Zhang, Ghassan Hamarneh, and Daniel Cohen-Or. A survey on shape correspondence. Computer Graphics Forum, 30(6):1681-1707, 2011.
Carola Wenk, Randall Salas, and Dieter Pfoser. Addressing the need for map-matching speed: Localizing global curve-matching algorithms. In Proceedings of the 18th International Conference on Scientific and Statistical Database Management., pages 879-888, 2006.
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The Farthest-Point Geodesic Voronoi Diagram of Points on the Boundary of a Simple Polygon
Given a set of sites (points) in a simple polygon, the farthest-point geodesic Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic metric. We present an O((n+m)loglogn)-time algorithm to compute the farthest-point geodesic Voronoi diagram for m sites lying on the boundary of a simple n-gon.
Geodesic distance
simple polygons
farthest-point Voronoi diagram
56:1-56:15
Regular Paper
Eunjin
Oh
Eunjin Oh
Luis
Barba
Luis Barba
Hee-Kap
Ahn
Hee-Kap Ahn
10.4230/LIPIcs.SoCG.2016.56
Alok Aggarwal, Leonidas J Guibas, James Saxe, and Peter W Shor. A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete &Computational Geometry, 4(6):591-604, 1989.
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. In Proceedings of the 31st Symposium on Compututaional Geometry, SoCG, pages 209-223, 2015.
Boris Aronov, Steven Fortune, and Gordon Wilfong. The furthest-site geodesic Voronoi diagram. Discrete &Computational Geometry, 9(3):217-255, 1993.
T. Asano and G.T. Toussaint. Computing the geodesic center of a simple polygon. Technical Report SOCS-85.32, McGill University, 1985.
Cecilia Bohler, Rolf Klein, and Chih-Hung Liu. Forest-like abstract Voronoi diagrams in linear time. In Proceedings of the 26th Canadian Conference on Computational Geometry, CCCG, pages 133-141, 2014.
B Chazelle. A theorem on polygon cutting with applications. In Proceedings 23rd Annual Symposium on Foundations of Computer Science, FOCS, pages 339-349, 1982.
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J. S. B. Mitchell. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, pages 633-701. Elsevier, 2000.
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Richard Pollack, Micha Sharir, and Günter Rote. Computing the geodesic center of a simple polygon. Discrete &Computational Geometry, 4(6):611-626, 1989.
Subhash Suri. Computing geodesic furthest neighbors in simple polygons. Journal of Computer and System Sciences, 39(2):220-235, 1989.
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Avoiding the Global Sort: A Faster Contour Tree Algorithm
We revisit the classical problem of computing the contour tree of a scalar field f:M to R, where M is a triangulated simplicial mesh in R^d. The contour tree is a fundamental topological structure that tracks the evolution of level sets of f and has numerous applications in data analysis and visualization.
All existing algorithms begin with a global sort of at least all critical values of f, which can require (roughly) Omega(n log n) time. Existing lower bounds show that there are pathological instances where this sort is required. We present the first algorithm whose time complexity depends on the contour tree structure, and avoids the global sort for non-pathological inputs. If C denotes the set of critical points in M, the running time is roughly O(sum_{v in C} log l_v), where l_v is the depth of v in the contour tree. This matches all existing upper bounds, but is a significant asymptotic improvement when the contour tree is short and fat. Specifically, our approach ensures that any comparison made is between nodes that are either adjacent in M or in the same descending path in the contour tree, allowing us to argue strong optimality properties of our algorithm.
Our algorithm requires several novel ideas: partitioning M in well-behaved portions, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis.
contour trees
computational topology
computational geometry
57:1-57:14
Regular Paper
Benjamin
Raichel
Benjamin Raichel
C.
Seshadhri
C. Seshadhri
10.4230/LIPIcs.SoCG.2016.57
C. Bajaj, M. van Kreveld, R. W. van Oostrum, V. Pascucci, and D. R. Schikore. Contour trees and small seed sets for isosurface traversal. Technical Report UU-CS-1998-25, Department of Information and Computing Sciences, Utrecht University, 1998.
K. Beketayev, G. Weber, M. Haranczyk, P.-T. Bremer, M. Hlawitschka, and B. Hamann. Visualization of topology of transformation pathways in complex chemical systems. In Computer Graphics Forum (EuroVis 2011), pages 663-672, 2011.
R. Boyell and H. Ruston. Hybrid techniques for real-time radar simulation. In Proceedings of Fall Joint Computer Conference, pages 445-458, 1963.
P.-T. Bremer, G. Weber, V. Pascucci, M. Day, and J. Bell. Analyzing and tracking burning structures in lean premixed hydrogen flames. IEEE Transactions on Visualization and Computer Graphics, 16(2):248-260, 2010.
P.-T. Bremer, G. Weber, J. Tierny, V. Pascucci, M. Day, and J. Bell. Analyzing and tracking burning structures in lean premixed hydrogen flames. IEEE Transactions on Visualization and Computer Graphics, 17(9):1307-1325, 2011.
H. Carr. Topological Manipulation of Isosurfaces. PhD thesis, University of British Columbia, 2004.
H. Carr, J. Snoeyink, and U. Axen. Computing contour trees in all dimensions. Computational Geometry: Theory and Applications, 24(2):75-94, 2003.
Y. Chiang, T. Lenz, X. Lu, and G. Rote. Simple and optimal output-sensitive construction of contour trees using monotone paths. Computational Geometry: Theory and Applications, 30(2):165-195, 2005. URL: http://dx.doi.org/10.1016/j.comgeo.2004.05.002.
http://dx.doi.org/10.1016/j.comgeo.2004.05.002
K. Cole-McLaughlin, H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci. Loops in reeb graphs of 2-manifolds. In Proceedings of the Symposium on Computational Geometry (SoCG), pages 344-350, 2003. URL: http://dx.doi.org/10.1145/777792.777844.
http://dx.doi.org/10.1145/777792.777844
H. Doraiswamy and V. Natarajan. Efficient algorithms for computing reeb graphs. Computational Geometry: Theory and Applications, 42:606-616, 2009.
H. Doraiswamy and V. Natarajan. Computing reeb graphs as a union of contour trees. IEEE Transactions on Visualization and Computer Graphics, 19(2):249-262, 2013.
H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer, 1987.
H. Freeman and S. Morse. On searching a contour map for a given terrain elevation profile. Journal of the Franklin Institute, 284(1):1-25, 1967.
W. Harvey, Y. Wang, and R. Wenger. A randomized o(m log m) time algorithm for computing reeb graph of arbitrary simplicial complexes. In Proceedings of the Symposium on Computational Geometry (SoCG), pages 267-276, 2010.
D. Laney, P.-T. Bremer, A. Macarenhas, P. Miller, and V. Pascucci. Understanding the structure of the turbulent mixing layer in hydrodynamic instabilities. IEEE Transactions on Visualization and Computer Graphics, 12(6):1053-1060, 2006.
A. Mascarenhas, R. Grout, P.-T. Bremer, V. Pascucci, E. Hawkes, and J. Chen. Topological feature extraction for comparison of length scales in terascale combustion simulation data. In Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, pages 229-240, 2011.
S. Parsa. A deterministic o(m log m) time algorithm for the reeb graph. In Proceedings of the Symposium on Computational Geometry (SoCG), pages 269-276, 2012.
V. Pascucci and K. Cole-McLaughlin. Efficient computation of the topology of level set. In IEEE Visualization, pages 187-194, 2002. URL: http://dx.doi.org/10.1109/VISUAL.2002.1183774.
http://dx.doi.org/10.1109/VISUAL.2002.1183774
V. Pascucci, G. Scorzelli, P.-T. Bremer, and A. Mascarenhas. Robust on-line computation of reeb graphs: simplicity and speed. ACM Transactions on Graphics, 26(58), 2007.
B. Raichel and C. Seshadhri. Avoiding the global sort: A faster contour tree algorithm. CoRR, abs/1411.2689, 2014.
Y. Shinagawa and T. Kunii. Constructing a reeb graph automatically from cross sections. IEEE Comput. Graphics Appl., 11(6):44-51, 1991.
S. Tarasov and M. Vyalyi. Construction of contour trees in 3d in O(n log n) steps. In Proceedings of the Symposium on Computational Geometry (SoCG), pages 68-75, 1998. URL: http://dx.doi.org/10.1145/276884.276892.
http://dx.doi.org/10.1145/276884.276892
J. Tierny, A. Gyulassy, E. Simon, and V. Pascucci. Loop surgery for volumetric meshes: Reeb graphs reduced to contour trees. IEEE Trans. on Visualization and Computer Graphics, 15(6):1177-1184, 2009.
M. van Kreveld, R. van Oostrum, C. Bajaj, V. Pascucci, and D. Schikore. Contour trees and small seed sets for isosurface traversal. In Proceedings of the Symposium on Computational Geometry (SoCG), pages 212-220, 1997. URL: http://dx.doi.org/10.1145/262839.269238.
http://dx.doi.org/10.1145/262839.269238
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Configurations of Lines in 3-Space and Rigidity of Planar Structures
Let L be a sequence (l_1,l_2,...,l_n) of n lines in C^3. We define the intersection graph G_L=([n],E) of L, where [n]:={1,..., n}, and with {i,j} in E if and only if i\neq j and the corresponding lines l_i and l_j intersect, or are parallel (or coincide). For a graph G=([n],E), we say that a sequence L is a realization of G if G subset G_L. One of the main results of this paper is to provide a combinatorial characterization of graphs G=([n],E) that have the following property: For every generic realization L of G, that consists of n pairwise distinct lines, we have G_L=K_n, in which case the lines of L are either all concurrent or all coplanar.
The general statements that we obtain about lines, apart from their independent interest, turns out to be closely related to the notion of graph rigidity. The connection is established due to the so-called Elekes-Sharir framework, which allows us to transform the problem into an incidence problem involving lines in three dimensions. By exploiting the geometry of contacts between lines in 3D, we can obtain alternative, simpler, and more precise characterizations of the rigidity of graphs.
Line configurations
Rigidity
Global Rigidity
Laman graphs
58:1-58:14
Regular Paper
Orit E.
Raz
Orit E. Raz
10.4230/LIPIcs.SoCG.2016.58
L. Asimow and B. Roth. The rigidity of graphs. Trans. Amer. Math. Soc., 245:279-289, 1978.
R. Connelly. Generic global rigidity. Discrete Comput. Geom., 33:549-563, 2005.
H. Crapo. Structural rigidity. Structural Topology, 1:26-45, 1979.
Gy. Elekes and M. Sharir. Incidences in three dimensions and distinct distances in the plane. Combinat. Probab. Comput., 20:571-608, 2011.
L. Guth and N. H. Katz. On the Erdős distinct distances problem in the plane. Annals Math., 18:155-190, 2015.
R. Hartshorne. Algebraic Geometry. Springer-Verlag, New York, 1977.
B. Hendrickson. Conditions for unique graph embeddings. Technical Report 88-950, Department of Computer Science, Cornell University, 1988.
B. Jackson and T. Jordán. Connected rigidity matroids and unique realizations of graphs. J. Combinat. Theory, Ser. B, 94:1-29, 2005.
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J.L. Martin. Geometry of graph varieties. Trans. Amer. Math. Soc., 355:4151-4169, 2003.
T. Tao. Lines in the Euclidean group SE(2). Blog post, available at URL: https://terrytao.wordpress.com/2011/03/05/lines-in-the-euclidean-group-se2/.
https://terrytao.wordpress.com/2011/03/05/lines-in-the-euclidean-group-se2/
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Weak 1/r-Nets for Moving Points
In this paper, we extend the weak 1/r-net theorem to a kinetic setting where the underlying set of points is moving polynomially with bounded description complexity. We establish that one can find a kinetic analog N of a weak 1/r-net of cardinality O(r^(d(d+1)/2)log^d r) whose points are moving with coordinates that are rational functions with bounded description complexity. Moreover, each member of N has one polynomial coordinate.
Hypergraphs
Weak epsilon-net
59:1-59:13
Regular Paper
Alexandre
Rok
Alexandre Rok
Shakhar
Smorodinsky
Shakhar Smorodinsky
10.4230/LIPIcs.SoCG.2016.59
N. Alon, I. Bárány, Z. Füredi, and D.J. Kleitman. Point selections and weak e-nets for convex hulls. Combinatorics, Probability & Computing, 1:189-200, 1992.
B. Aronov, B. Chazelle, and H. Edelsbrunner. Points and triangles in the plane and halving planes in space. Discrete & Computational Geometry, 6:435-442, 1991. URL: http://dx.doi.org/10.1007/BF02574700.
http://dx.doi.org/10.1007/BF02574700
B. Bukh, J. Matoušek, and G. Nivasch. Lower bounds for weak epsilon-nets and stair-convexity. Israel Journal of Mathematics, 182(1):199-228, 2011.
J.L. De Carufel, M. Katz, M. Korman, A. van Renssen, M. Roeloffzen, and S. Smorodinsky. On kinetic range spaces and their applications. CoRR, abs/1507.02130, 2015. URL: http://arxiv.org/abs/1507.02130.
http://arxiv.org/abs/1507.02130
B. Chazelle, H. Edelsbrunner, M. Grigni, L.J. Guibas, M. Sharir, and E. Welzl. Improved bounds on weak epsilon-nets for convex sets. Discrete & Computational Geometry, 13:1-15, 1995.
B. Chazelle, H. Edelsbrunner, M. Grigni, L.J. Guibas, M. Sharir, and E. Welzl. Improved bounds on weak epsilon-nets for convex sets. Discrete & Computational Geometry, 13:1-15, 1995. URL: http://dx.doi.org/10.1007/BF02574025.
http://dx.doi.org/10.1007/BF02574025
D. Haussler and E. Welzl. epsilon-nets and simplex range queries. Discrete & Computational Geometry, 2:127-151, 1987. URL: http://dx.doi.org/10.1007/BF02187876.
http://dx.doi.org/10.1007/BF02187876
J. Matoušek and U. Wagner. New constructions of weak epsilon-nets. Discrete & Computational Geometry, 32(2):195-206, 2004.
N.H. Mustafa and S. Ray. Weak epsilon-nets have basis of size o(1/epsilon log(1/epsilon)) in any dimension. Comput. Geom., 40(1):84-91, 2008. URL: http://dx.doi.org/10.1016/j.comgeo.2007.02.006.
http://dx.doi.org/10.1016/j.comgeo.2007.02.006
A. Ya. Chervonenkis V. N. Vapnik. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16:264-280, 1971.
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Applications of Incidence Bounds in Point Covering Problems
In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed.
First we give a non-parameterized algorithm for both problems in O*(2^n) (where the O*(.) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant. Our result improves the previous best times O*((k/1.35)^k) for Line Cover (where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving on the previous time of O*((k^2/1.3)^k).
Point Cover
Incidence Bounds
Inclusion Exclusion
Exponential Algorithm
60:1-60:15
Regular Paper
Peyman
Afshani
Peyman Afshani
Edvin
Berglin
Edvin Berglin
Ingo
van Duijn
Ingo van Duijn
Jesper
Sindahl Nielsen
Jesper Sindahl Nielsen
10.4230/LIPIcs.SoCG.2016.60
Peyman Afshani, Edvin Berglin, Ingo van Duijn, and Jesper Sindahl Nielsen. Applications of incidence bounds in point covering problems. arXiv:1603.07282, 2016.
Pankaj K Agarwal and Boris Aronov. Counting facets and incidences. Discrete &Computational Geometry, 7(1):359-369, 1992.
Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM Journal on Computing, 39(2):546-563, 2009.
Cheng Cao. Study on two optimization problems: Line cover and maximum genus embedding. Master’s thesis, Texas A&M University, 2012.
Herbert Edelsbrunner. Algorithms in Combinatorial Geometry. Springer Publishing Company, Incorporated, 1st edition, 2012.
Herbert Edelsbrunner, Leonidas Guibas, and Micha Sharir. The complexity of many cells in arrangements of planes and related problems. Discrete &Computational Geometry, 5(1):197-216, 1990.
György Elekes and Csaba D Tóth. Incidences of not-too-degenerate hyperplanes. In Proceedings of the twenty-first annual symposium on Computational geometry, pages 16-21. ACM, 2005.
Vladimir Estivill-Castro, Apichat Heednacram, and Francis Suraweera. FPT-algorithms for minimum-bends tours. International Journal of Computational Geometry &Applications, 21(02):189-213, 2011.
Jacob Fox, János Pach, Adam Sheffer, Andrew Suk, and Joshua Zahl. A semi-algebraic version of Zarankiewicz’s problem. arXiv preprint arXiv:1407.5705, 2014.
Magdalene Grantson and Christos Levcopoulos. Covering a set of points with a minimum number of lines. Springer, 2006.
Ben Joseph Green and Terence Tao. On sets defining few ordinary lines. Discrete & Computational Geometry, 50(2):409-468, 2013.
Leonidas J Guibas, Mark H Overmars, and Jean-Marc Robert. The exact fitting problem in higher dimensions. Computational geometry, 6(4):215-230, 1996.
LJ Guibas, Mark Overmars, and Jean-Marc Robert. The exact fitting problem for points. In Proc. 3rd Canadian Conference on Computational Geometry, pages 171-174, 1991.
Stefan Kratsch, Geevarghese Philip, and Saurabh Ray. Point line cover: The easy kernel is essentially tight. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1596-1606. SIAM, 2014.
VS Anil Kumar, Sunil Arya, and Hariharan Ramesh. Hardness of set cover with intersection 1. In Automata, Languages and Programming, pages 624-635. Springer, 2000.
Stefan Langerman and Pat Morin. Covering things with things. Discrete &Computational Geometry, 33(4):717-729, 2005.
Nimrod Megiddo and Arie Tamir. On the complexity of locating linear facilities in the plane. Operations research letters, 1(5):194-197, 1982.
János Pach and Micha Sharir. On the number of incidences between points and curves. Combinatorics, Probability and Computing, 7(01):121-127, 1998.
József Solymosi and Terence Tao. An incidence theorem in higher dimensions. Discrete &Computational Geometry, 48(2):255-280, 2012.
Endre Szemerédi and William T Trotter Jr. Extremal problems in discrete geometry. Combinatorica, 3(3-4):381-392, 1983.
Praveen Tiwari. On covering points with conics and strips in the plane. Master’s thesis, Texas A&M University, 2012.
Jianxin Wang, Wenjun Li, and Jianer Chen. A parameterized algorithm for the hyperplane-cover problem. Theoretical Computer Science, 411(44):4005-4009, 2010.
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Grouping Time-Varying Data for Interactive Exploration
We present algorithms and data structures that support the interactive analysis of the grouping structure of one-, two-, or higher-dimensional time-varying data while varying all defining parameters. Grouping structures characterise important patterns in the temporal evaluation of sets of time-varying data. We follow Buchin et al. [JoCG 2015] who define groups using three parameters: group-size, group-duration, and inter-entity distance. We give upper and lower bounds on the number of maximal groups over all parameter values, and show how to compute them efficiently. Furthermore, we describe data structures that can report changes in the set of maximal groups in an output-sensitive manner. Our results hold in R^d for fixed d.
Trajectory
Time series
Moving entity
Grouping
Algorithm
Data structure
61:1-61:16
Regular Paper
Arthur
van Goethem
Arthur van Goethem
Marc
van Kreveld
Marc van Kreveld
Maarten
Löffler
Maarten Löffler
Bettina
Speckmann
Bettina Speckmann
Frank
Staals
Frank Staals
10.4230/LIPIcs.SoCG.2016.61
P. Agarwal and J. Matoušek. On Range Searching with Semialgebraic Sets. Disc. &Comput. Geom., 11(4):393-418, 1994.
N. Amato, M. Goodrich, and E. Ramos. Computing the arrangement of curve segments: Divide-and-conquer algorithms via sampling. In Proc. 11th ACM-SIAM Symp. on Disc. Algorithms, pages 705-706, 2000.
G. Andrienko, N. Andrienko, and S. Wrobel. Visual analytics tools for analysis of movement data. ACM SIGKDD Explorations Newsletter, 9(2):38-46, 2007.
M. Bender and M. Farach-Colton. The LCA problem revisited. In LATIN 2000: Theoret. Informatics, volume 1776 of LNCS, pages 88-94. Springer, 2000.
M. Bender and M. Farach-Colton. The level ancestor problem simplified. Theoret. Computer Science, 321(1):5-12, 2004.
M. Benkert, B. Djordjevic, J. Gudmundsson, and T. Wolle. Finding popular places. Int. J. of Comput. Geom. &Appl., 20(1):19-42, 2010.
P. Bovet and S. Benhamou. Spatial analysis of animals' movements using a correlated random walk model. J. of Theoret. Biology, 131(4):419-433, 1988.
K. Buchin, M. Buchin, M. van Kreveld, M. Löffler, R. Silveira, C. Wenk, and L. Wiratma. Median trajectories. Algorithmica, 66(3):595-614, 2013.
K. Buchin, M. Buchin, M. van Kreveld, B. Speckmann, and F. Staals. Trajectory grouping structure. J. of Comput. Geom., 6(1):75-98, 2015.
T. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. J. of the ACM, 57(3):16:1-16:15, 2010.
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D. Eppstein, M. Goodrich, and J. Simons. Set-difference range queries. In Proc. 2013 Canadian Conf. on Comput. Geom., 2013.
A. Fujimura and K. Sugihara. Geometric analysis and quantitative evaluation of sport teamwork. Systems and Computers in Japan, 36(6):49-58, 2005.
J. Gudmundsson, M. van Kreveld, and B. Speckmann. Efficient detection of patterns in 2D trajectories of moving points. GeoInformatica, 11:195-215, 2007.
J. Gudmundsson, M. van Kreveld, and F. Staals. Algorithms for hotspot computation on trajectory data. In Proc. 21st ACM SIGSPATIAL GIS, pages 134-143, 2013.
D. Keim, G. Andrienko, J.-D. Fekete, C. Görg, J. Kohlhammer, and G. Melançon. Visual analytics: Definition, process, and challenges. In A. Kerren, J. Stasko, J.-D. Fekete, and C. North, editors, Information Visualization, volume 4950 of LNCS, pages 154-175. Springer, 2008.
I. Kostitsyna, M. van Kreveld, M. Löffler, B. Speckmann, and F. Staals. Trajectory grouping structure under geodesic distance. In Proc. 31th Symp. Computat. Geom. Lipics, 2015.
X. Li, X. Li, D. Tang, and X. Xu. Deriving features of traffic flow around an intersection from trajectories of vehicles. In Proc. IEEE 18th Int. Conf. Geoinformatics, pages 1-5, 2010.
J. Matoušek. Efficient partition trees. Disc. &Comput. Geom., 8(3):315-334, 1992.
M. Mirzargar, R. Whitaker, and R. Kirby. Curve Boxplot: generalization of Boxplot for ensembles of curves. IEEE Trans. on Vis. and Comp. Graphics, 20(12):2654-2663, 2014.
A. Stohl. Computation, accuracy and applications of trajectories - a review and bibliography. Atmospheric Environment, 32(6):947-966, 1998.
Arthur van Goethem, Marc van Kreveld, Maarten Löffler, Bettina Speckmann, and Frank Staals. Grouping time-varying data for interactive exploration. CoRR, abs/1603.06252, 2016.
A. Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comput., 11(4):721-736, 1982.
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On the Separability of Stochastic Geometric Objects, with Applications
In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let S=S_R U S_B be a given set of stochastic bichromatic points, and define n = min{|S_R|, |S_B|} and N = max{|S_R|, |S_B|}. We show that the separable-probability (SP) of S can be computed in O(nN^{d-1}) time for d >= 3 and O(min{nN log N, N^2}) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nN^d) time for d >= 2. In addition, we give an Omega(nN^{d-1}) witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in O(nN^d) and O(nN^{d+1}) time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems.
Stochastic objects
linear separability
separable-probability
expected separation-margin
convex hull
62:1-62:16
Regular Paper
Jie
Xue
Jie Xue
Yuan
Li
Yuan Li
Ravi
Janardan
Ravi Janardan
10.4230/LIPIcs.SoCG.2016.62
P.K. Agarwal, B. Aronov, S. Har-Peled, J.M. Phillips, K. Yi, and W. Zhang. Nearest neighbor searching under uncertainty II. In Proceedings of the 32nd Symposium on Principles of Database Systems, pages 115-126. ACM, 2013.
P.K. Agarwal, S-W. Cheng, Y. Tao, and K. Yi. Indexing uncertain data. In Proceedings of the Twenty-eighth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pages 137-146. ACM, 2009.
P.K. Agarwal, S-W. Cheng, and K. Yi. Range searching on uncertain data. ACM Transactions on Algorithms (TALG), 8(4):43, 2012.
P.K. Agarwal, S. Har-Peled, S. Suri, H. Yıldız, and W. Zhang. Convex hulls under uncertainty. In Algorithms-ESA 2014, pages 37-48. Springer, 2014.
C.C. Aggarwal and P.S. Yu. A survey of uncertain data algorithms and applications. IEEE Transactions on Knowledge and Data Engineering, 21(5):609-623, 2009.
A. Agrawal, Y. Li, J. Xue, and R. Janardan. The most-likely skyline problem for stochastic points. Submitted to a journal, 2015.
N. Dalvi, C. Ré, and D. Suciu. Probabilistic databases: diamonds in the dirt. Communications of the ACM, 52(7):86-94, 2009.
M. de Berg, A.D. Mehrabi, and F. Sheikhi. Separability of imprecise points. In Algorithm theory - SWAT 2014, volume 8503 of LNCS, pages 146-157. Springer, 2014.
M. de Berg, M. van Kreveld, M. Overmars, and O.C. Schwarzkopf. Computational Geometry. Springer, 2000.
H. Edelsbrunner and L.J. Guibas. Topologically sweeping an arrangement. In Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, pages 389-403, 1986.
M. Fink, J. Hershberger, N. Kumar, and Suri S. Hyperplane separability and convexity of probabilistic points. 32nd International Symposium on Computational Geometry, 2016.
P. Kamousi, T.M. Chan, and S. Suri. Stochastic minimum spanning trees in euclidean spaces. In Proceedings of the Twenty-seventh Annual Symposium on Computational geometry, pages 65-74. ACM, 2011.
Y. Li, J. Xue, A. Agrawal, and R. Janardan. On the arrangement of stochastic lines in ℝ². Submitted to a journal, 2015.
M. Löffler. Data imprecision in computational geometry. PhD thesis, Utrecht Univ., 2009.
M. Löffler and M. van Kreveld. Largest and smallest convex hulls for imprecise points. Algorithmica, 56(2):235-269, 2010.
S. Suri and K. Verbeek. On the most likely Voronoi Diagram and nearest neighbor searching. In Algorithms and Computation, pages 338-350. Springer, 2014.
S. Suri, K. Verbeek, and H. Yıldız. On the most likely convex hull of uncertain points. In Algorithms-ESA 2013, pages 791-802. Springer, 2013.
J. Xue, Y. Li, and R. Janardan. On the separability of stochastic geometric objects, with applications. (Full version). ArXiv:1603.07021 [cs.CG], 2016.
W. Zhang. Nearest-neighbor searching under uncertainty. PhD thesis, Duke Univ., 2012.
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Approximating Convex Shapes With Respect to Symmetric Difference Under Homotheties
The symmetric difference is a robust operator for measuring the error of approximating one shape by another. Given two convex shapes P and C, we study the problem of minimizing the volume of their symmetric difference under all possible scalings and translations of C. We prove that the problem can be solved by convex programming. We also present a combinatorial algorithm for convex polygons in the plane that runs in O((m+n) log^3(m+n)) expected time, where n and m denote the number of vertices of P and C, respectively.
shape matching
convexity
symmetric difference
homotheties
63:1-63:15
Regular Paper
Juyoung
Yon
Juyoung Yon
Sang Won
Bae
Sang Won Bae
Siu-Wing
Cheng
Siu-Wing Cheng
Otfried
Cheong
Otfried Cheong
Bryan T.
Wilkinson
Bryan T. Wilkinson
10.4230/LIPIcs.SoCG.2016.63
H.-K. Ahn, S.-W. Cheng, and I. Reinbacher. Maximum overlap of convex polytopes under translation. Computational Geometry: Theory and Applications, 46:552-565, 2013.
H. Alt, J. Blömer, M. Godau, and H. Wagener. Approximation of convex polygons. In Automata, Languages and Programming, volume 443 of LNCS, pages 703-716. Springer, 1990.
H. Alt, U. Fuchs, G. Rote, and G. Weber. Matching convex shapes with respect to the symmetric difference. Algorithmica, 21:89-103, 1998.
D. Avis, P. Bose, T. Shermer, J. Snoeyink, G. Toussaint, and B. Zhu. On the sectional area of convex polytopes. In Communication at the 12th Annual Symposium on Computational Geometry, page C, 1996.
P. Brass and M. Lassak. Problems on approximation by triangles. Geombinatorics, 10:103-115, 2001.
B. Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete &Computational Geometry, 9:145-158, 1993.
M. de Berg, O. Cheong, O. Devillers, M. van Kreveld, and M. Teillaud. Computing the maximum overlap of two convex polygons under translations. Theory of Computing Systems, 31:613-628, 1998.
R. Fleischer, K. Mehlhorn, G. Rote, E. Welzl, and C. Yap. Simultaneous inner and outer approximation of shapes. Algorithmica, 8:365-389, 1992.
G. N. Frederickson and D. B. Johnson. Generalized selection and ranking: sorted matrices. SIAM Journal on Computing, 13:14-30, 1984.
H. Groemer. On the symmetric difference metric for convex bodies. Beiträge zur Algebra und Geometrie, 41:107-114, 2000.
J. Matoušek. Randomized optimal algorithm for slope selection. Information Processing Letters, 39:183-187, 1991.
N. Megiddo. Applying parallel computation algorithms in the design of serial algorithms. Journal of ACM, 30:852-865, 1983.
R. Schneider. Convex Bodies - the Brunn-Minkowski theory. Cambridge University Press, 1993.
D. Schymura. An upper bound on the volume of the symmetric difference of a body and a congruent copy. Advances in Geometry, 14:287-298, 2014.
D. M. J. Tax and R. P. W. Duin. Support vector data description. Machine Learning, 54:45-66, 2004.
R. Veltkamp and M. Hagedoorn. Shape similarity measures, properties and constructions. In Advances in Visual Information Systems, volume 1929 of LNCS, pages 467-476. Springer, 2000.
Creative Commons Attribution 3.0 Unported license
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Interactive Geometric Algorithm Visualization in a Browser
We present an online, interactive tool for writing and presenting interactive geometry demos suitable for classroom demonstrations. Code for the demonstrations is written in JavaScript using p5.js, a JavaScript library based on Processing.
Computational Geometry
Processing
JavaScript
Visualisation
Incremental Algorithms
64:1-64:4
Regular Paper
Kirk
Gardner
Kirk Gardner
Lynn
Asselin
Lynn Asselin
Donald
Sheehy
Donald Sheehy
10.4230/LIPIcs.SoCG.2016.64
p5.js. Free Software Foundation, version 2.1. URL: http://p5js.org/.
http://p5js.org/
Ace code editor. Free Software Foundation, version 2.1., 2010. URL: https://ace.c9.io.
https://ace.c9.io
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Geometric Models for Musical Audio Data
We study the geometry of sliding window embeddings of audio features that summarize perceptual information about audio, including its pitch and timbre. These embeddings can be viewed as point clouds in high dimensions, and we add structure to the point clouds using a cover tree with adaptive thresholds based on multi-scale local principal component analysis to automatically assign points to clusters. We connect neighboring clusters in a scaffolding graph, and we use knowledge of stratified space structure to refine our estimates of dimension in each cluster, demonstrating in our music applications that choruses and verses have higher dimensional structure, while transitions between them are lower dimensional. We showcase our technique with an interactive web-based application powered by Javascript and WebGL which plays music synchronized with a principal component analysis embedding of the point cloud down to 3D. We also render the clusters and the scaffolding on top of this projection to visualize the transitions between different sections of the music.
Geometric Models
Audio Analysis
High Dimensional Data Analysis
Stratified Space Models
65:1-65:5
Regular Paper
Paul
Bendich
Paul Bendich
Ellen
Gasparovic
Ellen Gasparovic
John
Harer
John Harer
Christopher
Tralie
Christopher Tralie
10.4230/LIPIcs.SoCG.2016.65
Mark A. Bartsch and Gregory H. Wakefield. To catch a chorus: Using chroma-based representations for audio thumbnailing. In 2001 IEEE Workshop on the Applications of Signal Processing to Audio and Acoustics, pages 15-18. IEEE, 2001.
Paul Bendich, Ellen Gasparovic, John Harer, and Christopher Tralie. Scaffoldings and spines: Organizing high-dimensional data using cover trees, local principal component analysis, and persistent homology, 2016. http://arxiv.org/abs/1602.06245.
Alina Beygelzimer, Sham Kakade, and John Langford. Cover trees for nearest neighbor. In Proceedings of the 23rd International Conference on Machine Learning, pages 97-104. ACM, 2006.
Bruce P. Bogert, Michael J.R. Healy, and John W. Tukey. The quefrency analysis of time series for echoes: Cepstrum, pseudo-autocovariance, cross-cepstrum and saphe cracking. In Proceedings of the symposium on time series analysis, volume 15, pages 209-243. chapter, 1963.
Brian McFee, Colin Raffel, Dawen Liang, Daniel PW Ellis, Matt McVicar, Eric Battenberg, and Oriol Nieto. librosa: Audio and music signal analysis in Python. In Proceedings of the 14th Python in Science Conference, 2015.
George Tzanetakis and Perry Cook. Musical genre classification of audio signals. IEEE transactions on Speech and Audio Processing, 10(5):293-302, 2002.
Creative Commons Attribution 3.0 Unported license
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Visualizing Scissors Congruence
Consider two simple polygons with equal area. The Wallace-Bolyai-Gerwien theorem states that these polygons are scissors congruent, that is, they can be dissected into finitely many congruent polygonal pieces. We present an interactive application that visualizes this constructive proof.
polygonal congruence
geometry
rigid transformations
66:1-66:3
Regular Paper
Satyan
Devadoss
Satyan Devadoss
Ziv
Epstein
Ziv Epstein
Dmitriy
Smirnov
Dmitriy Smirnov
10.4230/LIPIcs.SoCG.2016.66
Creative Commons Attribution 3.0 Unported license
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Visualization of Geometric Spanner Algorithms
It is easier to understand an algorithm when it can be seen in interactive mode. The current study implemented four algorithms to construct geometric spanners; the path-greedy, gap-greedy, Theta-graph and Yao-graph algorithms. The data structure visualization framework (http://www.cs.usfca.edu/~galles/visualization/) developed by David Galles was used.
Two features were added to allow its use in spanner algorithm visualization: support point-based algorithms and export of the output to Ipe drawing software format. The interactive animations in the framework make steps of visualization beautiful and media controls are available to manage the animations. Visualization does not require extensions to be installed on the web browser. It is available at http://cs.yazd.ac.ir/cgalg/AlgsVis/.
geometric spanner networks
geometric spanner algorithms animations.
67:1-67:4
Regular Paper
Mohammad
Farshi
Mohammad Farshi
Seyed Hossein
Hosseini
Seyed Hossein Hosseini
10.4230/LIPIcs.SoCG.2016.67
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.
Sunil Arya and Michiel Smid. Efficient construction of a bounded-degree spanner with low weight. Algorithmica, 17(1):33-54, 1997.
Rom Aschner. Geometric spanners applet. http://www.cs.bgu.ac.il/~romas/greedy.html .
http://www.cs.bgu.ac.il/~romas/greedy.html
Ken Clarkson. Approximation algorithms for shortest path motion planning. In Proc. of the 19t Annual ACM Symp. on Theory of Computing, pages 56-65. ACM, 1987.
Mohammad Farshi and Seyed Hossein Hosseini. Geometric spanner algorithms visualizations. http://cs.yazd.ac.ir/cgalg/AlgsVis/ .
http://cs.yazd.ac.ir/cgalg/AlgsVis/
David Galles. Data structure visualizations. http://www.cs.usfca.edu/~galles/visualization/ .
http://www.cs.usfca.edu/~galles/visualization/
Giri Narasimhan and Michiel Smid. Geometric spanner networks. Cambridge University Press, 2007.
Daniel Russel and Leonidas J. Guibas. Exploring protein folding trajectories using geometric spanners. Pacific Symposium on Biocomputing, pages 40-51, 2005.
Petra Specht and Michiel Smid. Visualization of spanners. http://isgwww.cs.uni-magdeburg.de/tspanner/spanner.html .
http://isgwww.cs.uni-magdeburg.de/tspanner/spanner.html
Andrew Chi-Chih Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing, 11(4):721-736, 1982.
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Path Planning for Simple Robots using Soft Subdivision Search
The concept of resolution-exact path planning is a theoretically sound alternative to the standard exact algorithms, and provides much stronger guarantees than probabilistic or sampling algorithms. It opens the way for the introduction of soft predicates in the context of subdivision algorithm. Taking a leaf from the great success of the Probabilistic Road Map (PRM) framework, we formulate an analogous framework for subdivision, called Soft Subdivision Search (SSS). In this video, we illustrate the SSS framework for a trio of simple planar robots: disc, triangle and 2-links. These robots have, respectively, 2, 3 and 4 degrees of freedom. Our 2-link robot can also avoid self-crossing. These algorithms operate in realtime and are relatively easy to implement.
Robot Path Planning
Soft Predicates
Resolution-Exact Algorithm
Subdivision Search
68:1-68:5
Regular Paper
Ching-Hsiang
Hsu
Ching-Hsiang Hsu
John Paul
Ryan
John Paul Ryan
Chee
Yap
Chee Yap
10.4230/LIPIcs.SoCG.2016.68
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Exploring Circle Packing Algorithms
We present an interactive tool for visualizing and experimenting with different circle packing algorithms.
Computational Geometry
Processing
Javascript
Visualization
Incremental Algorithms
69:1-69:4
Regular Paper
Kevin
Pratt
Kevin Pratt
Connor
Riley
Connor Riley
Donald
Sheehy
Donald Sheehy
10.4230/LIPIcs.SoCG.2016.69
Alexander I. Bobenko and Boris A. Springborn. Variational principles for circle patterns and koebe’s theorem. Transactions of the American Mathematical Society, 356(2):659-689, 2003.
Yves Colin de Verdière. Un principe variationnel pour les empilements de cercles. Inventiones mathematicae, 104(1):655-669, 1991.
Jonathan A. Kelner. Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus. SIAM J. Comput., 35(4):882-902, 2006.
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, 1997.
Bojan Mohar. A polynomial time circle packing algorithm. Discrete Mathematics, 117(1-3):257-263, 1993.
Kenneth Stephenson. Introduction to Circle Packing. Cambridge University Press, New York, New York, 2005.
Creative Commons Attribution 3.0 Unported license
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The Explicit Corridor Map: Using the Medial Axis for Real-Time Path Planning and Crowd Simulation
We describe and demonstrate the Explicit Corridor Map (ECM), a navigation mesh for path planning and crowd simulation in virtual environments. For a bounded 2D environment with polygonal obstacles, the ECM is the medial axis of the free space annotated with nearest-obstacle information. It can be used to compute short and smooth paths for disk-shaped characters of any radius. It is also well-defined for multi-layered 3D environments that consist of connected planar layers. We highlight various operations on the ECM, such as dynamic updates, visibility queries, and the computation of paths (indicative routes).
We have implemented the ECM as the basis of a real-time crowd simulation framework with path following and collision avoidance. Our implementation has been successfully used to simulate real-life events involving large crowds of heterogeneous characters. The enclosed demo application displays various features of our software.
Medial axis
Navigation mesh
Path planning
Crowd simulation
70:1-70:5
Regular Paper
Wouter
van Toll
Wouter van Toll
Atlas F.
Cook IV
Atlas F. Cook IV
Marc
van Kreveld
Marc van Kreveld
Roland
Geraerts
Roland Geraerts
10.4230/LIPIcs.SoCG.2016.70
R. Geraerts. Planning short paths with clearance using Explicit Corridors. In Proceedings of the IEEE International Conference on Robotics and Automation, pages 1997-2004, 2010.
P. Hart, N. Nilsson, and B. Raphael. A formal basis for the heuristic determination of minimum cost paths. IEEE Transactions on Systems Science and Cybernetics, 4(2):100-107, 1968.
M. Held. VRONI and ArcVRONI: Software for and applications of Voronoi diagrams in science and engineering. In Proceedings of the 8th International Symposium on Voronoi Diagrams in Science and Engineering, pages 3-12, 2011.
K.E. Hoff III, T. Culver, J. Keyser, M. Lin, and D. Manocha. Fast computation of generalized Voronoi diagrams using graphics hardware. International Conference on Computer Graphics and Interactive Techniques, pages 277-286, 1999.
D.T. Lee and R.L. Drysdale III. Generalization of Voronoi diagrams in the plane. SIAM Journal on Computing, 10(1):73-87, 1981.
F. Preparata. The medial axis of a simple polygon. In Mathematical Foundations of Computer Science, volume 53, pages 443-450. Springer, 1977.
W. van Toll, N. Jaklin, and R. Geraerts. Towards believable crowds: A generic multi-level framework for agent navigation. In ASCI.OPEN, 2015.
W.G. van Toll, A.F. Cook IV, and R. Geraerts. Navigation meshes for realistic multi-layered environments. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 3526-3532, 2011.
W.G. van Toll, A.F. Cook IV, and R. Geraerts. A navigation mesh for dynamic environments. Computer Animation and Virtual Worlds, 23(6):535-546, 2012.
Creative Commons Attribution 3.0 Unported license
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High-Dimensional Geometry of Sliding Window Embeddings of Periodic Videos
We explore the high dimensional geometry of sliding windows of periodic videos. Under a reasonable model for periodic videos, we show that the sliding window is necessary to disambiguate all states within a period, and we show that a video embedding with a sliding window of an appropriate dimension lies on a topological loop along a hypertorus. This hypertorus has an independent ellipse for each harmonic of the motion. Natural motions with sharp transitions from foreground to background have many harmonics and are hence in higher dimensions, so linear subspace projections such as PCA do not accurately summarize the geometry of these videos. Noting this, we invoke tools from topological data analysis and cohomology to parameterize motions in high dimensions with circular coordinates after the embeddings. We show applications to videos in which there is obvious periodic motion and to videos in which the motion is hidden.
Video Processing
High Dimensional Geometry
Circular Coordinates
Nonlinear Time Series
71:1-71:5
Regular Paper
Christopher
Tralie
Christopher Tralie
10.4230/LIPIcs.SoCG.2016.71
Vin De Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Persistent cohomology and circular coordinates. Discrete &Computational Geometry, 45(4):737-759, 2011.
Herbert Edelsbrunner and John Harer. Computational topology: an introduction. American Mathematical Soc., 2010.
Holger Kantz and Thomas Schreiber. Nonlinear time series analysis, volume 7. Cambridge university press, 2004.
Jose A Perea and John Harer. Sliding windows and persistence: An application of topological methods to signal analysis. Foundations of Computational Mathematics, 15(3):799-838, 2013.
Arno Schödl, Richard Szeliski, David H Salesin, and Irfan Essa. Video textures. In Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 489-498. ACM Press/Addison-Wesley Publishing Co., 2000.
Neal Wadhwa, Michael Rubinstein, Frédo Durand, and William T Freeman. Phase-based video motion processing. ACM Transactions on Graphics (TOG), 32(4):80, 2013.
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Introduction to Persistent Homology
This video presents an introduction to persistent homology, aimed at a viewer who has mathematical aptitude but not necessarily knowledge of algebraic topology. Persistent homology is an algebraic method of discerning the topological features of complex data, which in recent years has found applications in fields such as image processing and biological systems. Using smooth animations, the video conveys the intuition behind persistent homology, while giving a taste of its key properties, applications, and mathematical underpinnings.
Persistent Homology
Topological Data Analysi
72:1-72:3
Regular Paper
Matthew L.
Wright
Matthew L. Wright
10.4230/LIPIcs.SoCG.2016.72
P. Bendich, J.S. Marron, E. Miller, A. Pieloch, and S. Skwerer. Persistent homology analysis of brain artery trees. Annals of Applied Statistics, to appear.
G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian. On the local behavior of spaces of natural images. International Journal of Computer Vision, 76(1):1-12, 2008.
J.M. Chan, G. Carlsson, and R. Rabadan. Topology of viral evolution. Proceedings of the National Academy of Sciences, 110(46), November 2013.
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete and Computational Geometry, 37(1):103-120, 2007.
V. de Silva and R. Ghrist. Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology, 7:339-358, 2007.
H. Edelsbrunner and J. Harer. Persistent homology: a survey. In Surveys on discrete and computational geometry: twenty years later: AMS-IMS-SIAM Joint Summer Research Conference, June 18-22, 2006, Snowbird, Utah, volume 453, page 257. American Mathematical Society, 2008.
H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete and Computational Geometry, 28(4):511-533, 2002.
J. Perea and G. Carlsson. A klein-bottle-based dictionary for texture representation. International Journal of Computer Vision, 107(1):75-97, 2014.
J. Perea, A. Deckard, S.B. Haase, and J. Harer. Sw1pers: Sliding windows and 1-persistence scoring; discovering periodicity in gene expression time series data. BMC Bioinformatics, 16(1), 2015.
K. Turner, S. Mukherjee, and Doug M Boyer. Sufficient statistics for shapes and surfaces. preprint, 2013.
A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete and Computational Geometry, 33(2):249-274, 2005.
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