31st International Symposium on Computational Geometry (SoCG 2015), SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands
SoCG 2015
June 22-25, 2015
Eindhoven, The Netherlands
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
Lars
Arge
Lars Arge
János
Pach
János Pach
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
34
2015
978-3-939897-83-5
https://www.dagstuhl.de/dagpub/978-3-939897-83-5
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
i-xx
Front Matter
Lars
Arge
Lars Arge
János
Pach
János Pach
10.4230/LIPIcs.SOCG.2015.i
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Combinatorial Discrepancy for Boxes via the gamma_2 Norm
The gamma_2 norm of a real m by n matrix A is the minimum number t such that the column vectors of A are contained in a 0-centered ellipsoid E that in turn is contained in the hypercube [-t, t]^m. This classical quantity is polynomial-time computable and was proved by the second author and Talwar to approximate the hereditary discrepancy: it bounds the hereditary discrepancy from above and from below, up to logarithmic factors. Here we provided a simplified proof of the upper bound and show that both the upper and the lower bound are asymptotically tight in the worst case.
We then demonstrate on several examples the power of the gamma_2 norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of log(n)^(d-1) (up to constant factors) for the d-dimensional Tusnady problem, asking for the combinatorial discrepancy of an n-point set in d-dimensional space with respect to axis-parallel boxes. For d>2, this improves the previous best lower bound, which was of order approximately log(n)^((d-1)/2), and it comes close to the best known upper bound of O(log(n)^(d+1/2)), for which we also obtain a new, very simple proof. Applications to lower bounds for dynamic range searching and lower bounds in differential privacy are given.
discrepancy theory
range counting
factorization norms
1-15
Regular Paper
Jirí
Matoušek
Jirí Matoušek
Aleksandar
Nikolov
Aleksandar Nikolov
10.4230/LIPIcs.SOCG.2015.1
T. van Aardenne-Ehrenfest. Proof of the impossibility of a just distribution of an infinite sequence of points. Nederl. Akad. Wet., Proc., 48:266-271, 1945. Also in Indag. Math. 7, 71-76 (1945).
T. van Aardenne-Ehrenfest. On the impossibility of a just distribution. Nederl. Akad. Wet., Proc., 52:734-739, 1949. Also in Indag. Math. 11, 264-269 (1949).
J. R. Alexander, J. Beck, and W. W. L. Chen. Geometric discrepancy theory and uniform distribution. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 10, pages 185-207. CRC Press LLC, Boca Raton, FL, 1997.
N. Bansal. Constructive algorithms for discrepancy minimization. http://arxiv.org/abs/1002.2259, also in FOCS'10: Proc. 51st IEEE Symposium on Foundations of Computer Science, pages 3-10, 2010.
http://arxiv.org/abs/1002.2259
J. Beck. Balanced two-colorings of finite sets in the square. I. Combinatorica, 1:327-335, 1981.
J. Beck. Balanced two-colorings of finite sets in the cube. Discrete Mathematics, 73:13-25, 1989.
J. Beck. A two-dimensional van Aardenne-Ehrenfest theorem in irregularities of distribution. Compositio Math., 72:269-339, 1989.
J. Beck and W. W. L. Chen. Irregularities of Distribution. Cambridge University Press, Cambridge, 1987.
Rajendra Bhatia. Matrix analysis, volume 169 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.
D. Bilyk and M. T. Lacey. On the small ball inequality in three dimensions. Duke Math. J., 143(1):81-115, 2008.
D. Bilyk, M. T. Lacey, and A. Vagharshakyan. On the small ball inequality in all dimensions. J. Funct. Anal., 254(9):2470-2502, 2008.
G. Bohus. On the discrepancy of 3 permutations. Random Struct. Algo., 1:215-220, 1990.
J. Bourgain and L. Tzafriri. Invertibility of large submatrices with applications to the geometry of banach spaces and harmonic analysis. Israel journal of mathematics, 57(2):137-224, 1987.
M. Charikar, A. Newman, and A. Nikolov. Tight hardness results for minimizing discrepancy. In Proc. 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), San Francisco, California, USA, pages 1607-1614, 2011.
B. Chazelle. The Discrepancy Method. Cambridge University Press, Cambridge, 2000.
B. Chazelle and A. Lvov. A trace bound for the hereditary discrepancy. Discrete Comput. Geom., 26(2):221-231, 2001.
B. Chazelle and A. Lvov. The discrepancy of boxes in higher dimension. Discrete Comput. Geom., 25(4):519-524, 2001.
J. G. van der Corput. Verteilungsfunktionen I. Akad. Wetensch. Amsterdam, Proc., 38:813-821, 1935.
J. G. van der Corput. Verteilungsfunktionen II. Akad. Wetensch. Amsterdam, Proc., 38:1058-1066, 1935.
M. Drmota and R. F. Tichy. Sequences, discrepancies and applications (Lecture Notes in Mathematics 1651). Springer-Verlag, Berlin etc., 1997.
Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Foundations and Trends in Theoretical Computer Science, 9(3-4):211-407, 2014.
Michael L. Fredman. The complexity of maintaining an array and computing its partial sums. J. ACM, 29(1):250-260, 1982.
J. H. Halton. On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math., 2:84-90, 1960.
J. M. Hammersley. Monte Carlo methods for solving multivariable problems. Ann. New York Acad. Sci., 86:844-874, 1960.
K. G. Larsen. On range searching in the group model and combinatorial discrepancy. SIAM Journal on Computing, 43(2):673-686, 2014.
Troy Lee, Adi Shraibman, and Robert Špalek. A direct product theorem for discrepancy. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, CCC 2008, 23-26 June 2008, College Park, Maryland, USA, pages 71-80. IEEE Computer Society, 2008.
Nati Linial, Shahar Mendelson, Gideon Schechtman, and Adi Shraibman. Complexity measures of sign matrices. Combinatorica, 27(4):439-463, 2007.
L. Lovász, J. Spencer, and K. Vesztergombi. Discrepancy of set-systems and matrices. European J. Combin., 7:151-160, 1986.
J. Matoušek. The determinant bound for discrepancy is almost tight. Proc. Amer. Math. Soc., 141(2):451-460, 2013.
J. Matoušek. On the discrepancy for boxes and polytopes. Monatsh. Math., 127(4):325-336, 1999.
J. Matoušek. Geometric Discrepancy (An Illustrated Guide), 2nd printing. Springer-Verlag, Berlin, 2010.
Jiří Matoušek and Aleksandar Nikolov. Combinatorial discrepancy for boxes via the ellipsoid-infinity norm. To appear in SoCG 15., 2014.
S. Muthukrishnan and A. Nikolov. Optimal private halfspace counting via discrepancy. In STOC '12: Proceedings of the 44th symposium on Theory of Computing, pages 1285-1292, New York, NY, USA, 2012. ACM.
A. Newman, O. Neiman, and A. Nikolov. Beck’s three permutations conjecture: A counterexample and some consequences. In Proc. 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 253-262, 2012.
A. Nikolov and K. Talwar. Approximating hereditary discrepancy via small width ellipsoids. Preprint arXiv:1311.6204, 2013.
A. Nikolov and K. Talwar. On the hereditary discrepancy of homogeneous arithmetic progressions. To Appear in Proceedings of AMS, 2013.
A. Nikolov, K. Talwar, and Li Zhang. The geometry of differential privacy: the sparse and approximate cases. In Proc. 45th ACM Symposium on Theory of Computing (STOC), Palo Alto, California, USA, pages 351-360, 2013. Full version to appear in SIAM Journal on Computing as The Geometry of Differential Privacy: the Small Database and Approximate Cases.
D. Pálvölgyi. Indecomposable coverings with concave polygons. Discrete Comput. Geom., 44(3):577-588, 2010.
K. F. Roth. On irregularities of distribution. Mathematika, 1:73-79, 1954.
Thomas Rothvoß. Constructive discrepancy minimization for convex sets. CoRR, abs/1404.0339, 2014. To Appear in FOCS 2014.
W. M. Schmidt. On irregularities of distribution VII. Acta Arith., 21:45-50, 1972.
A. Seeger. Calculus rules for combinations of ellipsoids and applications. Bull. Australian Math. Soc., 47(01):1-12, 1993.
J. Spencer. Ten Lectures on the Probabilistic Method. CBMS-NSF. SIAM, Philadelphia, PA, 1987.
A. Srinivasan. Improving the discrepancy bound for sparse matrices: better approximations for sparse lattice approximation problems. In Proc. 8th ACM-SIAM Symposium on Discrete Algorithms, pages 692-701, 1997.
G. Strang and S. MacNamara. Functions of difference matrices are Toeplitz plus Hankel. SIAM Review, 2014. To appear.
Nicole Tomczak-Jaegermann. Banach-Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.
R. Vershynin. John’s decompositions: Selecting a large part. Israel Journal of Mathematics, 122(1):253-277, 2001.
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Tilt: The Video - Designing Worlds to Control Robot Swarms with Only Global Signals
We present fundamental progress on the computational universality of swarms of micro- or nano-scale robots in complex environments, controlled not by individual navigation, but by a uniform global, external force. More specifically, we consider a 2D grid world, in which all obstacles and robots are unit squares, and for each actuation, robots move maximally until they collide with an obstacle or another robot. The objective is to control robot motion within obstacles, design obstacles in order to achieve desired permutation of robots, and establish controlled interaction that is complex enough to allow arbitrary computations. In this video, we illustrate progress on all these challenges: we demonstrate NP-hardness of parallel navigation, we describe how to construct obstacles that allow arbitrary permutations, and we establish the necessary logic gates for performing arbitrary in-system computations.
Particle swarms
global control
complexity
geometric computation
16-18
Regular Paper
Aaron T.
Becker
Aaron T. Becker
Erik D.
Demaine
Erik D. Demaine
Sándor P.
Fekete
Sándor P. Fekete
Hamed Mohtasham
Shad
Hamed Mohtasham Shad
Rose
Morris-Wright
Rose Morris-Wright
10.4230/LIPIcs.SOCG.2015.16
Aaron T. Becker, Erik D. Demaine, Sándor P. Fekete, Golnaz Habibi, and James McLurkin. Reconfiguring massive particle swarms with limited, global control. In 9th International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics (ALGOSENSORS), volume 8343 of Springer LNCS, pages 51-66, 2013.
Aaron T. Becker, Erik D. Demaine, Sándor P. Fekete, and James McLurkin. Particle computation: Designing worlds to control robot swarms with only global signals. In 2014 IEEE International Conference on Robotics and Automation, ICRA 2014, Hong Kong, China, May 31 - June 7, 2014, pages 6751-6756, 2014.
Erik D. Demaine, Martin L. Demaine, and Joseph O'Rourke. PushPush and Push-1 are NP-hard in 2D. In Proceedings of the 12th Annual Canadian Conference on Computational Geometry (CCCG), pages 211-219, August 2000.
Birgit Engels and Tom Kamphans. Randolphs robot game is NP-hard! Electronic Notes in Discrete Mathematics, 25:49-53, 2006.
Robert A. Hearn and Erik D. Demaine. PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. arXiv:cs/0205005, cs.CC/0205005, 2002.
Michael Hoffmann. Motion planning amidst movable square blocks: Push-* is NP-hard. In Canadian Conference on Computational Geometry, pages 205-210, June 2000.
Markus Holzer and Stefan Schwoon. Assembling molecules in ATOMIX is hard. Theoretical Computer Science, 313(3):447-462, 2004.
Hahmed Mohtasham Shad, Rose Morris-Wright, Erik D. Demaine, Sándor P. Fekete, and Aaron T. Becker. Particle computation: Device fan-out and binary memory. In 2015 IEEE International Conference on Robotics and Automation, ICRA Seattle, USA, May 26 - 30, 2015, page to appear, 2015.
ThinkFun. Tilt: Gravity fed logic maze. URL: http://www.thinkfun.com/tilt.
http://www.thinkfun.com/tilt
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Automatic Proofs for Formulae Enumerating Proper Polycubes
This video describes a general framework for computing formulae enumerating polycubes of size n which are proper in n-k dimensions (i.e., spanning all n-k dimensions), for a fixed value of k. (Such formulae are central in the literature of statistical physics in the study of percolation processes and collapse of branched polymers.) The implemented software re-affirmed the already-proven formulae for k <= 3, and proved rigorously, for the first time, the formula enumerating polycubes of size n that are proper in n-4 dimensions.
Polycubes
inclusion-exclusion
19-22
Regular Paper
Gill
Barequet
Gill Barequet
Mira
Shalah
Mira Shalah
10.4230/LIPIcs.SOCG.2015.19
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Visualizing Sparse Filtrations
Over the last few years, there have been several approaches to building sparser complexes that still give good approximations to the persistent homology. In this video, we have illustrated a geometric perspective on sparse filtrations that leads to simpler proofs, more general theorems, and a more visual explanation. We hope that as these techniques become easier to understand, they will also become easier to use.
Topological Data Analysis
Simplicial Complexes
Persistent Homology
23-25
Regular Paper
Nicholas J.
Cavanna
Nicholas J. Cavanna
Mahmoodreza
Jahanseir
Mahmoodreza Jahanseir
Donald R.
Sheehy
Donald R. Sheehy
10.4230/LIPIcs.SOCG.2015.23
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Visualizing Quickest Visibility Maps
Consider the following modification to the shortest path query problem in polygonal domains: instead of finding shortest path to a query point q, we find the shortest path to any point that sees q. We present an interactive visualization applet visualizing these quickest visibility paths. The applet also visualizes quickest visibility maps, that is the subdivision of the domain into cells by the quickest visibility path structure.
path planning
visibility
26-28
Regular Paper
Topi
Talvitie
Topi Talvitie
10.4230/LIPIcs.SOCG.2015.26
E. M. Arkin, A. Efrat, C. Knauer, J. S. B. Mitchell, V. Polishchuk, G. Rote, L. Schlipf, and T. Talvitie. Shortest path to a segment and quickest visibility queries. In SoCG, 2015.
R. Khosravi and M. Ghodsi. The fastest way to view a query point in simple polygons. In European Workshop on Computational Geometry, pages 187-190. Technische Universiteit Eindhoven, 2005.
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Sylvester-Gallai for Arrangements of Subspaces
In this work we study arrangements of k-dimensional subspaces V_1,...,V_n over the complex numbers. Our main result shows that, if every pair V_a, V_b of subspaces is contained in a dependent triple (a triple V_a, V_b, V_c contained in a 2k-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on k (and not on n). The theorem holds under the assumption that the subspaces are pairwise non-intersecting (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly's theorem for complex numbers), which proves the k=1 case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. One of the main ingredients in the proof is a strengthening of a theorem of Barthe (from the k=1 to k>1 case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem).
Sylvester-Gallai
Locally Correctable Codes
29-43
Regular Paper
Zeev
Dvir
Zeev Dvir
Guangda
Hu
Guangda Hu
10.4230/LIPIcs.SOCG.2015.29
Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff. Fractional Sylvester-Gallai theorems. Proceedings of the National Academy of Sciences, 110(48):19213-19219, 2013.
Boaz Barak, Zeev Dvir, Amir Yehudayoff, and Avi Wigderson. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC'11, pages 519-528, 2011.
Franck Barthe. On a reverse form of the Brascamp-Lieb inequality. Inventiones mathematicae, 134(2):335-361, 1998.
P. Borwein and W. O. J. Moser. A survey of Sylvester’s problem and its generalizations. Aequationes Mathematicae, 40(1):111-135, 1990.
Zeev Dvir. On matrix rigidity and locally self-correctable codes. computational complexity, 20(2):367-388, 2011.
Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Breaking the quadratic barrier for 3-LCC’s over the reals. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 784-793, 2014.
Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Improved rank bounds for design matrices and a new proof of Kelly’s theorem. Forum of Mathematics, Sigma, 2, 10 2014.
Noam Elkies, Lou M Pretorius, and Konrad J Swanepoel. Sylvester-Gallai theorems for complex numbers and quaternions. Discrete & Computational Geometry, 35(3):361-373, 2006.
P. Erdös, Richard Bellman, H. S. Wall, James Singer, and V. Thébault. Problems for solution: 4065-4069. The American Mathematical Monthly, 50(1):65-66, 1943.
Sten Hansen. A generalization of a theorem of Sylvester on the lines determined by a finite point set. Mathematica Scandinavica, 16:175-180, 1965.
L. M. Kelly. A resolution of the Sylvester-Gallai problem of J.-P. Serre. Discrete & Computational Geometry, 1(1):101-104, 1986.
Peter D. Lax. Linear algebra and its applications. Pure and Applied Mathematics. Wiley-Interscience, 2007.
E. Melchior. Uber vielseite der projektive ebene. Deutsche Math., 5:461-475, 1940.
J. J. Sylvester. Mathematical question 11851. Educational Times, 59:98, 1893.
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Computational Aspects of the Colorful Carathéodory Theorem
Let P_1,...,P_{d+1} be d-dimensional point sets such that the convex hull of each P_i contains the origin. We call the sets P_i color classes, and we think of the points in P_i as having color i. A colorful choice is a set with at most one point of each color. The colorful Caratheodory theorem guarantees the existence of a colorful choice whose convex hull contains the origin. So far, the computational complexity of finding such a colorful choice is unknown.
We approach this problem from two directions. First, we consider approximation algorithms: an m-colorful choice is a set that contains at most m points from each color class. We show that for any fixed epsilon > 0, an (epsilon d)-colorful choice containing the origin in its convex hull can be found in polynomial time. This notion of approximation has not been studied before, and it is motivated through the applications of the colorful Caratheodory theorem in the literature. In the second part, we present a natural generalization of the colorful Caratheodory problem: in the Nearest Colorful Polytope problem (NCP), we are given d-dimensional point sets P_1,...,P_n that do not necessarily contain the origin in their convex hulls. The goal is to find a colorful choice whose convex hull minimizes the distance to the origin. We show that computing local optima for the NCP problem is PLS-complete, while computing a global optimum is NP-hard.
colorful Carathéodory theorem
high-dimensional approximation
PLS
44-58
Regular Paper
Wolfgang
Mulzer
Wolfgang Mulzer
Yannik
Stein
Yannik Stein
10.4230/LIPIcs.SOCG.2015.44
Emile Aarts and Jan Karel Lenstra, editors. Local search in combinatorial optimization. Princeton University Press, 2003.
Jorge L. Arocha, Imre Bárány, Javier Bracho, Ruy Fabila, and Luis Montejano. Very colorful theorems. Discrete Comput. Geom., 42(2):142-154, 2009.
Imre Bárány. A generalization of Carathéodory’s theorem. Discrete Math., 40(2-3):141-152, 1982.
Imre Bárány and Shmuel Onn. Colourful linear programming and its relatives. Math. Oper. Res., 22(3):550-567, 1997.
David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? J. Comput. System Sci., 37(1):79-100, 1988.
Jiří Matoušek. Lectures on discrete geometry. Springer, 2002.
Frédéric Meunier and Antoine Deza. A further generalization of the colourful Carathéodory theorem. In Discrete geometry and optimization, volume 69 of Fields Inst. Commun., pages 179-190. Springer, New York, 2013.
Frédéric Meunier and Pauline Sarrabezolles. Colorful linear programming, Nash equilibrium, and pivots. arxiv:1409.3436, 2014.
Wil Michiels, Emile Aarts, and Jan Korst. Theoretical aspects of local search. Monographs in Theoretical Computer Science. Springer, Berlin, 2007.
Gary L. Miller and Donald R. Sheehy. Approximate centerpoints with proofs. Comput. Geom., 43(8):647-654, 2010.
Wolfgang Mulzer and Daniel Werner. Approximating Tverberg points in linear time for any fixed dimension. Discrete Comput. Geom., 50(2):520-535, 2013.
Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. System Sci., 48(3):498-532, 1994.
Karanbir S. Sarkaria. Tverberg’s theorem via number fields. Israel J. Math., 79(2-3):317-320, 1992.
Alejandro A. Schäffer and Mihalis Yannakakis. Simple local search problems that are hard to solve. SIAM J. Comput., 20(1):56-87, 1991.
Helge Tverberg. Further generalization of Radon’s theorem. J. London Math. Soc., 43:352-354, 1968.
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Semi-algebraic Ramsey Numbers
Given a finite set P of points from R^d, a k-ary semi-algebraic relation E on P is the set of k-tuples of points in P, which is determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. The Ramsey number R^{d,t}_k(s,n) is the minimum N such that any N-element point set P in R^d equipped with a k-ary semi-algebraic relation E, such that E has complexity at most t, contains s members such that every k-tuple induced by them is in E, or n members such that every k-tuple induced by them is not in E.
We give a new upper bound for R^{d,t}_k(s,n) for k=3 and s fixed. In particular, we show that for fixed integers d,t,s, R^{d,t}_3(s,n)=2^{n^{o(1)}}, establishing a subexponential upper bound on R^{d,t}_3(s,n). This improves the previous bound of 2^{n^C} due to Conlon, Fox, Pach, Sudakov, and Suk, where C is a very large constant depending on d,t, and s. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in R^d. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results.
Ramsey theory
semi-algebraic relation
one-sided hyperplanes
Schur numbers
59-73
Regular Paper
Andrew
Suk
Andrew Suk
10.4230/LIPIcs.SOCG.2015.59
H. L. Abbott and D. Hanson. A problem of schur and its generalizations. Acta Arith., 20:175-187, 1972.
H. L. Abbott and L. Moser. Sum-free sets of integers. Acta Arith., 11:392-396, 1966.
P. K. Agarwal and J. Erickson. Optimal partition trees. In In Proc. 26th Ann. ACM Sympos. Comput. Geom., pages 1-10, 2010.
P. K. Agarwal and J. Erickson. Geometric range searching and its relatives. In J. E. Goodman B. Chazelle and R. Pollack, editors, Advances in Dicsrete and Computational Geometry, pages 1-56, 1998.
M. Ajtai, J. Komlós, and E. Szemerédi. A note on ramsey numbers. J. Combin. Theory Ser. A, 29:354-360, 1980.
N. Alon, J. Pach, R. Pinchasi, R. Radoičić, and M. Sharir. Crossing patterns of semi-algebraic sets. J. Combin. Theory Ser. A, 111:310-326, 2005.
S. Basu, R. Pollack, and M. F. Roy. Algorithms in Real Algebraic Geometry. Springer-Verlag, Berlin, 2nd edition edition, 2006.
T. Bohman. The triangle-free process. Adv. Math., 221:1653-1677, 2009.
T. Bohman and P. Keevash. The early evolution of the h-free process. Invent. Math., 181:291-336, 2010.
B. Bukh and M. Matoušek. Erdős-Szekeres-type statements: Ramsey function and decidability in dimension 1. Duke Math. Journal, 63:2243-2270, 2014.
B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. A singly exponential stratification scheme for real semi-algebraic varieties and its applications. Theor. Comput. Sci., 84:77-105, 1991.
K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, ii. Discrete Comput. Geom., 4:387-421, 1989.
D. Conlon, J. Fox, J. Pach, B. Sudakov, and A. Suk. Ramsey-type results for semi-algebraic relations. Trans. Amer. Math. Soc., 366:5043-5065, 2014.
D. Conlon, J. Fox, and B. Sudakov. Hypergraph ramsey numbers. J. Amer. Math. Soc., 23:247-266, 2010.
V. Dujmović and S. Langerman. A center transversal theorem for hyperplanes and applications to graph drawing. In In Proc. 27th Ann. ACM Sympos. Comput. Geom., pages 117-124, 2011.
M. Eliáš, J. Matoušek, E. Roldán-Pensado, and Z. Safernová. Lower bounds on geometric Ramsey functions. SIAM J. Discrete Math, 28:1960-1970, 2014.
M. Eliáš and J. Matoušek. Higher-order Erdős-Szekeres theorems. Advances in Mathematics, 244:1-15, 2013.
P. Erdős. Some remarks on the theory of graphs. Bull. Amer. Math. Soc., 53:292-294, 1947.
P. Erdős, A. Hajnal, and R. Rado. Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hungar., 16:93-196, 1965.
P. Erdős and R. Rado. Combinatorial theorems on classifications of subsets of a given set. Proc. London Math. Soc., 3:417-439, 1952.
P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compos. Math., 2:463-470, 1935.
G. Exoo. A Lower Bound for Schur numbers and multicolor Ramsey numbers of K₃. Electronic J. Combinatorics, 1:1-3, 1994.
J. Fox, J. Pach, B. Sudakov, and A. Suk. Erdős-Szekeres-type theorems for monotone paths and convex bodies. Proceedings of the London Mathematical Society, 105:953-982, 2012.
H. Fredricksen and M. Sweet. Symmetric sum-free partitions and lower bounds for schur numbers. Electronic J. Combinatorics, 7:1-9, 2000.
J. H. Kim. The ramsey number r(3,t) has order of magnitude t²/log t. Random Structures Algorithms, 7:173-207, 1995.
V. Koltun. Almost tight upper bounds for vertical decompositions in four dimensions. J. ACM, 51:699-730, 2004.
J. Matoušek. Efficient partition trees. Discrete Comput. Geom., 8:315-334, 1992.
J. Matoušek and E. Welzl. Good splitters for counting points in triangles. J. Algorithms, 13:307-319, 1992.
J. Milnor. On the betti numbers of real varieties. Proc. Amer. Math. Soc., 15:275-280, 1964.
D. Mubayi and A. Suk. A ramsey-type result for geometric 𝓁-hypergraphs. European Journal of Combinatorics, 41:232-241, 2014.
I. Schur. Über die Kongruenz x^m + y^m = z^mmod p. Jahresber. Deutch. Math. Verein., 25:114-117, 1916.
M. J. Steele. Variations on the monotone subsequence theme of Erdős and Szekeres. In D. Aldous, editor, Discrete Probability and Algorithms, IMA Volumes in Mathematics and its Applications, pages 111-131, Berlin, 1995. Springer.
A. Suk. A note on order-type homogeneous point sets. Mathematika, 60:37-42, 2014.
R. Thom. Sur l'homologie des variétés algébriques réelles. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pages 255-265, Princeton, N.J., 1965. Princeton University.
H. E. Warren. Lower bounds for approximation by nonlinear manifold. Trans. Amer. Math. Soc., 133:167-178, 1968.
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A Short Proof of a Near-Optimal Cardinality Estimate for the Product of a Sum Set
In this note it is established that, for any finite set A of real numbers, there exist two elements a, b from A such that |(a + A)(b + A)| > c|A|^2 / log |A|, where c is some positive constant. In particular, it follows that |(A + A)(A + A)| > c|A|^2 / log |A|. The latter inequality had in fact already been established in an earlier work of the author and Rudnev, which built upon the recent developments of Guth and Katz in their work on the Erdös distinct distance problem. Here, we do not use those relatively deep methods, and instead we need just a single application of the Szemerédi-Trotter Theorem. The result is also qualitatively stronger than the corresponding sum-product estimate from the paper of the author and Rudnev, since the set (a + A)(b + A) is defined by only two variables, rather than four. One can view this as a solution for the pinned distance problem, under an alternative notion of distance, in the special case when the point set is a direct product A x A. Another advantage of this more elementary approach is that these results can now be extended for the first time to the case when A is a set of complex numbers.
Szemerédi-Trotter Theorem
pinned distances
sum-product estimates
74-80
Regular Paper
Oliver
Roche-Newton
Oliver Roche-Newton
10.4230/LIPIcs.SOCG.2015.74
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A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes
We derive tight expressions for the maximum number of k-faces, k=0,...,d-1, of the Minkowski sum, P_1+...+P_r, of r convex d-polytopes P_1,...,P_r in R^d, where d >= 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f- and h-vector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the f-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P_1+...+P_r as a section of the Cayley polytope C of the summands; bounding the k-faces of P_1+...+P_r reduces to bounding the subset of the (k+r-1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.
Convex polytopes
Minkowski sum
upper bound
81-95
Regular Paper
Menelaos I.
Karavelas
Menelaos I. Karavelas
Eleni
Tzanaki
Eleni Tzanaki
10.4230/LIPIcs.SOCG.2015.81
Karim A. Adiprasito and Raman Sanyal. Relative Stanley-Reisner theory and Upper Bound Theorems for Minkowski sums, 2014. URL: http://arxiv.org/abs/1405.7368v3.
http://arxiv.org/abs/1405.7368v3
G. Ewald and G. C. Shephard. Stellar Subdivisions of Boundary Complexes of Convex Polytopes. Mathematische Annalen, 210:7-16, 1974.
Günter Ewald. Combinatorial Convexity and Algebraic Geometry. Graduate Texts in Mathematics. Springer, 1996.
Efi Fogel, Dan Halperin, and Christophe Weibel. On the Exact Maximum Complexity of Minkowski Sums of Polytopes. Discrete Comput. Geom., 42:654-669, 2009.
Komei Fukuda and Christophe Weibel. f-vectors of Minkowski Additions of Convex Polytopes. Discrete Comput. Geom., 37(4):503-516, 2007.
R. L. Graham, M. Grotschel, and L. Lovasz. Handbook of Combinatorics, volume 2. MIT Press, North Holland, 1995.
Peter Gritzmann and Bernd Sturmfels. Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner bases. SIAM J. Disc. Math., 6(2):246-269, 1993.
Birkett Huber, Jörg Rambau, and Francisco Santos. The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc., 2(2):179-198, 2000.
Menelaos I. Karavelas, Christos Konaxis, and Eleni Tzanaki. The maximum number of faces of the Minkowski sum of three convex polytopes. J. Comput. Geom., 6(1):21-74, 2015.
Menelaos I. Karavelas and Eleni Tzanaki. The maximum number of faces of the Minkowski sum of two convex polytopes. In Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA'12), pages 11-28, 2012.
Menelaos I. Karavelas and Eleni Tzanaki. A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes, 2015. URL: http://arxiv.org/abs/1502.02265v2.
http://arxiv.org/abs/1502.02265v2
Jiří Matoušek. Lectures on Discrete Geometry. Graduate Texts in Mathematics. Springer-Verlag New York, Inc., New York, 2002.
B. Matschke, J. Pfeifle, and V. Pilaud. Prodsimplicial-neighborly polytopes. Discrete Comput. Geom., 46(1):100-131, 2011.
P. McMullen. The maximum numbers of faces of a convex polytope. Mathematika, 17:179-184, 1970.
Raman Sanyal. Topological obstructions for vertex numbers of Minkowski sums. J. Comb. Theory, Ser. A, 116(1):168-179, 2009.
Christophe Weibel. Maximal f-vectors of Minkowski Sums of Large Numbers of Polytopes. Discrete Comput. Geom., 47(3):519-537, 2012.
Günter M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.
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Two Proofs for Shallow Packings
We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A delta-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v in W is greater than delta, where delta > 0 is an integer parameter. The delta-packing number is then defined as the cardinality of the largest delta-separated subcollection of V. Haussler showed an asymptotically tight bound of Theta((n / delta)^d) on the delta-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X' of X of size m <= n and for any parameter 1 <= k <= m, the number of vectors of length at most k in the restriction of V to X' is only O(m^{d_1} k^{d-d_1}), for a fixed integer d > 0 and a real parameter 1 <= d_1 <= d (this generalizes the standard notion of bounded primal shatter dimension when d_1 = d). In this case when V is "k-shallow" (all vector lengths are at most k), we show that its delta-packing number is O(n^{d_1} k^{d-d_1} / delta^d), matching Haussler's bound for the special cases where d_1=d or k=n. We present two proofs, the first is an extension of Haussler's approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler's proof.
Set systems of bounded primal shatter dimension
delta-packing & Haussler’s approach
relative approximations
Clarkson-Shor random sampling approach
96-110
Regular Paper
Kunal
Dutta
Kunal Dutta
Esther
Ezra
Esther Ezra
Arijit
Ghosh
Arijit Ghosh
10.4230/LIPIcs.SOCG.2015.96
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Shortest Path in a Polygon using Sublinear Space
We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon P with n vertices in a read only memory, and additional working memory of size m, the new algorithm computes the shortest path (in P) in O(n^2 / m) expected time, assuming m = O(n / log^2 n). This requires several new tools, which we believe to be of independent interest.
Specifically, we show that violator space problems, an abstraction of low dimensional linear-programming (and LP-type problems), can be solved using constant space and expected linear time, by modifying Seidel's linear programming algorithm and using pseudo-random sequences.
Shortest path
violator spaces
limited space
111-125
Regular Paper
Sariel
Har-Peled
Sariel Har-Peled
10.4230/LIPIcs.SOCG.2015.111
T. Asano, K. Buchin, M. Buchin, M. Korman, W. Mulzer, G. Rote, and A. Schulz. Memory-constrained algorithms for simple polygons. Comput. Geom. Theory Appl., 46(8):959-969, 2013.
T. Asano, K. Buchin, M. Buchin, M. Korman, W. Mulzer, G. Rote, and A. Schulz. Reprint of: Memory-constrained algorithms for simple polygons. Comput. Geom. Theory Appl., 47(3):469-479, 2014.
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.
Y. Brise and B. Gärtner. Clarkson’s algorithm for violator spaces. Comput. Geom. Theory Appl., 44(2):70-81, 2011.
B. Chazelle, D. Liu, and A. Magen. Sublinear geometric algorithms. SIAM J. Comput., 35(3):627-646, 2005.
B. Chazelle and J. Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms, 21:579-597, 1996.
K. L. Clarkson. Las Vegas algorithms for linear and integer programming. J. Assoc. Comput. Mach., 42:488-499, 1995.
K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4:387-421, 1989.
S. J. Fortune. A sweepline algorithm for Voronoi diagrams. Algorithmica, 2:153-174, 1987.
B. Gärtner, J. Matoušek, L. Rüst, and P. Šavroň. Violator spaces: Structure and algorithms. In Proc. 14th Annu. European Sympos. Algorithms\CNFESA, pages 387-398, 2006.
B. Gärtner, J. Matoušek, L. Rüst, and P. Šavroň. Violator spaces: Structure and algorithms. Discrete Appl. Math., 156(11):2124-2141, 2008.
L. J. Guibas and J. Hershberger. Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci., 39(2):126-152, October 1989.
S. Har-Peled. Geometric Approximation Algorithms, volume 173 of Mathematical Surveys and Monographs. Amer. Math. Soc., Boston, MA, USA, 2011.
S. Har-Peled. Quasi-polynomial time approximation scheme for sparse subsets of polygons. In Proc. 30th Annu. Sympos. Comput. Geom.\CNFSoCG, pages 120-129, 2014.
S. Har-Peled. Shortest path in a polygon using sublinear space. CoRR, abs/1412.0779, 2014.
P. Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. Assoc. Comput. Mach., 53(3):307-323, 2006.
D. T. Lee and F. P. Preparata. Euclidean shortest paths in the presence of rectilinear barriers. Networks, 14:393-410, 1984.
N. Megiddo. Linear programming in linear time when the dimension is fixed. J. Assoc. Comput. Mach., 31:114-127, 1984.
K. Mulmuley. Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, Englewood Cliffs, NJ, 1994.
L. Y. Rüst. The P-Matrix Linear Complementarity Problem - Generalizations and Specializations. PhD thesis, ETH, 2007. Diss. ETH No. 17387.
P. Šavroň. Abstract models of optimization problems. PhD thesis, Charles University, 2007. URL: http://kam.mff.cuni.cz/~xofon/thesis/diplomka.pdf.
http://kam.mff.cuni.cz/~xofon/thesis/diplomka.pdf
R. Seidel. Small-dimensional linear programming and convex hulls made easy. Discrete Comput. Geom., 6:423-434, 1991.
M. Sharir and E. Welzl. A combinatorial bound for linear programming and related problems. In Proc. 9th Sympos. Theoret. Aspects Comput. Sci., volume 577 of Lect. Notes in Comp. Sci., pages 569-579, London, UK, 1992. Springer-Verlag.
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Optimal Morphs of Convex Drawings
We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.
Convex Drawings
Planar Graphs
Morphing
Geometric Representations
126-140
Regular Paper
Patrizio
Angelini
Patrizio Angelini
Giordano
Da Lozzo
Giordano Da Lozzo
Fabrizio
Frati
Fabrizio Frati
Anna
Lubiw
Anna Lubiw
Maurizio
Patrignani
Maurizio Patrignani
Vincenzo
Roselli
Vincenzo Roselli
10.4230/LIPIcs.SOCG.2015.126
S. Alamdari, P. Angelini, T. M. Chan, G. Di Battista, F. Frati, A. Lubiw, M. Patrignani, V. Roselli, S. Singla, and B. T. Wilkinson. Morphing planar graph drawings with a polynomial number of steps. In SODA, pages 1656-1667, 2013.
G. Aloupis, L. Barba, P. Carmi, V. Dujmovic, F. Frati, and P. Morin. Compatible connectivity-augmentation of planar disconnected graphs. In SODA, pages 1602-1615, 2015.
P. Angelini, G. Da Lozzo, G. Di Battista, F. Frati, M. Patrignani, and V. Roselli. Morphing planar graph drawings optimally. In ICALP, volume 8572 of LNCS, pages 126-137, 2014.
P. Angelini, F. Frati, M. Patrignani, and V. Roselli. Morphing planar graph drawings efficiently. In GD, volume 8242 of LNCS, pages 49-60, 2013.
I. Bárány and G. Rote. Strictly convex drawings of planar graphs. Documenta Mathematica, 11:369-391, 2006.
D. Barnette and B. Grünbaum. On Steinitz’s theorem concerning convex 3-polytopes and on some properties of planar graphs. In Many Facets of Graph Theory, volume 110 of Lecture Notes in Mathematics, pages 27-40. Springer, 1969.
F. Barrera-Cruz, P. Haxell, and A. Lubiw. Morphing planar graph drawings with unidirectional moves. Mexican Conference on Discr. Math. and Comput. Geom., 2013.
N. Bonichon, S. Felsner, and M. Mosbah. Convex drawings of 3-connected plane graphs. Algorithmica, 47(4):399-420, 2007.
S. Cairns. Deformations of plane rectilinear complexes. Am. Math. Mon., 51:247-252, 1944.
N. Chiba, T. Yamanouchi, and T. Nishizeki. Linear algorithms for convex drawings of planar graphs. In Progress in Graph Theory, pages 153-173. Academic Press, New York, NY, 1984.
M. Chrobak and G. Kant. Convex grid drawings of 3-connected planar graphs. Int. J. Comput. Geometry Appl., 7(3):211-223, 1997.
C. Erten, S. G. Kobourov, and C. Pitta. Intersection-free morphing of planar graphs. In GD, volume 2912 of LNCS, pages 320-331, 2004.
C. Friedrich and P. Eades. Graph drawing in motion. J. Graph Alg. Ap., 6:353-370, 2002.
C. Gotsman and V. Surazhsky. Guaranteed intersection-free polygon morphing. Computers & Graphics, 25(1):67-75, 2001.
B. Grunbaum and G.C. Shephard. The geometry of planar graphs. Camb. Univ. Pr., 1981.
S. H. Hong and H. Nagamochi. Convex drawings of hierarchical planar graphs and clustered planar graphs. J. Discrete Algorithms, 8(3):282-295, 2010.
S. H. Hong and H. Nagamochi. A linear-time algorithm for symmetric convex drawings of internally triconnected plane graphs. Algorithmica, 58(2):433-460, 2010.
M. S. Rahman, S. I. Nakano, and T. Nishizeki. Rectangular grid drawings of plane graphs. Comput. Geom., 10(3):203-220, 1998.
M. S. Rahman, T. Nishizeki, and S. Ghosh. Rectangular drawings of planar graphs. J. of Algorithms, 50:62-78, 2004.
J. M. Schmidt. Contractions, removals, and certifying 3-connectivity in linear time. SIAM J. Comput., 42(2):494-535, 2013.
V. Surazhsky and C. Gotsman. Controllable morphing of compatible planar triangulations. ACM Trans. Graph, 20(4):203-231, 2001.
V. Surazhsky and C. Gotsman. Intrinsic morphing of compatible triangulations. Internat. J. of Shape Model., 9:191-201, 2003.
C. Thomassen. Planarity and duality of finite and infinite graphs. J. Comb. Theory, Ser. B, 29(2):244-271, 1980.
C. Thomassen. Deformations of plane graphs. J. Comb. Th. Ser. B, 34(3):244-257, 1983.
C. Thomassen. Plane representations of graphs. In J. A. Bondy and U. S. R. Murty, editors, Progress in Graph Theory, pages 43-69. Academic Press, New York, NY, 1984.
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1-String B_2-VPG Representation of Planar Graphs
In this paper, we prove that every planar graph has a 1-string B_2-VPG representation - a string representation using paths in a rectangular grid that contain at most two bends. Furthermore, two paths representing vertices u, v intersect precisely once whenever there is an edge between u and v.
Graph drawing
string graphs
VPG graphs
planar graphs
141-155
Regular Paper
Therese
Biedl
Therese Biedl
Martin
Derka
Martin Derka
10.4230/LIPIcs.SOCG.2015.141
Takao Asano, Shunji Kikuchi, and Nobuji Saito. A linear algorithm for finding Hamiltonian cycles in 4-connected maximal planar graphs. Discr. Applied Mathematics, 7(1):1 - 15, 1984.
Therese C. Biedl and Martin Derka. 1-string B₂-VPG representation of planar graphs. CoRR, abs/1411.7277, 2014.
Jérémie Chalopin and Daniel Gonçalves. Every planar graph is the intersection graph of segments in the plane: extended abstract. In ACM Symposium on Theory of Computing, STOC 2009, pages 631-638. ACM, 2009.
Jérémie Chalopin, Daniel Gonçalves, and Pascal Ochem. Planar graphs are in 1-string. In ACM-SIAM Symposium on Discrete Algorithms, SODA '07, pages 609-617. SIAM, 2007.
Jérémie Chalopin, Daniel Gonçalves, and Pascal Ochem. Planar graphs have 1-string representations. Discrete & Computational Geometry, 43(3):626-647, 2010.
Steven Chaplick and Torsten Ueckerdt. Planar graphs as VPG-graphs. J. Graph Algorithms Appl., 17(4):475-494, 2013.
Natalia de Castro, Francisco Javier Cobos, Juan Carlos Dana, Alberto Márquez, and Marc Noy. Triangle-free planar graphs and segment intersection graphs. J. Graph Algorithms Appl., 6(1):7-26, 2002.
Hubert de Fraysseix, Patrice Ossona de Mendez, and János Pach. Representation of planar graphs by segments. Intuitive Geometry, 63:109-117, 1991.
Gideon Ehrlich, Shimon Even, and Robert Endre Tarjan. Intersection graphs of curves in the plane. J. Comb. Theory, Ser. B, 21(1):8-20, 1976.
Stefan Felsner, Kolja B. Knauer, George B. Mertzios, and Torsten Ueckerdt. Intersection graphs of L-shapes and segments in the plane. In Mathematical Foundations of Computer Science (MFCS'14), Part II, volume 8635 of Lecture Notes in Computer Science, pages 299-310. Springer, 2014.
Irith Ben-Arroyo Hartman, Ilan Newman, and Ran Ziv. On grid intersection graphs. Discrete Mathematics, 87(1):41-52, 1991.
Edward R. Scheinerman. Intersection Classes and Multiple Intersection Parameters of Graphs. PhD thesis, Princeton University, 1984.
Hassler Whitney. A theorem on graphs. The Annals of Mathematics, 32(2):387-390, 1931.
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Spanners and Reachability Oracles for Directed Transmission Graphs
Let P be a set of n points in d dimensions, each with an associated radius r_p > 0. The transmission graph G for P has vertex set P and an edge from p to q if and only if q lies in the ball with radius r_p around p. Let t > 1. A t-spanner H for G is a sparse subgraph of G such that for any two vertices p, q connected by a path of length l in G, there is a p-q-path of length at most tl in H. We show how to compute a t-spanner for G if d=2. The running time is O(n (log n + log Psi)), where Psi is the ratio of the largest and smallest radius of two points in P. We extend this construction to be independent of Psi at the expense of a polylogarithmic overhead in the running time. As a first application, we prove a property of the t-spanner that allows us to find a BFS tree in G for any given start vertex s of P in the same time.
After that, we deal with reachability oracles for G. These are data structures that answer reachability queries: given two vertices, is there a directed path between them? The quality of a reachability oracle is measured by the space S(n), the query time Q(n), and the preproccesing time. For d=1, we show how to compute an oracle with Q(n) = O(1) and S(n) = O(n) in time O(n log n). For d=2, the radius ratio Psi again turns out to be an important measure for the complexity of the problem. We present three different data structures whose quality depends on Psi: (i) if Psi < sqrt(3), we achieve Q(n) = O(1) with S(n) = O(n) and preproccesing time O(n log n); (ii) if Psi >= sqrt(3), we get Q(n) = O(Psi^3 sqrt(n)) and S(n) = O(Psi^5 n^(3/2)); and (iii) if Psi is polynomially bounded in n, we use probabilistic methods to obtain an oracle with Q(n) = O(n^(2/3)log n) and S(n) = O(n^(5/3) log n) that answers queries correctly with high probability. We employ our t-spanner to achieve a fast preproccesing time of O(Psi^5 n^(3/2)) and O(n^(5/3) log^2 n) in case (ii) and (iii), respectively.
Transmission Graphs
Reachability Oracles
Spanner
Intersection Graph
156-170
Regular Paper
Haim
Kaplan
Haim Kaplan
Wolfgang
Mulzer
Wolfgang Mulzer
Liam
Roditty
Liam Roditty
Paul
Seiferth
Paul Seiferth
10.4230/LIPIcs.SOCG.2015.156
Jochen Alber and Jirí Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. J. Algorithms, 52(2):134-151, 2004.
Azzedine Boukerche. Algorithms and Protocols for Wireless Sensor Networks. Wiley Series on Parallel and Distributed Computing). Wiley-IEEE Press, 1st edition, 2008.
Sergio Cabello and Miha Jejĉiĉ. Shortest paths in intersection graphs of unit disks. Comput. Geom., 48(4):360-367, 2015.
Paul Callahan and 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.
Paz Carmi, 2014. Personal communication.
Timothy M. Chan. A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries. J. ACM, 57(3):Art. 16, 15, 2010.
Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Math., 86(1-3):165-177, 1990.
Thomas Cormen, Charles Leiserson, Ronald Rivest, and Clifford Stein. Introduction to Algorithms. MIT Press, 2nd edition, 2001.
Martin Fürer and Shiva Prasad Kasiviswanathan. Spanners for geometric intersection graphs with applications. J. Comput. Geom., 3(1):31-64, 2012.
Sariel Har-Peled. Geometric Approximation Algorithms. AMS, 2011.
Jacob Holm, Eva Rotenberg, and Mikkel Thorup. Planar Reachability in Linear Space and Constant Time. CoRR, arXiv:1411.5867, 2014.
Hiroshi Imai, Masao Iri, and Kazuo Murota. Voronoi Diagram in the Laguerre Geometry and its Applications. SICOMP, 14(1):93-105, 1985.
D. Kirkpatrick. Optimal Search in Planar Subdivisions. SICOMP, 12(1):28-35, 1983.
G. Narasimhan and M. Smid. Geometric spanner networks. Cambridge Univ. Press, 2007.
David Peleg and Liam Roditty. Localized spanner construction for ad hoc networks with variable transmission range. TOSN, 7(3), 2010.
P. v. Rickenbach, R. Wattenhofer, and A. Zollinger. Algorithmic Models of Interference in Wireless Ad Hoc and Sensor Networks. IEEE ACM T NETWORK, 17(1):172-185, 2009.
Andrew Chi-Chih Yao. On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems. SICOMP, 11(4):721-736, 1982.
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Recognition and Complexity of Point Visibility Graphs
A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set.
We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs.
Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.
point visibility graphs
recognition
existential theory of the reals
171-185
Regular Paper
Jean
Cardinal
Jean Cardinal
Udo
Hoffmann
Udo Hoffmann
10.4230/LIPIcs.SOCG.2015.171
James Abello and Krishna Kumar. Visibility graphs and oriented matroids. Discrete & Computational Geometry, 28(4):449-465, 2002.
Karim A. Adiprasito, Arnau Padrol, and Louis Theran. Universality theorems for inscribed polytopes and Delaunay triangulations. ArXiv e-prints, 2014.
Daniel Bienstock. Some provably hard crossing number problems. Discrete and Computational Geometry, 6:443-459, 1991.
John Canny. Some algebraic and geometric computations in PSPACE. In STOC '88, pages 460-467. ACM, 1988.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2008 (third edition).
Subir K. Ghosh. On recognizing and characterizing visibility graphs of simple polygons. In 1st Scandinavian Workshop on Algorithm Theory (SWAT), pages 96-104, 1988.
Subir K. Ghosh. On recognizing and characterizing visibility graphs of simple polygons. Discrete & Computational Geometry, 17(2):143-162, 1997.
Subir K. Ghosh. Visibility Algorithms in the Plane. Cambridge University Press, 2007.
Subir K. Ghosh and Partha P. Goswami. Unsolved problems in visibility graphs of points, segments, and polygons. ACM Computing Surveys (CSUR), 46(2):22, 2013.
Subir K. Ghosh and Bodhayan Roy. Some results on point visibility graphs. In Algorithms and Computation (WALCOM), volume 8344 of Lecture Notes in Computer Science, pages 163-175. Springer, 2014.
Jacob E. Goodman, Richard Pollack, and Bernd Sturmfels. The intrinsic spread of a configuration in ℝ^d. Journal of the American Mathematical Society, pages 639-651, 1990.
Branko Grünbaum. Convex Polytopes, volume 221 (2nd ed.) of Graduate Texts in Mathematics. Springer-Verlag, 2003.
Michael Kapovich and John J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41:1051-1107, 2002.
Jan Kára, Attila Pór, and David R. Wood. On the chromatic number of the visibility graph of a set of points in the plane. Discrete & Computational Geometry, 34(3):497-506, 2005.
Jan Kratochvíl and Jirí Matoušek. Intersection graphs of segments. Journal of Combinatorial Theory. Series B, 62(2):289-315, 1994.
Jan Kynčl. Simple realizability of complete abstract topological graphs in P. Discrete and Computational Geometry, 45(3):383-399, 2011.
Tomás Lozano-Pérez and Michael A. Wesley. An algorithm for planning collision-free paths among polyhedral obstacles. Commun. ACM, 22(10):560-570, October 1979.
Jiří Matoušek. Intersection graphs of segments and ∃ℝ. ArXiv e-prints, 2014.
Colin McDiarmid and Tobias Müller. Integer realizations of disk and segment graphs. Journal of Combinatorial Theory, Series B, 103(1):114 - 143, 2013.
Nicolai E. Mnëv. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and geometry—Rohlin seminar, pages 527-543. Springer, 1988.
Joseph O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, 1987.
Joseph O'Rourke and Ileana Streinu. Vertex-edge pseudo-visibility graphs: Characterization and recognition. In Symposium on Computational Geometry, pages 119-128, 1997.
Michael S. Payne, Attila Pór, Pavel Valtr, and David R. Wood. On the connectivity of visibility graphs. Discrete & Computational Geometry, 48(3):669-681, 2012.
Attila Pór and David R. Wood. On visibility and blockers. JoCG, 1(1):29-40, 2010.
Jürgen Richter-Gebert. Mnëv’s universality theorem revisited. In Proceedings of the Séminaire Lotharingien de Combinatoire, pages 211-225, 1995.
Bodhayan Roy. Point visibility graph recognition is NP-hard. ArXiv e-prints, 2014.
Marcus Schaefer. Complexity of some geometric and topological problems. In 17th International Symposium on Graph Drawing (GD), pages 334-344, 2009.
Marcus Schaefer. Realizability of graphs and linkages. In Thirty Essays on Geometric Graph Theory. Springer, 2012.
Peter W. Shor. Stretchability of pseudolines is NP-hard. Applied Geometry and Discrete Mathematics-The Victor Klee Festschrift, 4:531-554, 1991.
Karl Georg Christian Staudt. Geometrie der Lage. Verlag von Bauer und Raspe, 1847.
Ileana Streinu. Non-stretchable pseudo-visibility graphs. Comput. Geom., 31(3):195-206, 2005.
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Geometric Spanners for Points Inside a Polygonal Domain
Let P be a set of n points inside a polygonal domain D. A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set P with respect to the geodesic distance function d where for any two points p and q, d(p,q) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., h=0), we construct a (sqrt(10)+eps)-spanner that has O(n log^2 n) edges where eps is the a given positive real number. For a case where there are h holes, our construction gives a (5+eps)-spanner with the size of O(sqrt(h) n log^2 n).
Moreover, we study t-spanners for the visibility graph of P (VG(P), for short) with respect to a hole-free polygonal domain D. The graph VG(P) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (3+eps)-spanner of size almost O(n^{4/3}). In addition, we show that there is a set P of n points such that any (3-eps)-spanner of VG(P) must contain almost n^2 edges.
Geometric Spanners
Polygonal Domain
Visibility Graph
186-197
Regular Paper
Mohammad Ali
Abam
Mohammad Ali Abam
Marjan
Adeli
Marjan Adeli
Hamid
Homapour
Hamid Homapour
Pooya Zafar
Asadollahpoor
Pooya Zafar Asadollahpoor
10.4230/LIPIcs.SOCG.2015.186
Mohammad Ali Abam, Paz Carmi, Mohammad Farshi, and Michiel Smid. On the power of the semi-separated pair decomposition. Computational Geometry, 46(6):631-639, 2013.
Mohammad Ali Abam, Mark De Berg, Mohammad Farshi, and Joachim Gudmundsson. Region-fault tolerant geometric spanners. Discrete & Computational Geometry, 41(4):556-582, 2009.
Mohammad Ali Abam, Mark De Berg, Mohammad Farshi, Joachim Gudmundsson, and Michiel Smid. Geometric spanners for weighted point sets. Algorithmica, 61(1):207-225, 2011.
Mohammad Ali Abam and Sariel Har-Peled. New constructions of sspds and their applications. Computational Geometry, 45(5):200-214, 2012.
Pankaj K Agarwal and Micha Sharir. Applications of a new space-partitioning technique. Discrete & Computational Geometry, 9(1):11-38, 1993.
Noga Alon, Paul Seymour, and Robin Thomas. Planar separators. SIAM Journal on Discrete Mathematics, 7(2):184-193, 1994.
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.
Prosenjit Bose, Jurek Czyzowicz, Evangelos Kranakis, Danny Krizanc, and Anil Maheshwari. Polygon cutting: Revisited. In Proceedings of Japanese Conference on Discrete & Computational Geometry (JCDCG'98), volume 1763 of LNCS, pages 81-92. Springer, 1998.
Paul B Callahan and S Rao Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. Journal of the ACM, 42(1):67-90, 1995.
Kenneth L Clarkson, Herbert Edelsbrunner, Leonidas J Guibas, Micha Sharir, and Emo Welzl. Combinatorial complexity bounds for arrangements of curves and spheres. Discrete & Computational Geometry, 5(1):99-160, 1990.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 2008.
Sariel Har-Peled and Manor Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM Journal on Computing, 35(5):1148-1184, 2006.
Giri Narasimhan and Michiel Smid. Geometric spanner networks. Cambridge University Press, 2007.
Kunal Talwar. Bypassing the embedding: algorithms for low dimensional metrics. In Proceedings of Annual ACM symposium on Theory of computing, pages 281-290, 2004.
Kasturi R Varadarajan. A divide-and-conquer algorithm for min-cost perfect matching in the plane. In Proceedings of Annual Symposium on Foundations of Computer Science, pages 320-331, 1998.
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An Optimal Algorithm for the Separating Common Tangents of Two Polygons
We describe an algorithm for computing the separating common tangents of two simple polygons using linear time and only constant workspace. A tangent of a polygon is a line touching the polygon such that all of the polygon lies to the same side of the line. A separating common tangent of two polygons is a tangent of both polygons where the polygons are lying on different sides of the tangent. Each polygon is given as a read-only array of its corners. If a separating common tangent does not exist, the algorithm reports that. Otherwise, two corners defining a separating common tangent are returned. The algorithm is simple and implies an optimal algorithm for deciding if the convex hulls of two polygons are disjoint or not. This was not known to be possible in linear time and constant workspace prior to this paper.
An outer common tangent is a tangent of both polygons where the polygons are on the same side of the tangent. In the case where the convex hulls of the polygons are disjoint, we give an algorithm for computing the outer common tangents in linear time using constant workspace.
planar computational geometry
simple polygon
common tangent
optimal algorithm
constant workspace
198-208
Regular Paper
Mikkel
Abrahamsen
Mikkel Abrahamsen
10.4230/LIPIcs.SOCG.2015.198
M. Abrahamsen. An optimal algorithm computing edge-to-edge visibility in a simple polygon. In Proceedings of the 25th Canadian Conference on Computational Geometry, CCCG, pages 157-162, 2013.
T. Asano, K. Buchin, M. Buchin, M. Korman, W. Mulzer, G. Rote, and A. Schulz. Memory-constrained algorithms for simple polygons. Computational Geometry: Theory and Applications, 46(8):959-969, 2013.
T. Asano, W. Mulzer, G. Rote, and Y. Wang. Constant-work-space algorithms for geometric problems. Journal of Computational Geometry, 2(1):46-68, 2011.
L. Barba, M. Korman, S. Langerman, and R.I. Silveira. Computing the visibility polygon using few variables. In Proceedings of the 22nd International Symposium on Algorithms and Computation, ISAAC, volume 7014 of Lecture Notes in Computer Science, pages 70-79. Springer, 2011.
G.S. Brodal and R. Jacob. Dynamic planar convex hull. In Proceedings of the 43rd annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 617-626, 2002.
R.L. Graham and F.F. Yao. Finding the convex hull of a simple polygon. Journal of Algorithms, 4(4):324-331, 1983.
Leonidas Guibas, John Hershberger, and Jack Snoeyink. Compact interval trees: A data structure for convex hulls. International Journal of Computational Geometry & Applications, 1(1):1-22, 1991.
J. Hershberger and S. Suri. Applications of a semi-dynamic convex hull algorithm. BIT Numerical Mathematics, 32(2):249-267, 1992.
D. Kirkpatrick and J. Snoeyink. Computing common tangents without a separating line. In Proceedings of the 4th International Workshop on Algorithms and Data Structures, WADS, volume 955 of Lecture Notes in Computer Science, pages 183-193. Springer, 1995.
A.A. Melkman. On-line construction of the convex hull of a simple polyline. Information Processing Letters, 25(1):11-12, 1987.
M.H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23(2):166-204, 1981.
F.P. Preparata and S.J. Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 20(2):87-93, 1977.
G.T. Toussaint. Solving geometric problems with the rotating calipers. In Proceedings of the IEEE Mediterranean Electrotechnical Conference, MELECON, pages A10.02/1-4, 1983.
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A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon
Let P be a closed simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P. In 1989, Pollack, Sharir and Rote [Disc. & Comput. Geom. 89] showed an O(n log n)-time algorithm that computes the geodesic center of P. Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.
Geodesic distance
facility location
1-center problem
simple polygons
209-223
Regular Paper
Hee Kap
Ahn
Hee Kap Ahn
Luis
Barba
Luis Barba
Prosenjit
Bose
Prosenjit Bose
Jean-Lou
De Carufel
Jean-Lou De Carufel
Matias
Korman
Matias Korman
Eunjin
Oh
Eunjin Oh
10.4230/LIPIcs.SOCG.2015.209
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. CoRR, abs/1501.00561, 2015.
Boris Aronov, Steven Fortune, and Gordon Wilfong. The furthest-site geodesic Voronoi diagram. Discrete & Computational Geometry, 9(1):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.
Sang Won Bae, Matias Korman, and Yoshio Okamoto. The geodesic diameter of polygonal domains. Discrete & Computational Geometry, 50(2):306-329, 2013.
Sang Won Bae, Matias Korman, and Yoshio Okamoto. Computing the geodesic centers of a polygonal domain. In Proceedings of CCCG, 2014.
Sang Won Bae, Matias Korman, Yoshio Okamoto, and Haitao Wang. Computing the L₁ geodesic diameter and center of a simple polygon in linear time. In Proceedings of LATIN, pages 120-131, 2014.
Bernard Chazelle. A theorem on polygon cutting with applications. In Proceedings of FOCS, pages 339-349, 1982.
Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete & Computational Geometry, 6(1):485-524, 1991.
H.N. Djidjev, A. Lingas, and J.-R. Sack. An O(nlog n) algorithm for computing the link center of a simple polygon. Discrete & Computational Geometry, 8:131-152, 1992.
Herbert Edelsbrunner and Ernst Peter Mücke. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. on Graphics, 9(1):66-104, 1990.
Leonidas Guibas, John Hershberger, Daniel Leven, Micha Sharir, and Robert E Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2(1-4):209-233, 1987.
Leonidas J Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of computer and system sciences, 39(2):126-152, 1989.
Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. SIAM Journal on Computing, 13(2):338-355, 1984.
John Hershberger and Subhash Suri. Matrix searching with the shortest-path metric. SIAM Journal on Computing, 26(6):1612-1634, 1997.
Y. Ke. An efficient algorithm for link-distance problems. In Proceedings of SoCG, pages 69-78, 1989.
Der-Tsai Lee and Franco P Preparata. Euclidean shortest paths in the presence of rectilinear barriers. Networks, 14(3):393-410, 1984.
Jiří Matoušek. Approximations and optimal geometric divide-and-conquer. Journal of Computer and System Sciences, 50(2):203-208, 1995.
Jiří Matoušek. Construction of epsilon nets. In Proc. of SoCG, pages 1-10. ACM, 1989.
Nimrod Megiddo. On the ball spanned by balls. Discrete & Computational Geometry, 4(1):605-610, 1989.
J. S. B. Mitchell. Geometric shortest paths and network optimization. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 633-701. Elsevier, 2000.
B.J. Nilsson and S. Schuierer. Computing the rectilinear link diameter of a polygon. In Proceedings of CG, pages 203-215, 1991.
B.J. Nilsson and S. Schuierer. An optimal algorithm for the rectilinear link center of a rectilinear polygon. Computational Geometry: Theory and Applications, 6:169-194, 1996.
Richard Pollack, Micha Sharir, and Günter Rote. Computing the geodesic center of a simple polygon. Discrete & Computational Geometry, 4(1):611-626, 1989.
S. Suri. Minimum Link Paths in Polygons and Related Problems. PhD thesis, Johns Hopkins Univ., 1987.
Subhash Suri. Computing geodesic furthest neighbors in simple polygons. Journal of Computer and System Sciences, 39(2):220-235, 1989.
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On the Smoothed Complexity of Convex Hulls
We establish an upper bound on the smoothed complexity of convex hulls in R^d under uniform Euclidean (L^2) noise. Specifically, let {p_1^*, p_2^*, ..., p_n^*} be an arbitrary set of n points in the unit ball in R^d and let p_i = p_i^* + x_i, where x_1, x_2, ..., x_n are chosen independently from the unit ball of radius r. We show that the expected complexity, measured as the number of faces of all dimensions, of the convex hull of {p_1, p_2, ..., p_n} is O(n^{2-4/(d+1)} (1+1/r)^{d-1}); the magnitude r of the noise may vary with n. For d=2 this bound improves to O(n^{2/3} (1+r^{-2/3})).
We also analyze the expected complexity of the convex hull of L^2 and Gaussian perturbations of a nice sample of a sphere, giving a lower-bound for the smoothed complexity. We identify the different regimes in terms of the scale, as a function of n, and show that as the magnitude of the noise increases, that complexity varies monotonically for Gaussian noise but non-monotonically for L^2 noise.
Probabilistic analysis
Worst-case analysis
Gaussian noise
224-239
Regular Paper
Olivier
Devillers
Olivier Devillers
Marc
Glisse
Marc Glisse
Xavier
Goaoc
Xavier Goaoc
Rémy
Thomasse
Rémy Thomasse
10.4230/LIPIcs.SOCG.2015.224
Robert M. Corless, Gaston H. Gonnet, D.E.G. Hare, David J. Jeffrey, and Donald E. Knuth. On the Lambert W function. Advances in Computational Mathematics, 5(1):329-359, 1996. http://dx.doi.org/10.1007/BF02124750.
Valentina Damerow and Christian Sohler. Extreme points under random noise. In Proc. 12th European Sympos. Algorithms, pages 264-274, 2004. http://dx.doi.org/10.1007/978-3-540-30140-0_25.
Mark de Berg, Herman Haverkort, and Constantinos P. Tsirogiannis. Visibility maps of realistic terrains have linear smoothed complexity. Journal of Computational Geometry, 1:57-71, 2010. http://jocg.org/index.php/jocg/article/view/12.
Olivier Devillers, Marc Glisse, and Xavier Goaoc. Complexity analysis of random geometric structures made simpler. In Symposium on Computational Geometry, pages 167-176, 2013. http://dx.doi.org/10.1145/2462356.2462362.
Marc Glisse, Sylvain Lazard, Julien Michel, and Marc Pouget. Silhouette of a random polytope. Research Report 8327, INRIA, 2013. http://hal.inria.fr/hal-00841374/.
Matthias Reitzner. Random polytopes. In New perspectives in stochastic geometry, pages 45-76. Oxford Univ. Press, Oxford, 2010.
Alfréd Rényi and Rolf Sulanke. Über die konvexe Hülle von n zufällig gewählten Punkten I. Z. Wahrsch. Verw. Gebiete, 2:75-84, 1963. http://dx.doi.org/10.1007/BF00535300.
Alfréd Rényi and Rolf Sulanke. Über die konvexe Hülle von n zufällig gewählten Punkten II. Z. Wahrsch. Verw. Gebiete, 3:138-147, 1964. http://dx.doi.org/10.1007/BF00535973.
Daniel A. Spielman and Shang-Hua Teng. Smoothed analysis: Why the simplex algorithm usually takes polynomial time. Journal of the ACM, 51:385-463, 2004. http://dx.doi.org/10.1145/990308.990310.
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Finding All Maximal Subsequences with Hereditary Properties
Consider a sequence s_1,...,s_n of points in the plane. We want to find all maximal subsequences with a given hereditary property P: find for all indices i the largest index j^*(i) such that s_i,...,s_{j^*(i)} has property P. We provide a general methodology that leads to the following specific results:
- In O(n log^2 n) time we can find all maximal subsequences with diameter at most 1.
- In O(n log n loglog n) time we can find all maximal subsequences whose convex hull has area at most 1.
- In O(n) time we can find all maximal subsequences that define monotone paths in some (subpath-dependent) direction.
The same methodology works for graph planarity, as follows. Consider a sequence of edges e_1,...,e_n over a vertex set V. In O(n log n) time we can find, for all indices i, the largest index j^*(i) such that (V,{e_i,..., e_{j^*(i)}}) is planar.
convex hull
diameter
monotone path
sequence of points
trajectory
240-254
Regular Paper
Drago
Bokal
Drago Bokal
Sergio
Cabello
Sergio Cabello
David
Eppstein
David Eppstein
10.4230/LIPIcs.SOCG.2015.240
Boris Aronov, Anne Driemel, Marc J. van Kreveld, Maarten Löffler, and Frank Staals. Segmentation of trajectories for non-monotone criteria. In SODA 2013, pages 1897-1911, 2013.
Michael J. Bannister, William E. Devanny, Michael T. Goodrich, and Joe Simons. Windows into geometric events. In CCCG 2014, 2014.
Michael J. Bannister, Christopher DuBois, David Eppstein, and Padhraic Smyth. Windows into relational events: Data structures for contiguous subsequences of edges. In SODA 2013, pages 856-864, 2013.
Kevin Buchin, Maike Buchin, Marc van Kreveld, Maarten Löffler, Rodrigo I. Silveira, Carola Wenk, and Lionov Wiratma. Median trajectories. Algorithmica, 66(3):595-614, 2013.
Kevin Buchin, Maike Buchin, Marc van Kreveld, and Jun Luo. Finding long and similar parts of trajectories. Comput. Geom., 44(9):465-476, 2011.
Maike Buchin, Anne Driemel, Marc J. van Kreveld, and Vera Sacristan. Segmenting trajectories: A framework and algorithms using spatiotemporal criteria. J. Spatial Information Science, 3(1):33-63, 2011.
Timothy M. Chan. Dynamic planar convex hull operations in near-logarithmic amortized time. J. ACM, 48(1):1-12, 2001.
Timothy M. Chan. A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries. J. ACM, 57(3), 2010.
Chen Chen, Hao Su, Qixing Huang, Lin Zhang, and Leonidas Guibas. Pathlet learning for compressing and planning trajectories. In SIGSPATIAL'13, pages 392-395, 2013.
Giuseppe Di Battista and Roberto Tamassia. On-Line planarity testing. SIAM J. Comput., 25(5):956-997, 1996.
David Eppstein. Dynamic Euclidean minimum spanning trees and extrema of binary functions. Discrete Comput. Geom., 13:111-122, 1995.
David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Thomas H. Spencer. Separator based sparsification. I. Planary testing and minimum spanning trees. J. Comput. Syst. Sci., 52(1):3-27, 1996.
David Eppstein, Michael T. Goodrich, and Maarten Löffler. Tracking moving objects with few handovers. In WADS 2011, volume 6844 of LNCS, pages 362-373. Springer, 2011.
Johannes Fischer and Volker Heun. Space-efficient preprocessing schemes for range minimum queries on static arrays. SIAM J. Comput., 40(2):465-492, 2011.
Zvi Galil, Giuseppe F. Italiano, and Neil Sarnak. Fully dynamic planarity testing with applications. J. ACM, 46(1):28-91, 1999.
Joachim Gudmundsson, Jyrki Katajainen, Damian Merrick, Cahya Ong, and Thomas Wolle. Compressing spatio-temporal trajectories. Comput. Geom., 42(9):825-841, 2009.
Joachim Gudmundsson, Marc van Kreveld, and Bettina Speckmann. Efficient detection of patterns in 2D trajectories of moving points. GeoInformatica, 11(2):195-215, 2007.
John Hopcroft and Robert Tarjan. Efficient planarity testing. J. ACM, 21(4):549-568, 1974.
Jakub Ła̧cki and Piotr Sankowski. Reachability in graph timelines. In ITCS 2013, pages 257-268, 2013.
Mark H. Overmars and Jan van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23(2):166-204, 1981.
Franco P. Preparata and Michael I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985.
Peter van Emde Boas. Preserving order in a forest in less than logarithmic time and linear space. Inf. Process. Lett., 6(3):80-82, 1977.
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Riemannian Simplices and Triangulations
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of the standard simplex under this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the "thickness" or "fatness" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.
Karcher means
barycentric coordinates
triangulation
Riemannian manifold
Riemannian simplices
255-269
Regular Paper
Ramsay
Dyer
Ramsay Dyer
Gert
Vegter
Gert Vegter
Mathijs
Wintraecken
Mathijs Wintraecken
10.4230/LIPIcs.SOCG.2015.255
M. Berger. A Panoramic View of Riemannian Geometry. Springer-Verlag, 2003.
J.-D. Boissonnat, R. Dyer, and A. Ghosh. Delaunay triangulation of manifolds. Research Report RR-8389, INRIA, 2013. (also: arXiv:1311.0117).
J.-D. Boissonnat, R. Dyer, and A. Ghosh. The stability of Delaunay triangulations. IJCGA, 23(04n05):303-333, 2013. (Preprint: arXiv:1304.2947).
J.-D. Boissonnat and A. Ghosh. Manifold reconstruction using tangential Delaunay complexes. Discrete and Computational Geometry, 51(1):221-267, 2014.
P. Buser and H. Karcher. Gromov’s almost flat manifolds, volume 81 of Astérique. Société mathématique de France, 1981.
S. S. Cairns. On the triangulation of regular loci. Annals of Mathematics. Second Series, 35(3):579-587, 1934.
S.-W. Cheng, T. K. Dey, H. Edelsbrunner, M. A. Facello, and S. H Teng. Sliver exudation. Journal of the ACM, 47(5):883-904, 2000.
S.-W. Cheng, T. K. Dey, and E. A. Ramos. Manifold reconstruction from point samples. In SODA, pages 1018-1027, 2005.
R. Dyer, G. Vegter, and M. Wintraecken. Riemannian simplices and triangulations. Geometriae Dedicata, 2015. To appear. (Preprint: arXiv:1406.3740).
H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30:509-541, 1977.
F. Labelle and J. R. Shewchuk. Isosurface stuffing: Fast tetrahedral meshes with good dihedral angles. ACM Trans. Graph., 26(3), 2007.
J. R. Munkres. Elementary differential topology. Princton University press, second edition, 1968.
R.M. Rustamov. Barycentric coordinates on surfaces. Eurographics Symposium of Geometry Processing, 29(5), 2010.
O. Sander. Geodesic finite elements on simplicial grids. International Journal for Numerical Methods in Engineering, 92(12):999-1025, 2012.
W. P. Thurston. Three-Dimensional Geometry and Topology. Princeton University Press, 1997.
S. W. von Deylen. Numerische Approximation in Riemannschen Mannigfaltigkeiten mithilfe des Karcher’schen Schwerpunktes. PhD thesis, Freie Universität Berlin, 2014 (to appear).
J. H. C. Whitehead. On C¹-complexes. Annals of Mathematics, 41(4), 1940.
H. Whitney. Geometric Integration Theory. Princeton University Press, 1957.
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An Edge-Based Framework for Enumerating 3-Manifold Triangulations
A typical census of 3-manifolds contains all manifolds (under various constraints) that can be triangulated with at most n tetrahedra. Although censuses are useful resources for mathematicians, constructing them is difficult: the best algorithms to date have not gone beyond n=12. The underlying algorithms essentially (i) enumerate all relevant 4-regular multigraphs on n nodes, and then (ii) for each multigraph G they enumerate possible 3-manifold triangulations with G as their dual 1-skeleton, of which there could be exponentially many. In practice, a small number of multigraphs often dominate the running times of census algorithms: for example, in a typical census on 10 tetrahedra, almost half of the running time is spent on just 0.3% of the graphs.
Here we present a new algorithm for stage (ii), which is the computational bottleneck in this process. The key idea is to build triangulations by recursively constructing neighbourhoods of edges, in contrast to traditional algorithms which recursively glue together pairs of tetrahedron faces. We implement this algorithm, and find experimentally that whilst the overall performance is mixed, the new algorithm runs significantly faster on those "pathological" multigraphs for which existing methods are extremely slow. In this way the old and new algorithms complement one another, and together can yield significant performance improvements over either method alone.
triangulations
enumeration
graph theory
270-284
Regular Paper
Benjamin A.
Burton
Benjamin A. Burton
William
Pettersson
William Pettersson
10.4230/LIPIcs.SOCG.2015.270
Benjamin A. Burton. Face pairing graphs and 3-manifold enumeration. J. Knot Theory Ramifications, 13(8):1057-1101, 2004.
Benjamin A. Burton. Enumeration of non-orientable 3-manifolds using face-pairing graphs and union-find. Discrete & Computational Geometry, 38(3):527-571, 2007.
Benjamin A. Burton. Detecting genus in vertex links for the fast enumeration of 3-manifold triangulations. In ISSAC 2011: Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, pages 59-66. ACM, 2011.
Benjamin A. Burton. The cusped hyperbolic census is complete. arXiv:1405.2695, 2014.
Benjamin A. Burton, Ryan Budney, and William Pettersson. Regina: Software for 3-manifold topology and normal surface theory. Licensed under GPLv2, 1999-2013.
Patrick J. Callahan, Martin V. Hildebrand, and Jeffrey R. Weeks. A census of cusped hyperbolic 3-manifolds. Math. Comp., 68(225):321-332, 1999. With microfiche supplement.
Martin Hildebrand and Jeffrey Weeks. A computer generated census of cusped hyperbolic 3-manifolds. In Computers and mathematics, pages 53-59. Springer, New York, 1989.
William Jaco and J. Hyam Rubinstein. 0-efficient triangulations of 3-manifolds. J. Differential Geom., 65(1):61-168, 2003.
Bruno. Martelli and Carlo. Petronio. Three-manifolds having complexity at most 9. Experimental Mathematics, 10(2):207-236, 2001.
Bruno Martelli and Carlo Petronio. A new decomposition theorem for 3-manifolds. Illinois J. Math., 46:755-780, 2002.
Sergei V. Matveev. Computer recognition of three-manifolds. Experiment. Math., 7(2):153-161, 1998.
Sergei V. Matveev. Algorithmic topology and classification of 3-manifolds, volume 9 of Algorithms and Computation in Mathematics. Springer, Berlin, second edition, 2007.
Sergei V. Matveev et al. Manifold recognizer, 2014. URL: http://www.matlas.math.csu.ru/?page=recognizer.
http://www.matlas.math.csu.ru/?page=recognizer
Edwin E. Moise. Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung. Ann. of Math. (2), 56:96-114, 1952.
Robert Sedgewick. Algorithms in C++. Addison-Wesley, Reading, MA, 1992.
Morwen Thistlethwaite. Cusped hyperbolic manifolds with 8 tetrahedra. http:// www. math. utk. edu/ ~morwen/ 8tet/, October 2010.
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Order on Order Types
Given P and P', equally sized planar point sets in general position, we call a bijection from P to P' crossing-preserving if crossings of connecting segments in P are preserved in P' (extra crossings may occur in P'). If such a mapping exists, we say that P' crossing-dominates P, and if such a mapping exists in both directions, P and P' are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.)
We argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossing-dominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomial-time algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points.
point set
order type
planar graph
crossing-free geometric graph
285-299
Regular Paper
Alexander
Pilz
Alexander Pilz
Emo
Welzl
Emo Welzl
10.4230/LIPIcs.SOCG.2015.285
Oswin Aichholzer, Franz Aurenhammer, and Hannes Krasser. Enumerating order types for small point sets with applications. Order, 19(3):265-281, 2002.
Oswin Aichholzer, Jean Cardinal, Vincent Kusters, Stefan Langerman, and Pavel Valtr. Reconstructing point set order types from radial orderings. In ISAAC 2014, volume 8889 of LNCS, pages 15-26. Springer, 2014.
Oswin Aichholzer, Thomas Hackl, Matias Korman, Marc van Kreveld, Maarten Löffler, Alexander Pilz, Bettina Speckmann, and Emo Welzl. Packing plane spanning trees and paths in complete geometric graphs. In Proc. 26th Canadian Conference on Computational Geometry (CCCG 2014), 2014.
Oswin Aichholzer, Thomas Hackl, Birgit Vogtenhuber, Clemens Huemer, Ferran Hurtado, and Hannes Krasser. On the number of plane graphs. In Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pages 504-513. ACM Press, 2006.
Oswin Aichholzer and Hannes Krasser. Abstract order type extension and new results on the rectilinear crossing number. Comput. Geom., 36(1):2-15, 2007.
Boris Aronov, Paul Erdős, Wayne Goddard, Daniel J. Kleitman, Michael Klugerman, János Pach, and Leonard J. Schulman. Crossing families. Combinatorica, 14(2):127-134, 1994.
Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, and David R. Wood. Partitions of complete geometric graphs into plane trees. Comput. Geom., 34(2):116-125, 2006.
Jean Cardinal, Michael Hoffmann, and Vincent Kusters. On universal point sets for planar graphs. In Computational Geometry and Graphs - Thailand-Japan Joint Conference (TJJCCGG 2012), volume 8296 of LNCS, pages 30-41. Springer, 2012.
Jacob E. Goodman and Richard Pollack. Multidimensional sorting. SIAM J. Comput., 12(3):484-507, 1983.
Jacob E. Goodman and Richard Pollack. Semispaces of configurations, cell complexes of arrangements. J. Combin. Theory Ser. A, 37(3):257-293, 1984.
Jan Kynčl. Enumeration of simple complete topological graphs. Eur. J. Comb., 30(7):1676-1685, 2009.
Nicolai E. Mnëv. The universality theorems on the classification problem of configuration varieties and convex polytope varieties. In Topology and Geometry - Rohlin Seminar, volume 1346 of Lecture Notes in Math., pages 527-544. Springer, 1988.
Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Crossing numbers of graphs with rotation systems. Algorithmica, 60(3):679-702, 2011.
Gerhard Ringel. Extremal problems in the theory of graphs. In Theory of Graphs and its Applications (Smolenice, 1963), pages 85-90. Publ. House Czechoslovak Acad. Sci., Prague, 1964.
Marcus Schaefer. Complexity of some geometric and topological problems. In Graph Drawing, volume 5849 of LNCS, pages 334-344. Springer, 2009.
Géza Tóth and Pavel Valtr. Geometric graphs with few disjoint edges. In Symposium on Computational Geometry, pages 184-191, 1998.
Benjamín Tovar, Luigi Freda, and Steven M. LaValle. Learning combinatorial map information from permutations of landmarks. I. J. Robotic Res., 30(9):1143-1156, 2011.
Stephen K. Wismath. Point and line segment reconstruction from visibility information. Int. J. Comput. Geometry Appl., 10(2):189-200, 2000.
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Limits of Order Types
The notion of limits of dense graphs was invented, among other reasons, to attack problems in extremal graph theory. It is straightforward to define limits of order types in analogy with limits of graphs, and this paper examines how to adapt to this setting two approaches developed to study limits of dense graphs.
We first consider flag algebras, which were used to open various questions on graphs to mechanical solving via semidefinite programming. We define flag algebras of order types, and use them to obtain, via the semidefinite method, new lower bounds on the density of 5- or 6-tuples in convex position in arbitrary point sets, as well as some inequalities expressing the difficulty of sampling order types uniformly.
We next consider graphons, a representation of limits of dense graphs that enable their study by continuous probabilistic or analytic methods. We investigate how planar measures fare as a candidate analogue of graphons for limits of order types. We show that the map sending a measure to its associated limit is continuous and, if restricted to uniform measures on compact convex sets, a homeomorphism. We prove, however, that this map is not surjective. Finally, we examine a limit of order types similar to classical constructions in combinatorial geometry (Erdos-Szekeres, Horton...) and show that it cannot be represented by any somewhere regular measure; we analyze this example via an analogue of Sylvester's problem on the probability that k random points are in convex position.
order types
Limits of discrete structures
Flag algebras
Erdos-Szekeres
Sylvester’s problem
300-314
Regular Paper
Xavier
Goaoc
Xavier Goaoc
Alfredo
Hubard
Alfredo Hubard
Rémi
de Joannis de Verclos
Rémi de Joannis de Verclos
Jean-Sébastien
Sereni
Jean-Sébastien Sereni
Jan
Volec
Jan Volec
10.4230/LIPIcs.SOCG.2015.300
B. Abrego, S. Fernandez-Merchant, and G. Salazar. The rectilinear crossing number of k_n: Closing in (or are we?). In Thirty Essays on Geometric Graph Theory, pages 5-18. Springer, 2013.
O. Aichholzer, F. Aurenhammer, and H. Krasser. Enumerating Order Types for Small Point Sets with Applications. In Proc. 17^th Ann. ACM Symp. Computational Geometry, pages 11-18, Medford, Massachusetts, USA, 2001.
N. Alon. The number of polytopes, configurations and real matroids. Mathematika, 33:62- 71, 1986.
G. Aloupis, J. Iacono, S. Langerman, Ö. Özkan, and S. Wuhrer. The complexity of order type isomorphism. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'14, pages 405-415, 2014.
B. Borchers. CSDP, A C library for semidefinite programming. Optimization Methods and Software, 11(1-4):613-623, 1999.
P. Brass, W. Moser, and J. Pach. Research Problems in Discrete Geometry. Springer, 2005.
P. Erdős and G. Szekeres. On some extremum problems in elementary geometry. Eotvos Sect. Math, 3-4:53-62, 1962.
J. D. Horton. Sets with no empty convex 7-gons. Canad. Math. Bull., 26:482-484, 1983.
L. Lovász and B. Szegedy. Limits of dense graph sequences. J. Combin. Theory Ser. B, 96(6):933-957, 2006.
E. Lubetzky and Y. Zhao. On replica symmetry of large deviations in random graphs. Random Structures & Algorithms, 2014. URL: http://dx.doi.org/10.1002/rsa.20536.
http://dx.doi.org/10.1002/rsa.20536
A. A. Razborov. Flag algebras. J. Symbolic Logic, 72(4):1239-1282, 2007.
E. Scheinerman and H. Wilf. The rectilinear crossing number of a complete graph and sylvester’s "four point problem" of geometric probability. Amer. Math. Monthly, 101:939-943, 1994.
P. W. Shor. Stretchability of pseudolines is NP-hard. In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift, volume 4 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 531-554. Amer. Math. Soc., 1991.
W. A. Stein et al. Sage Mathematics Software (Version 6.1). The Sage Development Team, 2013. URL: http://www.sagemath.org.
http://www.sagemath.org
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Combinatorial Redundancy Detection
The problem of detecting and removing redundant constraints is fundamental in optimization. We focus on the case of linear programs (LPs) in dictionary form, given by n equality constraints in n+d variables, where the variables are constrained to be nonnegative. A variable x_r is called redundant, if after removing its nonnegativity constraint the LP still has the same feasible region. The time needed to solve such an LP is denoted by LP(n,d).
It is easy to see that solving n+d LPs of the above size is sufficient to detect all redundancies. The currently fastest practical method is the one by Clarkson: it solves n+d linear programs, but each of them has at most s variables, where s is the number of nonredundant constraints.
In the first part we show that knowing all of the finitely many dictionaries of the LP is sufficient for the purpose of redundancy detection. A dictionary is a matrix that can be thought of as an enriched encoding of a vertex in the LP. Moreover - and this is the combinatorial aspect - it is enough to know only the signs of the entries, the actual values do not matter. Concretely we show that for any variable x_r one can find a dictionary, such that its sign pattern is either a redundancy or nonredundancy certificate for x_r.
In the second part we show that considering only the sign patterns of the dictionary, there is an output sensitive algorithm of running time of order d (n+d) s^{d-1} LP(s,d) + d s^{d} LP(n,d) to detect all redundancies. In the case where all constraints are in general position, the running time is of order s LP(n,d) + (n+d) LP(s,d), which is essentially the running time of the Clarkson method. Our algorithm extends naturally to a more general setting of arrangements of oriented topological hyperplane arrangements.
system of linear inequalities
redundancy removal
linear programming
output sensitive algorithm
Clarkson’s method
315-328
Regular Paper
Komei
Fukuda
Komei Fukuda
Bernd
Gärtner
Bernd Gärtner
May
Szedlák
May Szedlák
10.4230/LIPIcs.SOCG.2015.315
A. Björner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler. Oriented Matroids. Cambridge University Press, 1993.
T. M. Chan. Output-sensitive results on convex hulls, extreme points, and related problems. Discrete & Computational Geometry, 16(4):369-387, 1996.
V. Chvatal. Linear Programming. W. H. Freeman, 1983.
K. L. Clarkson. More output-sensitive geometric algorithms. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., pages 695-702, 1994.
G. B. Dantzig. Linear Programming and Extensions. Princeton University Press, Princeton, NJ, 1963.
J. H. Dulá, R. V. Helgason, and N. Venugopal. An algorithm for identifying the frame of a pointed finite conical hull. INFORMS J. Comput., 10(3):323-330, 1998.
K. Fukuda. Introduction to optimization. http://www.ifor.math.ethz.ch/teaching/Courses/Fall_2011/intro_fall_11, 2011.
http://www.ifor.math.ethz.ch/teaching/Courses/Fall_2011/intro_fall_11
K. Fukuda. Walking on the arrangement, not on the feasible region. Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?, IPAM, UCLA, 2011. presentation slides available as http://helper.ipam.ucla.edu/publications/sm2011/sm2011_9630.pdf. .
http://helper.ipam.ucla.edu/publications/sm2011/sm2011_9630.pdf
K. Fukuda. Lecture: Polyhedral computation. http://www-oldurls.inf.ethz.ch/personal/fukudak/lect/pclect/notes2015/, 2015.
http://www-oldurls.inf.ethz.ch/personal/fukudak/lect/pclect/notes2015/
K. Fukuda, B. Gärtner, and M. Szedlák. Combinatorial redundancy removal. Preprint: arXiv:1412.1241, 2014.
K. Fukuda and T. Terlaky. Criss-cross methods: A fresh view on pivot algorithms. Mathematical Programming, 79:369-395, 1997.
V. Klee and G. J. Minty. How good is the simplex algorithm? In O. Shisha, editor, Inequalities III, pages 159-175. Academic Press, 1972.
J. Matoušek, M. Sharir, and E. Welzl. A subexponential bound for linear programming. Algorithmica, 16:498-516, 1996.
Th. Ottmann, S. Schuierer, and S. Soundaralakshmi. Enumerating extreme points in higher dimensions. In E.W. Mayer and C. Puech, editors, STACS 95: 12th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 900, pages 562-570. Springer-Verlag, 1995.
C. Roos. An exponential example for Terlaky’s pivoting rule for the criss-cross simplex method. Mathematical Programming, 46:79-84, 1990.
T. Terlaky. A finite criss-cross method for the oriented matroids. Journal of Combinatorial Theory Series B, 42:319-327, 1987.
Z. Wang. A finite conformal-elimination free algorithm over oriented matroid programming. Chinese Annals of Math., 8B:120-125, 1987.
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Effectiveness of Local Search for Geometric Optimization
What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude 1/epsilon^c is an approximation scheme for the following problems in the Euclidean plane: TSP with random inputs, Steiner tree with random inputs, uniform facility location (with worst case inputs), and bicriteria k-median (also with worst case inputs). The randomness assumption is necessary for TSP.
Local Search
PTAS
Facility Location
k-Median
TSP
Steiner Tree
329-344
Regular Paper
Vincent
Cohen-Addad
Vincent Cohen-Addad
Claire
Mathieu
Claire Mathieu
10.4230/LIPIcs.SOCG.2015.329
S. Arora. Polynomial time approximation schemes for euclidean TSP and other geometric problems. In Symp. on Foundations of Computer Science, FOCS'96, Burlington, Vermont, USA, 14-16 October, 1996, pages 2-11, 1996.
S. Arora. Nearly linear time approximation schemes for euclidean TSP and other geometric problems. In Symp. on Foundations of Computer Science, FOCS'97, Miami Beach, Florida, USA, October 19-22, 1997, pages 554-563, 1997.
V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, and V. Pandit. Local search heuristics for k-median and facility location problems. SIAM J. Comput., 33(3):544-562, 2004.
J. Beardwood, J. H. Halton, and J. M. Hammersley. The shortest path through many points. Mathematical Proc. of the Cambridge Philosophical Society, 55:299-327, 1959.
T. M. Chan and S. Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. In Proc. of the Symp. on Computational Geometry, SCG'09, pages 333-340. ACM, 2009.
B. Chandra, H. J. Karloff, and C. A. Tovey. New results on the old k-opt algorithm for the TSP. In Proc. of the ACM-SIAM Symp. on Discrete Algorithms. 23-25 January 1994, Arlington, Virginia., pages 150-159, 1994.
M. Charikar and S. Guha. Improved combinatorial algorithms for facility location problems. SIAM J. Comput., 34(4):803-824, 2005.
F. A. Chudak and D. P. Williamson. Improved approximation algorithms for capacitated facility location problems. Math. Program., 102(2):207-222, March 2005.
G. A. Croes. A method for solving traveling salesman problems. Operations Research, 6(6):791-812, 1958.
M. Gibson and I. A. Pirwani. Algorithms for dominating set in disk graphs: Breaking the logn barrier - (extended abstract). In Algorithms - ESA 2010, European Symp., Liverpool, UK, September 6-8, 2010. Proc., Part I, pages 243-254, 2010.
D. S Johnson and L. A McGeoch. The traveling salesman problem : A case study in local optimization. Local Search in Combinatorial Optimization, 1:215-310, 1997.
T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman, and A. Y. Wu. A local search approximation algorithm for k-means clustering. Comput. Geom., 28(2-3):89-112, 2004.
R. M. Karp. Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Mathematics of Operations Research, 2(3):209-224, 1977.
S. G. Kolliopoulos and S. Rao. A nearly linear-time approximation scheme for the euclidean k-median problem. SIAM J. Comput., 37(3):757-782, 2007.
M. R. Korupolu, C. G. Plaxton, and R. Rajaraman. Analysis of a local search heuristic for facility location problems. J. Algorithms, 37(1):146-188, 2000.
E. Krohn, M. Gibson, G. Kanade, and K. R. Varadarajan. Guarding terrains via local search. JoCG, 5(1):168-178, 2014.
S. Lin. Computer solutions of the traveling salesman problem. Bell System Technical Journal, The, 44(10):2245-2269, 1965.
S. Lin and B. W Kernighan. An effective heuristic algorithm for the traveling-salesman problem. Operations research, 21(2):498-516, 1973.
N. Megiddo and K. J. Supowit. On the complexity of some common geometric location problems. SIAM J. Comput., 13(1):182-196, 1984.
O. Mersmann, B. Bischl, J. Bossek, H. Trautmann, M. Wagner, and F. Neumann. Local search and the traveling salesman problem: A feature-based characterization of problem hardness. In Learning and Intelligent Optimization - 6th International Conference, LION 6, Paris, France, January 16-20, 2012, Revised Selected Papers, pages 115-129, 2012.
J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric tsp, k-mst, and related problems. SIAM J. Comput., 28(4):1298-1309, 1999.
N. H. Mustafa and S. Ray. PTAS for geometric hitting set problems via local search. In Proc. of the ACM Symp. on Computational Geometry, Aarhus, Denmark, pages 17-22, 2009.
S. Rao and W. D. Smith. Approximating geometrical graphs via "spanners" and "banyans". In Proc. of the ACM Symp. on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 540-550, 1998.
G. Robins and A. Zelikovsky. Improved steiner tree approximation in graphs. In Proc. of the ACM-SIAM Symp. on Discrete Algorithms, SODA, pages 770-779. SIAM, 2000.
D. J. Rosenkrantz, R. Edwin Stearns, and P. M. Lewis II. An analysis of several heuristics for the traveling salesman problem. SIAM J. Comput., 6(3):563-581, 1977.
J. Vygen. Approximation algorithms for facility location problems. Technical Report 05950, Research Institute for Discrete Mathematics, University of Bonn, 2005.
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On the Shadow Simplex Method for Curved Polyhedra
We study the simplex method over polyhedra satisfying certain "discrete curvature" lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al. (SOCG 2012), Brunsch and Röglin (ICALP 2013), and Eisenbrand and Vempala (2014) have improved our understanding of such polyhedra.
We develop a new type of dual analysis of the shadow simplex method which provides a clean and powerful tool for improving all previously mentioned results. Our methods are inspired by the recent work of Bonifas and the first named author, who analyzed a remarkably similar process as part of an algorithm for the Closest Vector Problem with Preprocessing.
For our first result, we obtain a constructive diameter bound of O((n^2 / delta) ln (n / delta)) for n-dimensional polyhedra with curvature parameter delta in (0, 1]. For the class of polyhedra arising from totally unimodular constraint matrices, this implies a bound of O(n^3 ln n). For linear optimization, given an initial feasible vertex, we show that an optimal vertex can be found using an expected O((n^3 / delta) ln (n / delta)) simplex pivots, each requiring O(mn) time to compute. An initial feasible solution can be found using O((mn^3 / delta) ln (n / delta)) pivot steps.
Optimization
Linear Programming
Simplex Method
Diameter of Polyhedra
345-359
Regular Paper
Daniel
Dadush
Daniel Dadush
Nicolai
Hähnle
Nicolai Hähnle
10.4230/LIPIcs.SOCG.2015.345
Karim Alexander Adiprasito and Bruno Benedetti. The Hirsch conjecture holds for normal flag complexes. Arxiv Report 1303.3598, 2014.
M. L. Balinski. The Hirsch conjecture for dual transportation polyhedra. Math. Oper. Res., 9(4):629-633, 1984.
David Barnette. An upper bound for the diameter of a polytope. Discrete Math., 10:9-13, 1974.
Nicolas Bonifas, Marco Di Summa, Friedrich Eisenbrand, Nicolai Hähnle, and Martin Niemeier. On sub-determinants and the diameter of polyhedra. Discrete Comput. Geom., 52(1):102-115, 2014. Preliminary version in SOCG 12.
Karl-Heinz Borgwardt. The simplex method: A probabilistic analysis, volume 1 of Algorithms and Combinatorics: Study and Research Texts. Springer-Verlag, Berlin, 1987.
Graham Brightwell, Jan van den Heuvel, and Leen Stougie. A linear bound on the diameter of the transportation polytope. Combinatorica, 26(2):133-139, 2006.
Tobias Brunsch and Heiko Röglin. Finding short paths on polytopes by the shadow vertex algorithm. In Automata, languages, and programming. Part I, volume 7965 of Lecture Notes in Comput. Sci., pages 279-290. Springer, Heidelberg, 2013.
Daniel Dadush and Nicolas Bonifas. Short paths on the voronoi graph and closest vector problem with preprocessing. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 295-314. SIAM, 2015.
Daniel Dadush and Nicolai Hähnle. On the shadow simplex method for curved polyhedra (draft of full paper). Arxiv Report 1412.6705, 2014.
Jesús A. De Loera, Edward D. Kim, Shmuel Onn, and Francisco Santos. Graphs of transportation polytopes. J. Combin. Theory Ser. A, 116(8):1306-1325, 2009.
Martin Dyer and Alan Frieze. Random walks, totally unimodular matrices, and a randomised dual simplex algorithm. Math. Programming, 64(1, Ser. A):1-16, 1994.
Friedrich Eisenbrand and Santosh Vempala. Geometric random edge. Arxiv Report 1404.1568, 2014.
Gil Kalai. The diameter of graphs of convex polytopes and f-vector theory. In Applied geometry and discrete mathematics, volume 4 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 387-411. Amer. Math. Soc., Providence, RI, 1991.
Gil Kalai and Daniel J. Kleitman. A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Amer. Math. Soc. (N.S.), 26(2):315-316, 1992.
D. G. Larman. Paths of polytopes. Proc. London Math. Soc. (3), 20:161-178, 1970.
Benjamin Matschke, Francisco Santos, and Christophe Weibel. The width of 5-dimensional prismatoids. Arxiv Report 1202.4701, 2013.
Denis Naddef. The Hirsch conjecture is true for (0,1)-polytopes. Math. Programming, 45(1, Ser. B):109-110, 1989.
Francisco Santos. A counterexample to the Hirsch conjecture. Ann. of Math. (2), 176(1):383-412, 2012.
Daniel A. Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM, 51(3):385-463 (electronic), 2004.
Michael J. Todd. An improved Kalai-Kleitman bound for the diameter of a polyhedron. Arxiv Report 1402.3579, 2014.
Roman Vershynin. Beyond Hirsch conjecture: walks on random polytopes and smoothed complexity of the simplex method. SIAM J. Comput., 39(2):646-678, 2009.
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Pattern Overlap Implies Runaway Growth in Hierarchical Tile Systems
We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap.
This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly and avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem.
self-assembly
hierarchical
pumping
360-373
Regular Paper
Ho-Lin
Chen
Ho-Lin Chen
David
Doty
David Doty
Ján
Manuch
Ján Manuch
Arash
Rafiey
Arash Rafiey
Ladislav
Stacho
Ladislav Stacho
10.4230/LIPIcs.SOCG.2015.360
Gagan Aggarwal, Qi Cheng, Michael H. Goldwasser, Ming-Yang Kao, Pablo Moisset de Espanés, and Robert T. Schweller. Complexities for generalized models of self-assembly. SIAM Journal on Computing, 34:1493-1515, 2005. Preliminary version appeared in SODA 2004.
Robert D. Barish, Rebecca Schulman, Paul W. K. Rothemund, and Erik Winfree. An information-bearing seed for nucleating algorithmic self-assembly. Proceedings of the National Academy of Sciences, 106(15):6054-6059, March 2009.
Sarah Cannon, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Matthew J. Patitz, Robert T. Schweller, Scott M. Summers, and Andrew Winslow. Two hands are better than one (up to constant factors). In STACS 2013: Proceedings of the Thirtieth International Symposium on Theoretical Aspects of Computer Science, pages 172-184, 2013.
Ho-Lin Chen and David Doty. Parallelism and time in hierarchical self-assembly. In SODA 2012: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1163-1182, 2012.
Erik D. Demaine, Martin L. Demaine, Sándor P. Fekete, Matthew J. Patitz, Robert T. Schweller, Andrew Winslow, and Damien Woods. One tile to rule them all: Simulating any Turing machine, tile assembly system, or tiling system with a single puzzle piece. In ICALP 2014: Proceedings of the 41st International Colloquium on Automata, Languages, and Programming, 2014.
Erik D. Demaine, Matthew J. Patitz, Trent Rogers, Robert T. Schweller, Scott M. Summers, and Damien Woods. The two-handed tile assembly model is not intrinsically universal. In ICALP 2013: Proceedings of the 40th International Colloquium on Automata, Languages and Programming, July 2013.
David Doty. Theory of algorithmic self-assembly. Communications of the ACM, 55(12):78-88, December 2012.
David Doty. Producibility in hierarchical self-assembly. In UCNC 2014: Proceedings of 13th Unconventional Computation and Natural Computation, 2014.
David Doty, Matthew J. Patitz, Dustin Reishus, Robert T. Schweller, and Scott M. Summers. Strong fault-tolerance for self-assembly with fuzzy temperature. In FOCS 2010: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pages 417-426. IEEE, 2010.
David Doty, Matthew J. Patitz, and Scott M. Summers. Limitations of self-assembly at temperature 1. Theoretical Computer Science, 412(1-2):145-158, January 2011. Preliminary version appeared in DNA 2009.
Pierre Étienne Meunier, Matthew J. Patitz, Scott M. Summers, Guillaume Theyssier, Andrew Winslow, and Damien Woods. Intrinsic universality in tile self-assembly requires cooperation. In SODA 2014: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 752-771, 2014.
Kei Goto, Yoko Hinob, Takayuki Kawashima, Masahiro Kaminagab, Emiko Yanob, Gaku Yamamotob, Nozomi Takagic, and Shigeru Nagasec. Synthesis and crystal structure of a stable S-nitrosothiol bearing a novel steric protection group and of the corresponding S-nitrothiol. Tetrahedron Letters, 41(44):8479-8483, 2000.
Wilfried Heller and Thomas L. Pugh. "Steric protection" of hydrophobic colloidal particles by adsorption of flexible macromolecules. Journal of Chemical Physics, 22(10):1778, 1954.
Wilfried Heller and Thomas L. Pugh. "Steric" stabilization of colloidal solutions by adsorption of flexible macromolecules. Journal of Polymer Science, 47(149):203-217, 1960.
Chris Luhrs. Polyomino-safe DNA self-assembly via block replacement. Natural Computing, 9(1):97-109, March 2010. Preliminary version appeared in DNA 2008.
Ján Maňuch, Ladislav Stacho, and Christine Stoll. Two lower bounds for self-assemblies at temperature 1. Journal of Computational Biology, 17(6):841-852, 2010.
Matthew J. Patitz. An introduction to tile-based self-assembly. In UCNC 2012: Proceedings of the 11th international conference on Unconventional Computation and Natural Computation, pages 34-62, Berlin, Heidelberg, 2012. Springer-Verlag.
John H. Reif and Tianqi Song. Complexity and computability of temperature-1 tilings. Poster at DNA 2013: 19th International Meeting on DNA Computing and Molecular Programming, 2013.
Paul W. K. Rothemund and Erik Winfree. The program-size complexity of self-assembled squares (extended abstract). In STOC 2000: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 459-468, 2000.
Rebecca Schulman and Erik Winfree. Synthesis of crystals with a programmable kinetic barrier to nucleation. Proceedings of the National Academy of Sciences, 104(39):15236-15241, 2007.
Rebecca Schulman and Erik Winfree. Programmable control of nucleation for algorithmic self-assembly. SIAM Journal on Computing, 39(4):1581-1616, 2009. Preliminary version appeared in DNA 2004.
Leroy G. Wade. Organic Chemistry. Prentice Hall, 2nd edition, 1991.
Erik Winfree. Simulations of computing by self-assembly. Technical Report CaltechCSTR:1998.22, California Institute of Technology, 1998.
Erik Winfree. Self-healing tile sets. In Junghuei Chen, Natasa Jonoska, and Grzegorz Rozenberg, editors, Nanotechnology: Science and Computation, Natural Computing Series, pages 55-78. Springer, 2006.
Erik Winfree, Furong Liu, Lisa A. Wenzler, and Nadrian C. Seeman. Design and self-assembly of two-dimensional DNA crystals. Nature, 394(6693):539-44, 1998.
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Space Exploration via Proximity Search
We investigate what computational tasks can be performed on a point set in R^d, if we are only given black-box access to it via nearest-neighbor search. This is a reasonable assumption if the underlying point set is either provided implicitly, or it is stored in a data structure that can answer such queries. In particular, we show the following:
(A) One can compute an approximate bi-criteria k-center clustering of the point set, and more generally compute a greedy permutation of the point set.
(B) One can decide if a query point is (approximately) inside the convex-hull of the point set.
We also investigate the problem of clustering the given point set, such that meaningful proximity queries can be carried out on the centers of the clusters, instead of the whole point set.
Proximity search
implicit point set
probing
374-389
Regular Paper
Sariel
Har-Peled
Sariel Har-Peled
Nirman
Kumar
Nirman Kumar
David M.
Mount
David M. Mount
Benjamin
Raichel
Benjamin Raichel
10.4230/LIPIcs.SOCG.2015.374
L.-E. Andersson and N. F. Stewart. Introduction to the Mathematics of Subdivision Surfaces. SIAM, 2010.
G. Binnig, C. F. Quate, and Ch. Gerber. Atomic force microscope. Phys. Rev. Lett., 56:930-933, Mar 1986.
J. F. Blinn. A generalization of algebraic surface drawing. ACM Trans. Graphics, 1:235-256, 1982.
J.-D. Boissonnat, L. J. Guibas, and S. Oudot. Learning smooth shapes by probing. Comput. Geom. Theory Appl., 37(1):38-58, 2007.
K. L. Clarkson. Coresets, sparse greedy approximation, and the frank-wolfe algorithm. ACM Trans. Algo., 6(4), 2010.
R. Cole and C. K. Yap. Shape from probing. J. Algorithms, 8(1):19-38, 1987.
T. Feder and D. H. Greene. Optimal algorithms for approximate clustering. In Proc. 20th Annu. ACM Sympos. Theory Comput.\CNFSTOC, pages 434-444, 1988.
A. Goel, P. Indyk, and K. R. Varadarajan. Reductions among high dimensional proximity problems. In Proc. 12th ACM-SIAM Sympos. Discrete Algs.\CNFSODA, pages 769-778, 2001.
T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293-306, 1985.
S. Har-Peled. Geometric Approximation Algorithms, volume 173 of Mathematical Surveys and Monographs. Amer. Math. Soc., 2011.
S. Har-Peled, P. Indyk, and R. Motwani. Approximate nearest neighbors: Towards removing the curse of dimensionality. Theory Comput., 8:321-350, 2012. Special issue in honor of Rajeev Motwani.
S. Har-Peled, N. Kumar, D. Mount, and B. Raichel. Space exploration via proximity search. CoRR, abs/1412.1398, 2014.
S. Har-Peled and M. Mendel. Fast construction of nets in low dimensional metrics, and their applications. SIAM J. Comput., 35(5):1148-1184, 2006.
P. Indyk. Nearest neighbors in high-dimensional spaces. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 39, pages 877-892. CRC Press LLC, 2nd edition, 2004.
B. Kalantari. A characterization theorem and an algorithm for A convex hull problem. CoRR, abs/1204.1873, 2012.
B. B. Mandelbrot. The fractal geometry of nature. Macmillan, 1983.
J. M. Mulvey and M. P. Beck. Solving capacitated clustering problems. Euro. J. Oper. Res., 18:339-348, 1984.
F. Panahi, A. Adler, A. F. van der Stappen, and K. Goldberg. An efficient proximity probing algorithm for metrology. In Proc. IEEE Int. Conf. Autom. Sci. Engin. (CASE), pages 342-349, 2013.
S. S. Skiena. Problems in geometric probing. Algorithmica, 4:599-605, 1989.
S. S. Skiena. Geometric reconstruction problems. In J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 26, pages 481-490. CRC Press LLC, Boca Raton, FL, 1997.
R. M. Smelik, K. J. De Kraker, S. A. Groenewegen, T. Tutenel, and R. Bidarra. A survey of procedural methods for terrain modelling. In Proc. of the CASA Work. 3D Adv. Media Gaming Simul., 2009.
Wikipedia. Atomic force microscopy - wikipedia, the free encyclopedia, 2014.
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Star Unfolding from a Geodesic Curve
There are two known ways to unfold a convex polyhedron without overlap: the star unfolding and the source unfolding, both of which use shortest paths from vertices to a source point on the surface of the polyhedron. Non-overlap of the source unfolding is straightforward; non-overlap of the star unfolding was proved by Aronov and O'Rourke in 1992. Our first contribution is a much simpler proof of non-overlap of the star unfolding.
Both the source and star unfolding can be generalized to use a simple geodesic curve instead of a source point. The star unfolding from a geodesic curve cuts the geodesic curve and a shortest path from each vertex to the geodesic curve. Demaine and Lubiw conjectured that the star unfolding from a geodesic curve does not overlap. We prove a special case of the conjecture. Our special case includes the previously known case of unfolding from a geodesic loop. For the general case we prove that the star unfolding from a geodesic curve can be separated into at most two non-overlapping pieces.
unfolding
convex polyhedra
geodesic curve
390-404
Regular Paper
Stephen
Kiazyk
Stephen Kiazyk
Anna
Lubiw
Anna Lubiw
10.4230/LIPIcs.SOCG.2015.390
Pankaj K. Agarwal, Boris Aronov, and Catherine A. Schevon. Star unfolding of a polytope with applications. SIAM Journal on Computing, 26:1689-1713, 1997.
Boris Aronov and Joseph O'Rourke. Nonoverlap of the star unfolding. Discrete & Computational Geometry, 8(3):219-250, 1992.
Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Arlo Shallit, and Jonah Shallit. Zipper unfoldings of polyhedral complexes. In Proceedings of the 22nd Annual Canadian Conference on Computational Geometry (CCCG), pages 219-222, August 2010.
Erik D. Demaine and Anna Lubiw. A generalization of the source unfolding of convex polyhedra. In Revised Papers from the 14th Spanish Meeting on Computational Geometry (EGC 2011), volume 7579 of Lecture Notes in Computer Science, pages 185-199, Alcalá de Henares, Spain, June 27-30 2012.
Erik D. Demaine and Joseph O'Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, New York, NY, USA, 2007.
Albrecht Dürer. The Painter’s Manual: A Manual of Measurement of Lines, Areas, and Solids by Means of Compass and Ruler Assembled by Albrecht Dürer for the Use of All Lovers of Art with Appropriate Illustrations Arranged to be Printed in the Year MDXXV. The literary remains of Albrecht Dürer. Abaris Books, 1977.
Kouki Ieiri, Jin-ichi Itoh, and Costin Vîlcu. Quasigeodesics and farthest points on convex surfaces. Advances in Geometry, 11(4):571-584, 2011.
Jin-Ichi Itoh, Joseph O'Rourke, and Costin Vîlcu. Source unfoldings of convex polyhedra with respect to certain closed polygonal curves. In Proceedings of the 25th European Workshop on Computational Geometry (EuroCG), pages 61-64, 2009.
Jin-ichi Itoh, Joseph O'Rourke, and Costin Vîlcu. Star unfolding convex polyhedra via quasigeodesic loops. Discrete & Computational Geometry, 44(1):35-54, 2010.
Stephen Kiazyk. The star unfolding from a geodesic curve. Master’s thesis, Cheriton School of Computer Science, University of Waterloo, 2014.
Joseph O'Rourke and Costin Vîlcu. Development of curves on polyhedra via conical existence. Computational Geometry, 47(2, Part A):149-163, 2014.
Micha Sharir and Amir Schorr. On shortest paths in polyhedral spaces. SIAM Journal on Computing, 15(1):193-215, 1986.
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The Dirac-Motzkin Problem on Ordinary Lines and the Orchard Problem (Invited Talk)
Suppose you have n points in the plane, not all on a line. A famous theorem of Sylvester-Gallai asserts that there is at least one ordinary line, that is to say a line passing through precisely two of the n points. But how many ordinary lines must there be? It turns out that the answer is at least n/2 (if n is even) and roughly 3n/4 (if n is odd), provided that n is sufficiently large. This resolves a conjecture of Dirac and Motzkin from the 1950s. We will also discuss the classical orchard problem, which asks how to arrange n trees so that there are as many triples of colinear trees as possible, but no four in a line. This is joint work with Terence Tao and reports on the results of [Green and Tao, 2013].
combinatorial geometry
incidences
405-405
Invited Talk
Ben J.
Green
Ben J. Green
10.4230/LIPIcs.SOCG.2015.405
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On the Beer Index of Convexity and Its Variants
Let S be a subset of R^d with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate a relationship between these two natural measures of convexity of S.
We show that every subset S of the plane with simply connected components satisfies b(S) <= alpha c(S) for an absolute constant alpha, provided b(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. asserting that this estimate holds for simple polygons.
We also consider higher-order generalizations of b(S). For 1 <= k <= d, the k-index of convexity b_k(S) of a subset S of R^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every d >= 2 there is a constant beta(d) > 0 such that every subset S of R^d satisfies b_d(S) <= beta c(S), provided b_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma(d) > 0 such that for every epsilon from (0,1] there is a subset S of R^d of Lebesgue measure one satisfying c(S) <= epsilon and b_d(S) >= (gamma epsilon)/log_2(1/epsilon) >= (gamma c(S))/log_2(1/c(S)).
Beer index of convexity
convexity ratio
convexity measure
visibility
406-420
Regular Paper
Martin
Balko
Martin Balko
Vít
Jelínek
Vít Jelínek
Pavel
Valtr
Pavel Valtr
Bartosz
Walczak
Bartosz Walczak
10.4230/LIPIcs.SOCG.2015.406
M. Balko, V. Jelínek, P. Valtr, and B. Walczak. On the Beer index of convexity and its variants. full version, URL: http://arxiv.org/abs/1412.1769.
http://arxiv.org/abs/1412.1769
G. Beer. Continuity properties of the visibility function. Michigan Math. J., 20:297-302, 1973.
G. Beer. The index of convexity and the visibility function. Pacific J. Math., 44(1):59-67, 1973.
G. Beer. The index of convexity and parallel bodies. Pacific J. Math., 53(2):337-345, 1974.
W. Blaschke. Über affine Geometrie III: Eine Minimumeigenschaft der Ellipse. Ber. Verh. Kön. Sächs. Ges. Wiss. Leipzig Math.-Phys. Kl., 69:3-12, 1917.
P. G. Bradford and V. Capoyleas. Weak ε-nets for points on a hypersphere. Discrete Comput. Geom., 18(1):83-91, 1997.
S. Cabello, J. Cibulka, J. Kynčl, M. Saumell, and P. Valtr. Peeling potatoes near-optimally in near-linear time. In Proceedings of the 30th Annual Symposium on Computational Geometry, pages 224-231, 2014.
H. T. Croft, K. J. Falconer, and R. K. Guy. Unsolved Problems in Geometry. Unsolved Problems in Intuitive Mathematics. Springer New York, 2nd edition, 1991.
J. E. Goodman. On the largest convex polygon contained in a non-convex n-gon, or how to peel a potato. Geom. Dedicata, 11(1):99-106, 1981.
D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput. Geom., 2(2):127-151, 1987.
F. John. Extremum problems with inequalities as subsidiary conditions. In Studies and Essays, presented to R. Courant on his 60th birthday, January 8, 1948, pages 187-204, 1948.
R. Lang. A note on the measurability of convex sets. Arch. Math. (Basel), 47:90-92, 1986.
M. Lassak. Approximation of convex bodies by inscribed simplices of maximum volume. Beitr. Algebra Geom., 52(2):389-394, 2011.
J. Matoušek. Lectures on Discrete Geometry, volume 212 of Graduate Texts in Mathematics. Springer New York, 2002.
J. Pach and G. Tardos. Piercing quasi-rectangles - on a problem of Danzer and Rogers. J. Combin. Theory Ser. A, 119(7):1391-1397, 2012.
V. V. Prasolov. Elements of combinatorial and differential topology, volume 74 of Graduate Studies in Mathematics. American Mathematical Society, 2006.
G. Rote. The degree of convexity. In Abstracts of the 29th European Workshop on Computational Geometry, pages 69-72, 2013.
E. Sas. Über eine Extremumeigenschaft der Ellipsen. Compositio Math., 6:468-470, 1939.
H. I. Stern. Polygonal entropy: a convexity measure for polygons. Pattern Recogn. Lett., 10(4):229-235, 1989.
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Tight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Art Galleries
The chromatic art gallery problem asks for the minimum number of "colors" t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility - two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is Theta(loglog n). By contrast, the best upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is Theta(log n). Finally, our techniques also help us establish the first non-trivial lower bound of Omega(loglog n / logloglog n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.
Orthogonal polygons
art gallery problem
hypergraph coloring
421-435
Regular Paper
Frank
Hoffmann
Frank Hoffmann
Klaus
Kriegel
Klaus Kriegel
Subhash
Suri
Subhash Suri
Kevin
Verbeek
Kevin Verbeek
Max
Willert
Max Willert
10.4230/LIPIcs.SOCG.2015.421
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Low-Quality Dimension Reduction and High-Dimensional Approximate Nearest Neighbor
The approximate nearest neighbor problem (epsilon-ANN) in Euclidean settings is a fundamental question, which has been addressed by two main approaches: Data-dependent space partitioning techniques perform well when the dimension is relatively low, but are affected by the curse of dimensionality. On the other hand, locality sensitive hashing has polynomial dependence in the dimension, sublinear query time with an exponent inversely proportional to (1+epsilon)^2, and subquadratic space requirement.
We generalize the Johnson-Lindenstrauss Lemma to define "low-quality" mappings to a Euclidean space of significantly lower dimension, such that they satisfy a requirement weaker than approximately preserving all distances or even preserving the nearest neighbor. This mapping guarantees, with high probability, that an approximate nearest neighbor lies among the k approximate nearest neighbors in the projected space. These can be efficiently retrieved while using only linear storage by a data structure, such as BBD-trees. Our overall algorithm, given n points in dimension d, achieves space usage in O(dn), preprocessing time in O(dn log n), and query time in O(d n^{rho} log n), where rho is proportional to 1 - 1/loglog n, for fixed epsilon in (0, 1). The dimension reduction is larger if one assumes that point sets possess some structure, namely bounded expansion rate. We implement our method and present experimental results in up to 500 dimensions and 10^6 points, which show that the practical performance is better than predicted by the theoretical analysis. In addition, we compare our approach with E2LSH.
Approximate nearest neighbor
Randomized embeddings
Curse of dimensionality
Johnson-Lindenstrauss Lemma
Bounded expansion rate
Experimental study
436-450
Regular Paper
Evangelos
Anagnostopoulos
Evangelos Anagnostopoulos
Ioannis Z.
Emiris
Ioannis Z. Emiris
Ioannis
Psarros
Ioannis Psarros
10.4230/LIPIcs.SOCG.2015.436
I. Abraham, Y. Bartal, and O. Neiman. Local embeddings of metric spaces. In Proc. 39th ACM Symposium on Theory of Computing, pages 631-640. ACM Press, 2007.
D. Achlioptas. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. J. Comput. Syst. Sci., 66(4):671-687, 2003.
A. Andoni and P. Indyk. E²LSH 0.1 User Manual, Implementation of LSH: E2LSH, http://www.mit.edu/~andoni/LSH, 2005.
http://www.mit.edu/~andoni/LSH
A. Andoni and P. Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Commun. ACM, 51(1):117-122, 2008.
A. Andoni and I. Razenshteyn. Optimal data-dependent hashing for approximate near neighbors. In arXiv:1501.01062, to appear in the Proc. 47th ACM Symp. Theory of Computing, STOC'15, 2015.
S. Arya, G. D. da Fonseca, and D. M. Mount. Approximate polytope membership queries. In Proc. 43rd Annual ACM Symp. Theory of Computing, STOC'11, pages 579-586, 2011.
S. Arya, T. Malamatos, and D. M. Mount. Space-time tradeoffs for approximate nearest neighbor searching. J. ACM, 57(1):1:1-1:54, 2009.
S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman, and A. Y. Wu. An optimal algorithm for approximate nearest neighbor searching fixed dimensions. J. ACM, 45(6):891-923, 1998.
A. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbor. In Proc. 23rd Intern. Conf. Machine Learning, ICML'06, pages 97-104, 2006.
S. Dasgupta and Y. Freund. Random projection trees and low dimensional manifolds. In Proc. 40th Annual ACM Symp. Theory of Computing, STOC'08, pages 537-546, 2008.
S. Dasgupta and A. Gupta. An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Algorithms, 22(1):60-65, 2003.
M. Datar, N. Immorlica, P. Indyk, and V. S. Mirrokni. Locality-sensitive hashing scheme based on p-stable distributions. In Proc. 20th Annual Symp. Computational Geometry, SCG'04, pages 253-262, 2004.
A. Gupta, R. Krauthgamer, and J. R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In Proc. 44th Annual IEEE Symp. Foundations of Computer Science, FOCS'03, pages 534-, 2003.
S. Har-Peled and M. Mendel. Fast construction of nets in low dimensional metrics, and their applications. In Proc. 21st Annual Symp. Computational Geometry, SCG'05, pages 150-158, 2005.
P. Indyk and R. Motwani. Approximate nearest neighbors: Towards removing the curse of dimensionality. In Proc. 30th Annual ACM Symp. Theory of Computing, STOC'98, pages 604-613, 1998.
P. Indyk and A. Naor. Nearest-neighbor-preserving embeddings. ACM Trans. Algorithms, 3(3), 2007.
H. Jegou, M. Douze, and C. Schmid. Product quantization for nearest neighbor search. IEEE Trans. on Pattern Analysis and Machine Intelligence, 33(1):117-128, 2011.
W. B. Johnson and J. Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. Contemporary Mathematics, 26:189-206, 1984.
D. R. Karger and M. Ruhl. Finding nearest neighbors in growth-restricted metrics. In Proc. 34th Annual ACM Symp. Theory of Computing, STOC'02, pages 741-750, 2002.
R. Krauthgamer and J. R. Lee. Navigating nets: Simple algorithms for proximity search. In Proc. 15th Annual ACM-SIAM Symp. Discrete Algorithms, SODA'04, pages 798-807, 2004.
S. Meiser. Point location in arrangements of hyperplanes. Inf. Comput., 106(2):286-303, 1993.
D. M. Mount. ANN programming manual: http://www.cs.umd.edu/~mount/ANN/, 2010.
http://www.cs.umd.edu/~mount/ANN/
R. O'Donnell, Yi Wu, and Y. Zhou. Optimal lower bounds for locality-sensitive hashing (except when q is tiny). ACM Trans. Comput. Theory, 6(1):5:1-5:13, 2014.
R. Panigrahy. Entropy based nearest neighbor search in high dimensions. In Proc. 17th Annual ACM-SIAM Symp. Discrete Algorithms, SODA'06, pages 1186-1195, 2006.
I. Psarros. Low quality embeddings and approximate nearest neighbors, MSc Thesis, Dept. of Informatics & Telecommunications, University of Athens, 2014.
S. Vempala. Randomly-oriented k-d trees adapt to intrinsic dimension. In Proc. Foundations of Software Technology & Theor. Computer Science, pages 48-57, 2012.
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Restricted Isometry Property for General p-Norms
The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an m x n matrix satisfies RIP of order k for the L_p norm, if |Ax|_p is approximately |x|_p for every x with at most k non-zero coordinates.
For every 1 <= p < infty we obtain almost tight bounds on the minimum number of rows m necessary for the RIP property to hold. Prior to this work, only the cases p = 1, 1 + 1/log(k), and 2 were studied. Interestingly, our results show that the case p=2 is a "singularity" point: the optimal number of rows m is Theta(k^p) for all p in [1, infty)-{2}, as opposed to Theta(k) for k=2.
We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.
compressive sensing
dimension reduction
linear algebra
high-dimensional geometry
451-460
Regular Paper
Zeyuan
Allen-Zhu
Zeyuan Allen-Zhu
Rati
Gelashvili
Rati Gelashvili
Ilya
Razenshteyn
Ilya Razenshteyn
10.4230/LIPIcs.SOCG.2015.451
Zeyuan Allen-Zhu, Rati Gelashvili, Silvio Micali, and Nir Shavit. Johnson-Lindenstrauss Compression with Neuroscience-Based Constraints. ArXiv e-prints, abs/1411.5383, November 2014. Also appeared in the Proceedings of the National Academy of Sciences of the USA, vol 111, no 47.
Zeyuan Allen-Zhu, Rati Gelashvili, and Ilya Razenshteyn. Restricted Isometry Property for General p-Norms. ArXiv e-prints, abs/1407.2178v3, February 2015.
Richard Baraniuk, Mark Davenport, Ronald DeVore, and Michael Wakin. A simple proof of the restricted isometry property for random matrices. Constructive Approximation, 28(3):253-263, 2008.
Radu Berinde, Anna C. Gilbert, Piotr Indyk, Howard Karloff, and Martin J. Strauss. Combining geometry and combinatorics: A unified approach to sparse signal recovery. In Proceedings of the 46th Annual Allerton Conference on Communication, Control, and Computing (Allerton 2008), pages 798-805, 2008.
Harry Buhrman, Peter Bro Miltersen, Jaikumar Radhakrishnan, and Srinivasan Venkatesh. Are bitvectors optimal? SIAM Journal on Computing, 31(6):1723-1744, 2002.
Emmanuel Candès, Justin Romberg, and Terence Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 59(8):1207-1223, 2006.
Emmanuel Candès and Terence Tao. Decoding by linear programming. IEEE Transactions on Information Theory, 51(12):4203-4215, 2005.
Emmanuel J. Candès. The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique, 346(9-10):589-592, 2008.
Venkat B. Chandar. Sparse Graph Codes for Compression, Sensing, and Secrecy. PhD thesis, Massachusetts Institute of Technology, 2010.
Khanh Do Ba, Piotr Indyk, Eric Price, and David P. Woodruff. Lower bounds for sparse recovery. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'10), pages 1190-1197, 2010.
David L. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289-1306, 2006.
Anna C. Gilbert and Piotr Indyk. Sparse recovery using sparse matrices. Proceedings of IEEE, 98(6):937-947, 2010.
Anna C. Gilbert, Martin J. Strauss, Joel A. Tropp, and Roman Vershynin. One sketch for all: fast algorithms for compressed sensing. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC 2007), pages 237-246, 2007.
Piotr Indyk and Ilya Razenshteyn. On model-based RIP-1 matrices. In Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP'13), pages 564-575, 2013.
Kumar Joag-Dev and Frank Proschan. Negative association of random variables with applications. Annals of Statistics, 11(1):286-295, 1983.
Raghunandan M. Kainkaryam, Angela Bruex, Anna C. Gilbert, John Schiefelbein, and Peter J. Woolf. poolMC: Smart pooling of mRNA samples in microarray experiments. BMC Bioinformatics, 11(299), 2010.
Rafał Latała. Estimation of moments of sums of independent real random variables. Annals of Probability, 25(3):1502-1513, 1997.
James R. Lee, Manor Mendel, and Assaf Naor. Metric structures in L₁: dimension, snowflakes, and average distortion. European Journal of Combinatorics, 26(8):1180-1190, 2005.
S. Muthukrishnan. Data streams: Algorithms and applications. Foundations and Trends in Theoretical Computer Science, 1(2):117-236, 2005.
Mergen Nachin. Lower bounds on the column sparsity of sparse recovery matrices. undergraduate thesis, MIT, 2010.
A.V. Nagaev. Integral limit theorems taking large deviations into account when Cramér’s condition does not hold. I. Theory of Probability and Its Applications, 14(1):51-64, 1969.
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Jelani Nelson and Huy L. Nguyễn. Sparsity lower bounds for dimensionality reducing maps. In Proceedings of the 45th ACM Symposium on the Theory of Computing (STOC'13), pages 101-110, 2013.
Holger Rauhut. Compressive sensing and structured random matrices. Theoretical foundations and numerical methods for sparse recovery, 9:1-92, 2010.
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Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs
The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has been commonly used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several metrics on the set of Reeb graphs have been proposed. In this paper, we focus on two: the functional distortion distance and the interleaving distance. The former is based on the Gromov-Hausdorff distance, while the latter utilizes the equivalence between Reeb graphs and a particular class of cosheaves. However, both are defined by constructing a near-isomorphism between the two graphs of study. In this paper, we show that the two metrics are strongly equivalent on the space of Reeb graphs. Our result also implies the bottleneck stability for persistence diagrams in terms of the Reeb graph interleaving distance.
Reeb graph
interleaving distance
functional distortion distance
461-475
Regular Paper
Ulrich
Bauer
Ulrich Bauer
Elizabeth
Munch
Elizabeth Munch
Yusu
Wang
Yusu Wang
10.4230/LIPIcs.SOCG.2015.461
Ulrich Bauer, Xiaoyin Ge, and Yusu Wang. Measuring distance between Reeb graphs. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, New York, NY, USA, 2014. ACM.
Silvia Biasotti, Daniela Giorgi, Michela Spagnuolo, and Bianca Falcidieno. Reeb graphs for shape analysis and applications. Theoretical Computer Science, 392(1-3):5-22, February 2008.
Kevin Buchin, Maike Buchin, Marc van Kreveld, Bettina Speckmann, and Frank Staals. Trajectory grouping structure. In Frank Dehne, Roberto Solis-Oba, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures, volume 8037 of Lecture Notes in Computer Science, pages 219-230. Springer Berlin Heidelberg, 2013.
Frédéric Chazal and Jian Sun. Gromov-Hausdorff approximation of filament structure using Reeb-type graph. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 491:491-491:500, New York, NY, USA, 2014. ACM.
Justin Curry. Sheaves, Cosheaves and Applications. PhD thesis, University of Pennsylvania, December 2014.
Vin de Silva, Elizabeth Munch, and Amit Patel. Categorified Reeb graphs, January 2015.
Tamal K. Dey, Fengtao Fan, and Yusu Wang. An efficient computation of handle and tunnel loops via Reeb graphs. ACM Trans. Graph., 32(4):32:1-32:10, July 2013.
Barbara Di Fabio and Claudia Landi. The edit distance for Reeb graphs of surfaces, November 2014. URL: http://arxiv.org/abs/1411.1544.
http://arxiv.org/abs/1411.1544
Harish Doraiswamy and Vijay Natarajan. Output-Sensitive construction of Reeb graphs. Visualization and Computer Graphics, IEEE Transactions on, 18(1):146-159, January 2012.
Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. Amer. Math. Soc., Providence, Rhode Island, 2010.
Francisco Escolano, Edwin R. Hancock, and Silvia Biasotti. Complexity fusion for indexing Reeb digraphs. In Richard Wilson, Edwin Hancock, Adrian Bors, and William Smith, editors, Computer Analysis of Images and Patterns, volume 8047 of Lecture Notes in Computer Science, pages 120-127. Springer Berlin Heidelberg, 2013.
Xiaoyin Ge, Issam I. Safa, Mikhail Belkin, and Yusu Wang. Data skeletonization via Reeb graphs. Advances in Neural Information Processing Systems, 24:837-845, 2011.
William Harvey, Yusu Wang, and Rephael Wenger. A randomized O(m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes. In Proceedings of the Twenty Sixth Annual Symposium on Computational Geometry, SoCG'10, pages 267-276, New York, NY, USA, 2010. ACM.
Franck Hétroy and Dominique Attali. Topological quadrangulations of closed triangulated surfaces using the Reeb graph. Graphical Models, 65(1-3):131-148, May 2003.
Masaki Hilaga, Yoshihisa Shinagawa, Taku Kohmura, and Tosiyasu L. Kunii. Topology matching for fully automatic similarity estimation of 3D shapes. In Proceedings of the 28th annual conference on Computer graphics and interactive techniques, SIGGRAPH'01, pages 203-212, New York, NY, USA, 2001. ACM.
Ernest Michael. Continuous selections II. Annals of Mathematics, 64(3):pp. 562-580, 1956.
Dmitriy Morozov, Kenes Beketayev, and Gunther Weber. Interleaving distance between merge trees. Manuscript, 2013.
Mattia Natali, Silvia Biasotti, Giuseppe Patanè, and Bianca Falcidieno. Graph-based representations of point clouds. Graphical Models, 73(5):151-164, September 2011.
Monica Nicolau, Arnold J. Levine, and Gunnar Carlsson. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings of the National Academy of Sciences, 108(17):7265-7270, 2011.
Salman Parsa. A deterministic O(m log m) time algorithm for the Reeb graph. Discrete & Computational Geometry, 49(4):864-878, 2013.
Georges Reeb. Sur les points singuliers d'une forme de Pfaff complèment intégrable ou d'une fonction numérique. Comptes Rendus de L'Académie ses Séances, 222:847-849, 1946.
Dus̆an Repovs̆ and Pavel V. Semenov. Continuous Selections of Multivalued Mappings. Kluwer Academic Publishers, 1998.
Yoshihisa Shinagawa, Tosiyasu L. Kunii, and Yannick L. Kergosien. Surface coding based on Morse theory. IEEE Comput. Graph. Appl., 11(5):66-78, September 1991.
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.
Zoë Wood, Hugues Hoppe, Mathieu Desbrun, and Peter Schröder. Removing excess topology from isosurfaces. ACM Transactions on Graphics, 23(2):190-208, April 2004.
Yuan Yao, Jian Sun, Xuhui Huang, Gregory R. Bowman, Gurjeet Singh, Michael Lesnick, Leonidas J. Guibas, Vijay S. Pande, and Gunnar Carlsson. Topological methods for exploring low-density states in biomolecular folding pathways. The Journal of Chemical Physics, 130:144115, 2009.
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On Generalized Heawood Inequalities for Manifolds: A Van Kampen-Flores-type Nonembeddability Result
The fact that the complete graph K_5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M), where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2.
Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on surfaces as well as the Van Kampen--Flores theorem.
In the spirit of Kuhnel's conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.
Heawood Inequality
Embeddings
Van Kampen–Flores
Manifolds
476-490
Regular Paper
Xavier
Goaoc
Xavier Goaoc
Isaac
Mabillard
Isaac Mabillard
Pavel
Paták
Pavel Paták
Zuzana
Patáková
Zuzana Patáková
Martin
Tancer
Martin Tancer
Uli
Wagner
Uli Wagner
10.4230/LIPIcs.SOCG.2015.476
K. Appel and W. Haken. Every planar map is four colorable. I. Discharging. Illinois J. Math., 21(3):429-490, 1977.
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U. Brehm and W. Kühnel. 15-vertex triangulations of an 8-manifold. Math. Ann., 294(1):167-193, 1992.
T. K. Dey. On counting triangulations in d dimensions. Comput. Geom., 3(6):315-325, 1993.
A. I. Flores. Über die Existenz n-dimensionaler Komplexe, die nicht in den ℝ^2n topologisch einbettbar sind. Ergeb. Math. Kolloqu., 5:17-24, 1933.
X. Goaoc, P. Paták, Z. Patáková, M. Tancer, and U. Wagner. Bounding Helly numbers via Betti numbers. Preprint, arXiv:1310.4613, 2013.
B. Grünbaum. Imbeddings of simplicial complexes. Comment. Math. Helv., 44:502-513, 1969.
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Comparing Graphs via Persistence Distortion
Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the non-linear structure hidden behind the data. In this paper, we propose a new distance between two finite metric graphs, called the persistence-distortion distance, which draws upon a topological idea. This topological perspective along with the metric space viewpoint provide a new angle to the graph matching problem. Our persistence-distortion distance has two properties not shared by previous methods: First, it is stable against the perturbations of the input graph metrics. Second, it is a continuous distance measure, in the sense that it is defined on an alignment of the underlying spaces of input graphs, instead of merely their nodes. This makes our persistence-distortion distance robust against, for example, different discretizations of the same underlying graph.
Despite considering the input graphs as continuous spaces, that is, taking all points into account, we show that we can compute the persistence-distortion distance in polynomial time. The time complexity for the discrete case where only graph nodes are considered is much faster.
Graph matching
metric graphs
persistence distortion
topological method
491-506
Regular Paper
Tamal K.
Dey
Tamal K. Dey
Dayu
Shi
Dayu Shi
Yusu
Wang
Yusu Wang
10.4230/LIPIcs.SOCG.2015.491
M. Aanjaneya, F. Chazal, D. Chen, M. Glisse, L. Guibas, and D. Morozov. Metric graph reconstruction from noisy data. Int. J. Comput. Geom. Appl., pages 305-325, 2012.
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Ulrich Bauer, Xiaoyin Ge, and Yusu Wang. Measuring distance bewteen Reeb graphs. In Proc. 30th SoCG, pages 464-473, 2014.
D. Burago, Y. Burago, and S. Ivanov. A course in metric geometry. volume 33 of AMS Graduate Studies in Math. American Mathematics Society, 2001.
Frédéric Chazal and Jian Sun. Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph. In Proc. 30th SoCG, pages 491-500, 2014.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, 2007.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Extending persistence using Poincaré and Lefschetz duality. Foundations of Computational Mathematics, 9(1):79-103, 2009.
David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov. Vines and vineyards by updating persistence in linear time. In Proc. 22nd SoCG, pages 119-126, 2006.
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T. K. Dey and R. Wenger. Stability of critical points with interval persistence. Discrete Comput. Geom., 38:479-512, 2007.
Tamal K. Dey, Dayu Shi, and Yusu Wang. Comparing graphs via persistence distortion, 2015. arXiv:1503.07414.
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H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511-533, 2002.
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N. Hu, R.M. Rustamov, and L. Guibas. Graph Matching with Anchor Nodes: A Learning Approach. In IEEE Conference on CVPR, pages 2906-2913, 2013.
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M. Leordeanu, M. Hebert, and R. Sukthankar. An Integer Projected Fixed Point Method for Graph Matching and MAP Inference. In Proc. NIPS. Springer, December 2009.
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Facundo Mémoli. On the use of Gromov-Hausdorff Distances for Shape Comparison. In Symposium on Point Based Graphics, pages 81-90, 2007.
U. Ozertem and D. Erdogmus. Locally defined principal curves and surfaces. Journal of Machine Learning Research, 12:1249-1286, 2011.
T. Sousbie, C. Pichon, and H. Kawahara. The persistent cosmic web and its filamentary structure - II. Illustrations. Mon. Not. R. Astron. Soc., 414:384-403, 2011.
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Zhiping Zeng, Anthony K. H. Tung, Jianyong Wang, Jianhua Feng, and Lizhu Zhou. Comparing stars: On approximating graph edit distance. Proc. VLDB Endow., 2(1):25-36, August 2009.
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Bounding Helly Numbers via Betti Numbers
We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b,d) such that the following holds. If F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). These topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent interest.
Helly-type theorem
Ramsey’s theorem
Embedding of simplicial complexes
Homological almost-embedding
Betti numbers
507-521
Regular Paper
Xavier
Goaoc
Xavier Goaoc
Pavel
Paták
Pavel Paták
Zuzana
Patáková
Zuzana Patáková
Martin
Tancer
Martin Tancer
Uli
Wagner
Uli Wagner
10.4230/LIPIcs.SOCG.2015.507
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X. Goaoc, P. Paták, Z. Patáková, M. Tancer, and U. Wagner. Bounding Helly numbers via Betti numbers. URL: http://arxiv.org/abs/1310.4613.
http://arxiv.org/abs/1310.4613
A. Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, UK, 2002.
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H. Maehara. Helly-type theorems for spheres. Discrete & Computational Geometry, 4(1):279-285, 1989.
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S. A. Melikhov. The van Kampen obstruction and its relatives. Proc. Steklov Inst. Math., 266(1):142-176, 2009.
L. Montejano. A new topological Helly theorem and some transversal results. Discrete & Computational Geometry, 52(2):390-398, 2014.
J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, Menlo Park, CA, 1984.
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R. I. Soare. Computability theory and differential geometry. Bull. Symbolic Logic, 10(4):457-486, 2004.
M. Tancer. Intersection patterns of convex sets via simplicial complexes: A survey. In J. Pach, editor, Thirty Essays on Geometric Graph Theory, pages 521-540. Springer New York, 2013.
E. R. van Kampen. Komplexe in euklidischen Räumen. Abh. Math. Sem. Univ. Hamburg, 9:72-78, 1932.
U. Wagner. Minors in random and expanding hypergraphs. In Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG), pages 351 - 360, 2011.
R. Wenger. Helly-type theorems and geometric transversals. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete & Computational Geometry, chapter 4, pages 73-96. CRC Press LLC, Boca Raton, FL, 2nd edition, 2004.
W.-T. Wu. On the realization of complexes in euclidean spaces. I, II, III. Acta Math. Sinica (English transl. of I and III in Sci. Sinica), 5:505-552, 1955.
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Polynomials Vanishing on Cartesian Products: The Elekes-Szabó Theorem Revisited
Let F in Complex[x,y,z] be a constant-degree polynomial, and let A,B,C be sets of complex numbers with |A|=|B|=|C|=n. We show that F vanishes on at most O(n^{11/6}) points of the Cartesian product A x B x C (where the constant of proportionality depends polynomially on the degree of F), unless F has a special group-related form. This improves a theorem of Elekes and Szabo [ES12], and generalizes a result of Raz, Sharir, and Solymosi [RSS14a]. The same statement holds over R. When A, B, C have different sizes, a similar statement holds, with a more involved bound replacing O(n^{11/6}).
This result provides a unified tool for improving bounds in various Erdos-type problems in combinatorial geometry, and we discuss several applications of this kind.
Combinatorial geometry
incidences
polynomials
522-536
Regular Paper
Orit E.
Raz
Orit E. Raz
Micha
Sharir
Micha Sharir
Frank
de Zeeuw
Frank de Zeeuw
10.4230/LIPIcs.SOCG.2015.522
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Bisector Energy and Few Distinct Distances
We introduce the bisector energy of an n-point set P in the real plane, defined as the number of quadruples (a,b,c,d) from P such that a and b determine the same perpendicular bisector as c and d. If no line or circle contains M(n) points of P, then we prove that the bisector energy is O(M(n)^{2/5}n^{12/5} + M(n)n^2). We also prove the lower bound M(n)n^2, which matches our upper bound when M(n) is large. We use our upper bound on the bisector energy to obtain two rather different results:
(i) If P determines O(n / sqrt(log n)) distinct distances, then for any 0 < a < 1/4, either there exists a line or circle that contains n^a points of P, or there exist n^{8/5 - 12a/5} distinct lines that contain sqrt(log n) points of P. This result provides new information on a conjecture of Erdös regarding the structure of point sets with few distinct distances.
(ii) If no line or circle contains M(n) points of P, then the number of distinct perpendicular bisectors determined by P is min{M(n)^{-2/5}n^{8/5}, M(n)^{-1}n^2}). This appears to be the first higher-dimensional example in a framework for studying the expansion properties of polynomials and rational functions over the real numbers, initiated by Elekes and Ronyai.
Combinatorial geometry
distinct distances
incidence geometry
537-552
Regular Paper
Ben
Lund
Ben Lund
Adam
Sheffer
Adam Sheffer
Frank
de Zeeuw
Frank de Zeeuw
10.4230/LIPIcs.SOCG.2015.537
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Incidences between Points and Lines in Three Dimensions
We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R^3, so that no plane contains more than s lines, is O(m^{1/2}n^{3/4} + m^{2/3}n^{1/3}s^{1/3} + m + n) (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between m and n).
This bound, originally obtained by Guth and Katz as a major step in their solution of Erdos's distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth. The present paper presents a different and simpler derivation, with better bounds than those in Guth, and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.
Combinatorial Geometry
Algebraic Geometry
Incidences
The Polynomial Method
553-568
Regular Paper
Micha
Sharir
Micha Sharir
Noam
Solomon
Noam Solomon
10.4230/LIPIcs.SOCG.2015.553
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The Number of Unit-Area Triangles in the Plane: Theme and Variations
We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n^{20/9}), improving the earlier bound O(n^{9/4}) of Apfelbaum and Sharir. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans can be Omega(n^2), for any triple of lines (it is always O(n^2) in this case). (ii) We show that if S is a convex grid of the form A x B, where A, B are convex sets of n^{1/2} real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n^{31/14}) unit-area triangles.
Combinatorial geometry
incidences
repeated configurations
569-583
Regular Paper
Orit E.
Raz
Orit E. Raz
Micha
Sharir
Micha Sharir
10.4230/LIPIcs.SOCG.2015.569
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On the Number of Rich Lines in Truly High Dimensional Sets
We prove a new upper bound on the number of r-rich lines (lines with at least r points) in a 'truly' d-dimensional configuration of points v_1,...,v_n over the complex numbers. More formally, we show that, if the number of r-rich lines is significantly larger than n^2/r^d then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor r^d can be replaced with a tight r^{d+1}. If true, this would generalize the classic Szemeredi-Trotter theorem which gives a bound of n^2/r^3 on the number of r-rich lines in a planar configuration. This conjecture was shown to hold in R^3 in the seminal work of Guth and Katz and was also recently proved over R^4 (under some additional restrictions) by Solomon and Sharir. For the special case of arithmetic progressions (r collinear points that are evenly distanced) we give a bound that is tight up to lower order terms, showing that a d-dimensional grid achieves the largest number of r-term progressions.
The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree r-2 Veronese embedding takes r-collinear points to r linearly dependent images. Hence, each collinear r-tuple of points, gives us a dependent r-tuple of images. We then use the design-matrix method of Barak et al. to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.
Incidences
Combinatorial Geometry
Designs
Polynomial Method
Additive Combinatorics
584-598
Regular Paper
Zeev
Dvir
Zeev Dvir
Sivakanth
Gopi
Sivakanth Gopi
10.4230/LIPIcs.SOCG.2015.584
B. Barak, Z. Dvir, A. Wigderson, and A. Yehudayoff. Fractional Sylvester-Gallai theorems. Proceedings of the National Academy of Sciences, 2012.
B. Barak, Z. Dvir, A. Yehudayoff, and A. Wigderson. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC '11, pages 519-528, New York, NY, USA, 2011. ACM.
Saugata Basu and Martin Sombra. Polynomial partitioning on varieties and point-hypersurface incidences in four dimensions. arXiv preprint arXiv:1406.2144, 2014.
Z. Dvir. Incidence Theorems and Their Applications. Foundations and Trends in Theoretical Computer Science, 6(4):257-393, 2012.
Z. Dvir, S. Saraf, and A. Wigderson. Improved rank bounds for design matrices and a new proof of Kelly’s theorem. Forum of Mathematics, Sigma, 2, 10 2014.
L. Guth and N. Katz. Algebraic methods in discrete analogs of the Kakeya problem. Advances in Mathematics, 225(5):2828 - 2839, 2010.
Larry Guth and Nets Hawk Katz. On the Erdős distinct distances problem in the plane. Annals of Mathematics, 181(1):155-190, 2015.
Marton Hablicsek and Zachary Scherr. On the number of rich lines in high dimensional real vector spaces. arXiv preprint arXiv:1412.7025, 2014.
J. Kollar. Szemerédi-Trotter-type theorems in dimension 3. arXiv:1405.2243, 2014.
Izabella Laba and József Solymosi. Incidence theorems for pseudoflats. Discrete & Computational Geometry, 37(2):163-174, 2007.
M. Rudnev. On the number of incidences between planes and points in three dimensions. arXiv:1407.0426v2, 2014.
J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980.
Micha Sharir, Adam Sheffer, and Joshua Zahl. Improved bounds for incidences between points and circles. In Proceedings of the Twenty-ninth Annual Symposium on Computational Geometry, SoCG '13, pages 97-106, New York, NY, USA, 2013. ACM.
Micha Sharir and Noam Solomon. Incidences between points and lines in ℝ⁴. arXiv preprint arXiv:1411.0777, 2014.
József Solymosi and VH Vu. Distinct distances in high dimensional homogeneous sets. Contemporary Mathematics, 342:259-268, 2004.
József Solymosi and Terence Tao. An incidence theorem in higher dimensions. Discrete and Computational Geometry, 48(2):255-280, 2012.
E. Szemerédi and W. T. Trotter. Extremal problems in discrete geometry. Combinatorica, 3(3):381-392, 1983.
C. Toth. The Szemerédi-Trotter theorem in the complex plane. arXiv:math/0305283v4, 2003.
Joshua Zahl. A Szemerédi-Trotter type theorem in ℝ⁴. CoRR, abs/1203.4600, 2012.
Joshua Zahl. A note on rich lines in truly high dimensional sets. arXiv preprint arXiv:1503.01729, 2015.
R. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, pages 216-226. Springer-Verlag, 1979.
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Realization Spaces of Arrangements of Convex Bodies
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. On one hand, we show that every combinatorial type can be realized by an arrangement of convex bodies and (under mild assumptions) its realization space is contractible. On the other hand, we prove a universality theorem that says that the restriction of the realization space to arrangements of convex polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.
Oriented matroids
Convex sets
Realization spaces
Mnev’s universality theorem
599-614
Regular Paper
Michael Gene
Dobbins
Michael Gene Dobbins
Andreas
Holmsen
Andreas Holmsen
Alfredo
Hubard
Alfredo Hubard
10.4230/LIPIcs.SOCG.2015.599
Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler. Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1999.
Raghavan Dhandapani, Jacob E. Goodman, Andreas Holmsen, and Richard Pollack. Interval sequences and the combinatorial encoding of planar families of pairwise disjoint convex sets. Rev. Roum. Math. Pures Appl, 50(5-6):537-553, 2005.
Michael Gene Dobbins, Andreas Holmsen, and Alfredo Hubard. Regular systems of paths and families of convex sets in convex position. To appear in Transactions of the AMS.
Michael Gene Dobbins, Andreas Holmsen, and Alfredo Hubard. The Erdős-Szekeres problem for non-crossing convex sets. Mathematika, 60(2):463-484, 2014.
Stefan Felsner and Pavel Valtr. Coding and counting arrangements of pseudolines. Discrete & Computational Geometry, 46(3):405-416, 2011.
Jon Folkman and Jim Lawrence. Oriented matroids. Journal of Combinatorial Theory, Series B, 25(2):199-236, 1978.
Jacob E. Goodman. Proof of a conjecture of Burr, Grünbaum, and Sloane. Discrete Mathematics, 32(1):27-35, 1980.
Jacob E. Goodman and Richard Pollack. On the combinatorial classification of nondegenerate configurations in the plane. Journal of Combinatorial Theory, Series A, 29(2):220-235, 1980.
Jacob E. Goodman and Richard Pollack. Semispaces of configurations, cell complexes of arrangements. Journal of Combinatorial Theory, Series A, 37(3):257-293, 1984.
Jacob E. Goodman and Richard Pollack. Upper bounds for configurations and polytopes in ℝ^d. Discrete & Computational Geometry, 1(1):219-227, 1986.
Jacob E. Goodman and Richard Pollack. The combinatorial encoding of disjoint convex sets in the plane. Combinatorica, 28(1):69-81, 2008.
Helmut Groemer. Geometric applications of Fourier series and spherical harmonics, volume 61 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1996.
Branko Grünbaum. Arrangements and spreads. American Mathematical Society, 1972.
Luc Habert and Michel Pocchiola. Arrangements of double pseudolines. In Proceedings of the 25th Annual Symposium on Computational Geometry, pages 314-323. ACM, 2009.
Alfredo Hubard. Erdős-Szekeres para convexos. Bachelor’s thesis, UNAM, 2005.
Alfredo Hubard, Luis Montejano, Emiliano Mora, and Andrew Suk. Order types of convex bodies. Order, 28(1):121-130, 2011.
Michael Kapovich and John J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41(6):1051-1107, 2002.
Donald E. Knuth. Axioms and hulls, volume 606 of Lecture Notes in Computer Science. Springer-Verlag, 1992.
Friedrich Levi. Die teilung der projektiven ebene durch gerade oder pseudogerade. Ber. Math.-Phys. Kl. Sächs. Akad. Wiss, 78:256-267, 1926.
Nicolai E. Mnev. Varieties of combinatorial types of projective configurations and convex polytopes. Doklady Akademii Nauk SSSR, 283(6):1312-1314, 1985.
Nicolai E. Mnev. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and Geometry: Rohlin seminar, pages 527-543. Springer, 1988.
Mordechai Novick. Allowable interval sequences and line transversals in the plane. Discrete & Computational Geometry, 48(4):1058-1073, 2012.
Mordechai Novick. Allowable interval sequences and separating convex sets in the plane. Discrete & Computational Geometry, 47(2):378-392, 2012.
János Pach and Géza Tóth. Families of convex sets not representable by points. In Algorithms, architectures and information systems security, volume 3, page 43. World Scientific, 2008.
Arnau Padrol and Louis Theran. Delaunay triangulations with disconnected realization spaces. In Proceedings of the 30th Annual Symposium on Computational Geometry, pages 163-170. ACM, 2014.
Jürgen Richter-Gebert. Realization spaces of polytopes, volume 1643 of Lecture Notes in Mathematics. Springer Verlag, 1996.
Gerhard Ringel. Teilungen der ebene durch geraden oder topologische geraden. Mathematische Zeitschrift, 64(1):79-102, 1956.
Peter W. Shor. Stretchability of pseudolines is NP-hard. In Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift, volume 4, pages 531-554. American Mathematical Society, 1991.
Yasuyuki Tsukamoto. New examples of oriented matroids with disconnected realization spaces. Discrete & Computational Geometry, 49(2):287-295, 2013.
Ravi Vakil. Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Inventiones Mathematicae, 164(3):569-590, 2006.
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Computing Teichmüller Maps between Polygons
By the Riemann mapping theorem, one can bijectively map the interior of an n-gon P to that of another n-gon Q conformally (i.e., in an angle preserving manner). However, when this map is extended to the boundary it need not necessarily map the vertices of P to those of Q. For many applications it is important to find the "best" vertex-preserving mapping between two polygons, i.e., one that minimizes the maximum angle distortion (the so-called dilatation). Such maps exist, are unique, and are known as extremal quasiconformal maps or Teichmüller maps.
There are many efficient ways to approximate conformal maps, and the recent breakthrough result by Bishop computes a (1+epsilon)-approximation of the Riemann map in linear time. However, only heuristics have been studied in the case of Teichmüller maps.
We present two results in this paper. One studies the problem in the continuous setting and another in the discrete setting.
In the continuous setting, we solve the problem of finding a finite time procedure for approximating Teichmüller maps. Our construction is via an iterative procedure that is proven to converge in O(poly(1/epsilon)) iterations to a (1+epsilon)-approximation of the Teichmuller map. Our method uses a reduction of the polygon mapping problem to the marked sphere problem, thus solving a more general problem.
In the discrete setting, we reduce the problem of finding an approximation algorithm for computing Teichmüller maps to two basic subroutines, namely, computing discrete 1) compositions and 2) inverses of discretely represented quasiconformal maps. Assuming finite-time solvers for these subroutines we provide a (1+epsilon)-approximation algorithm.
Teichmüller maps
Surface registration
Extremal Quasiconformal maps
Computer vision
615-629
Regular Paper
Mayank
Goswami
Mayank Goswami
Xianfeng
Gu
Xianfeng Gu
Vamsi P.
Pingali
Vamsi P. Pingali
Gaurish
Telang
Gaurish Telang
10.4230/LIPIcs.SOCG.2015.615
L. V. Ahlfors. Lectures on quasiconformal mappings, volume 38 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard.
C. Bishop. Conformal mapping in linear time. Discrete and Comput. Geometry, 44(2):330-428, 2010.
Christopher Bishop. Analysis of conformal and quasiconformal maps. Results from prior NSF support, 2012. URL: http://www.math.sunysb.edu/~bishop/vita/nsf12.pdf.
http://www.math.sunysb.edu/~bishop/vita/nsf12.pdf
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P. Daripa and M. Goswami, 2014. Private communication.
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T. A. Driscoll and L. N. Trefethen. Schwarz-Christoffel Mapping. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, 2002.
T. A. Driscoll and S. A. Vavasis. Numerical conformal mapping using cross-ratios and delaunay triangulation. SIAM J. Sci. Comput, 19:1783-1803, 1998.
D. Gaidashev and D. Khmelev. On numerical algorithms for the solution of a beltrami equation. SIAM Journal on Numerical Analysis, 46(5):2238-2253, 2008.
F. P. Gardiner and N. Lakic. Quasiconformal Teichmüler theory. American Mathematical Society, 1999.
M. Goswami, X. Gu, V. Pingali, and G. Telang. Computing Teichmüller maps between polygons. arXiv:1401.6395 - http://arxiv.org/abs/1401.6395, 2014.
http://arxiv.org/abs/1401.6395
H. Grötzsch. Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhngender Bereiche. Leipz. Ber., 82:69-80, 1930.
X. Gu, Y. Wang, T. F. Chan, P. M. Thompson, and S. T. Yau. Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Transactions on Medical Imaging, 23(7):949-958, 2004.
X. Gu and S.T. Yau. Global surface conformal parameterization. In Symposium on Geometry Processing (SGP'03), volume 43, pages 127-137, 2003.
J. H. Hubbard. Teichmüller theory and applications to geometry, topology, and dynamics. Matrix Editions, 2006.
Ldmm - the large deformation diffeomorphic metric mapping tool. URL: http://cis.jhu.edu/software/lddmm-volume/tutorial.php.
http://cis.jhu.edu/software/lddmm-volume/tutorial.php
L. Lui, K. Lam, S. Yau, and X. Gu. Teichmüller Mapping (T-Map) and Its Applications to Landmark Matching Registration. SIAM Journal on Imaging Sciences, 7(1):391-426, 2014.
L. M. Lui, Xianfeng Gu, and Shing Tung Yau. Convergence of an iterative // algorithm for Teichmüller maps via generalized harmonic maps. arXiv:1307.2679 - http://arxiv.org/abs/1307.2679, 2014.
http://arxiv.org/abs/1307.2679
Lok Ming Lui, Tsz Wai Wong, Wei Zeng, Xianfeng Gu, Paul M. Thompson, Tony F. Chan, and Shing-Tung Yau. Optimization of surface registrations using beltrami holomorphic flow. Journal of Scientific Computing, 50(3):557-585, 2012.
P.M. Pardalos and M.G.C. Resende. Handbook of applied optimization, volume 1. Oxford University Press New York, 2002.
J. Ruppert. A delaunay refinement algorithm for quality 2-dimensional mesh generation. J. Algorithms, 18(3):548-585, 1995.
O. Teichmüller. Extremale quasikonforme Abbildungen und quadratische Differentiale. Preuss. Akad. Math.-Nat., 1, 1940.
O. Teichmüller. Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen. Preuss. Akad. Math.-Nat., 4, 1943.
Y. Wang, M. Gupta, S. Zhang, S. Wang, X. Gu, D. Samaras, and P. Huang. High Resolution Tracking of Non-Rigid Motion of Densely Sampled 3D Data Using Harmonic Maps. International Journal of Computer Vision, 76(3):283-300, 2008.
Y. Wang, J. Shi, X. Yin, X. Gu, T. F. Chan, S. T. Yau, A. W. Toga, and P. M. Thompson. Brain surface conformal parameterization with the ricci flow. IEEE Transactions on Medical Imaging, 31(2):251-264, 2012.
O. Weber, A. Myles, and D. Zorin. Computing extremal quasiconformal maps. Comp. Graph. Forum, 31(5):1679-1689, 2012.
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On-line Coloring between Two Lines
We study on-line colorings of certain graphs given as intersection graphs of objects "between two lines", i.e., there is a pair of horizontal lines such that each object of the representation is a connected set contained in the strip between the lines and touches both. Some of the graph classes admitting such a representation are permutation graphs (segments), interval graphs (axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability graphs (simple curves). We present an on-line algorithm coloring graphs given by convex sets between two lines that uses O(w^3) colors on graphs with maximum clique size w.
In contrast intersection graphs of segments attached to a single line may force any on-line coloring algorithm to use an arbitrary number of colors even when w=2.
The left-of relation makes the complement of intersection graphs of objects between two lines into a poset. As an aside we discuss the relation of the class C of posets obtained from convex sets between two lines with some other classes of posets: all 2-dimensional posets and all posets of height 2 are in C but there is a 3-dimensional poset of height 3 that does not belong to C.
We also show that the on-line coloring problem for curves between two lines is as hard as the on-line chain partition problem for arbitrary posets.
intersection graphs
cocomparability graphs
on-line coloring
630-641
Regular Paper
Stefan
Felsner
Stefan Felsner
Piotr
Micek
Piotr Micek
Torsten
Ueckerdt
Torsten Ueckerdt
10.4230/LIPIcs.SOCG.2015.630
Bartłomiej Bosek, Stefan Felsner, Kamil Kloch, Tomasz Krawczyk, Grzegorz Matecki, and Piotr Micek. On-line chain partitions of orders: a survey. Order, 29(1):49-73, 2012.
Bartłomiej Bosek, Henry A. Kierstead, Tomasz Krawczyk, Grzegorz Matecki, and Matthew E Smith. An improved subexponential bound for on-line chain partitioning. arXiv preprint arXiv:1410.3247, 2014.
Bartłomiej Bosek and Tomasz Krawczyk. The sub-exponential upper bound for on-line chain partitioning. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science FOCS 2010, pages 347-354. IEEE Computer Soc., Los Alamitos, CA, 2010.
Bartlomiej Bosek, Tomasz Krawczyk, and Edward Szczypka. First-Fit algorithm for the on-line chain partitioning problem. SIAM J. Discrete Math., 23(4):1992-1999, 2010.
Derek G. Corneil and P. A. Kamula. Extensions of permutation and interval graphs. Congr. Numer., 58:267-275, 1987. Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Fla., 1987).
Ido Dagan, Martin Charles Golumbic, and Ron Yair Pinter. Trapezoid graphs and their coloring. Discrete Appl. Math., 21(1):35-46, 1988.
Vida Dujmović, Gwenaël Joret, and David R. Wood. An improved bound for First-Fit on posets without two long incomparable chains. SIAM J. Discrete Math., 26(3):1068-1075, 2012.
Thomas Erlebach and Jiri Fiala. On-line coloring of geometric intersection graphs. Comput. Geom., 23(2):243-255, 2002.
Martin Charles Golumbic and Clyde L. Monma. A generalization of interval graphs with tolerances. In Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982), volume 35, pages 321-331, 1982.
Martin Charles Golumbic, Doron Rotem, and Jorge Urrutia. Comparability graphs and intersection graphs. Discrete Math., 43(1):37-46, 1983.
Henry A. Kierstead, George F. McNulty, and William T. Trotter, Jr. A theory of recursive dimension for ordered sets. Order, 1(1):67-82, 1984.
Henry A. Kierstead, Stephen G. Penrice, and William T. Trotter, Jr. On-line coloring and recursive graph theory. SIAM J. Discrete Math., 7(1):72-89, 1994.
Henry A. Kierstead, Stephen G. Penrice, and William T. Trotter, Jr. On-line and First-Fit coloring of graphs that do not induce P₅. SIAM J. Discrete Math., 8(4):485-498, 1995.
Henry A. Kierstead and Karin R. Saoub. First-Fit coloring of bounded tolerance graphs. Discrete Appl. Math., 159(7):605-611, 2011.
Henry A. Kierstead and William T. Trotter, Jr. An extremal problem in recursive combinatorics. In Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Vol. II (Baton Rouge, La., 1981), volume 33, pages 143-153, 1981.
Seog-Jin Kim, Alexandr Kostochka, and Kittikorn Nakprasit. On the chromatic number of intersection graphs of convex sets in the plane. Electron. J. Combin., 11(1):R52, 2004.
Lásló Lovász. Perfect graphs. In Selected topics in graph theory, 2, pages 55-87. Academic Press, London, 1983.
George Mertzios. The recognition of simple-triangle graphs and of linear-interval orders is polynomial. In Proceedings of the 21st European Symposium on Algorithms (ESA), Sophia Antipolis, France, September 2013, pp. 719-730.
Sriram V. Pemmaraju, Rajiv Raman, and Kasturi Varadarajan. Buffer minimization using max-coloring. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 562-571. ACM, New York, 2004.
Alexandre Rok and Bartosz Walczak. Outerstring graphs are χ-bounded. In Siu-Wing Cheng and Olivier Devillers, editors, 30th Annual Symposium on Computational Geometry (SoCG 2014), pages 136-143. ACM, New York, 2014.
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Building Efficient and Compact Data Structures for Simplicial Complexes
The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the Simplex Tree while retaining its functionalities. In addition, we propose two new data structures called Maximal Simplex Tree (MxST) and Simplex Array List (SAL). We analyze the compressed Simplex Tree, the Maximal Simplex Tree, and the Simplex Array List under various settings.
Simplicial complex
Compact data structures
Automaton
NP-hard
642-657
Regular Paper
Jean-Daniel
Boissonnat
Jean-Daniel Boissonnat
Karthik C.
S.
Karthik C. S.
Sébastien
Tavenas
Sébastien Tavenas
10.4230/LIPIcs.SOCG.2015.642
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Shortest Path to a Segment and Quickest Visibility Queries
We show how to preprocess a polygonal domain with a fixed starting point s in order to answer efficiently the following queries: Given a point q, how should one move from s in order to see q as soon as possible? This query resembles the well-known shortest-path-to-a-point query, except that the latter asks for the fastest way to reach q, instead of seeing it. Our solution methods include a data structure for a different generalization of shortest-path-to-a-point queries, which may be of independent interest: to report efficiently a shortest path from s to a query segment in the domain.
path planning
visibility
query structures and complexity
persistent data structures
continuous Dijkstra
658-673
Regular Paper
Esther M.
Arkin
Esther M. Arkin
Alon
Efrat
Alon Efrat
Christian
Knauer
Christian Knauer
Joseph S. B.
Mitchell
Joseph S. B. Mitchell
Valentin
Polishchuk
Valentin Polishchuk
Günter
Rote
Günter Rote
Lena
Schlipf
Lena Schlipf
Topi
Talvitie
Topi Talvitie
10.4230/LIPIcs.SOCG.2015.658
Esther M. Arkin, Joseph S. B. Mitchell, and Subhash Suri. Optimal link path queries in a simple polygon. In Proc. 3rd Ann. ACM-SIAM Symp. Discrete Algorithms (SODA'92), pages 269-279, 1992.
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Francisc Bungiu, Michael Hemmer, John Hershberger, Kan Huang, and Alexander Kröller. Efficient computation of visibility polygons. In 30th Europ. Workshop on Comput. Geom. (EuroCG'14), 2014.
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CGAL. Computational Geometry Algorithms Library. URL: http://www.cgal.org.
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Bernard Chazelle, Herbert Edelsbrunner, Michelangelo Grigni, Leonidas Guibas, John Hershberger, Micha Sharir, and Jack Snoeyink. Ray shooting in polygons using geodesic triangulations. In Javier Leach Albert, Burkhard Monien, and Mario Rodríguez Artalejo, editors, Automata, Languages and Programming (ICALP), volume 510 of Lecture Notes in Computer Science, pages 661-673. Springer, 1994.
Danny Z. Chen and Haitao Wang. Visibility and ray shooting queries in polygonal domains. In Proc. 13th Int. Conf. Algorithms Data Struct. (WADS'13), LNCS, pages 244-255, 2013.
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Moshe Dror, Alon Efrat, Anna Lubiw, and Joseph S. B. Mitchell. Touring a sequence of polygons. In Proc. 35th Symposium on Theory of Computing (STOC'03), pages 473-482, 2003.
Adrian Dumitrescu and Csaba D. Tóth. Watchman tours for polygons with holes. Comput. Geom. Theory Appl., 45(7):326-333, 2012.
S. Eriksson-Bique, J. Hershberger, V. Polishchuk, B. Speckmann, S. Suri, T. Talvitie, K. Verbeek, and H. Yıldız. Geometric k shortest paths. In Sanjeev Khanna, editor, Proc. 26th Ann. ACM-SIAM Symp. Discrete Algorithms, (SODA'15), pages 1616-1625. SIAM, 2015.
Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Computational Geometry: Theory and Applications, 5:165-185, 1995.
Subir Ghosh. Visibility Algorithms in the Plane. Cambridge University Press, 2007.
Subir Kumar Ghosh and David M. Mount. An output-sensitive algorithm for computing visibility graphs. SIAM J. Comput., 20(5):888-910, 1991.
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Allan Gr\danskOnlund and Seth Pettie. Threesomes, degenerates, and love triangles. In Proc. 55th Ann. Sympos. Found. Comput. Sci. (FOCS'14), pages 621-630. IEEE, 2014.
Leonidas J. Guibas, J. Hershberger, D. Leven, Micha Sharir, and R. E. Tarjan. Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2:209-233, 1987.
Leonidas J. Guibas, Rajeev Motwani, and Prabhakar Raghavan. The robot localization problem. SIAM J. Comput., 26(4):1120-1138, August 1997.
Olaf Andrew Hall-Holt. Kinetic Visibility. PhD thesis, Stanford University, 2002.
D. Harel and R. E. Tarjan. Fast algorithms for finding nearest common ancestors. SIAM J. Comput., 13(2):338-355, 1984.
P. J. Heffernan and Joseph S. B. Mitchell. An optimal algorithm for computing visibility in the plane. SIAM J. Comput., 24(1):184-201, 1995.
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Lin Lu, Chenglei Yang, and Jiaye Wang. Point visibility computing in polygons with holes. Journal of Information & Computational Science, 8(16):4165-4173, 2011.
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Trajectory Grouping Structure under Geodesic Distance
In recent years trajectory data has become one of the main types of geographic data, and hence algorithmic tools to handle large quantities of trajectories are essential. A single trajectory is typically represented as a sequence of time-stamped points in the plane. In a collection of trajectories one wants to detect maximal groups of moving entities and their behaviour (merges and splits) over time. This information can be summarized in the trajectory grouping structure.
Significantly extending the work of Buchin et al. [WADS 2013] into a realistic setting, we show that the trajectory grouping structure can be computed efficiently also if obstacles are present and the distance between the entities is measured by geodesic distance. We bound the number of critical events: times at which the distance between two subsets of moving entities is exactly epsilon, where epsilon is the threshold distance that determines whether two entities are close enough to be in one group. In case the n entities move in a simple polygon along trajectories with tau vertices each we give an O(tau n^2) upper bound, which is tight in the worst case. In case of well-spaced obstacles we give an O(tau(n^2 + m lambda_4(n))) upper bound, where m is the total complexity of the obstacles, and lambda_s(n) denotes the maximum length of a Davenport-Schinzel sequence of n symbols of order s. In case of general obstacles we give an O(tau min(n^2 + m^3 lambda_4(n), n^2m^2)) upper bound. Furthermore, for all cases we provide efficient algorithms to compute the critical events, which in turn leads to efficient algorithms to compute the trajectory grouping structure.
moving entities
trajectories
grouping
computational geometry
674-688
Regular Paper
Irina
Kostitsyna
Irina Kostitsyna
Marc
van Kreveld
Marc van Kreveld
Maarten
Löffler
Maarten Löffler
Bettina
Speckmann
Bettina Speckmann
Frank
Staals
Frank Staals
10.4230/LIPIcs.SOCG.2015.674
Marc Benkert, Joachim Gudmundsson, Florian Hübner, and Thomas Wolle. Reporting flock patterns. Computational Geometry, 41(3):111-125, 2008.
Kevin Buchin, Maike Buchin, Marc van Kreveld, Bettina Speckmann, and Frank Staals. Trajectory grouping structure. In Proc. 2013 WADS Algorithms and Data Structures Symposium, LNCS, pages 219-230. Springer, 2013.
Maike Buchin, Somayeh Dodge, and Bettina Speckmann. Context-aware similarity of trajectories. In Geographic Information Science, volume 7478 of LNCS, pages 43-56. Springer, 2012.
Maike Buchin, Anne Driemel, and Bettina Speckmann. Computing the Fréchet distance with shortcuts is NP-hard. In Symposium on Computational Geometry, page 367. ACM, 2014.
Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete Comput. Geom., 6(5):485-524, 1991.
Herbert Edelsbrunner and John L. Harer. Computational Topology - an introduction. American Mathematical Society, 2010.
Joachim Gudmundsson and Marc van Kreveld. Computing longest duration flocks in trajectory data. In Proc. 14th ACM International Symposium on Advances in Geographic Information Systems, GIS'06, pages 35-42. ACM, 2006.
Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989.
John Hershberger and Subhash Suri. An Optimal Algorithm for Euclidean Shortest Paths in the Plane. SIAM Journal on Computing, 28(6):2215-2256, 1999.
Hoyoung Jeung, Man Lung Yiu, Xiaofang Zhou, Christian S. Jensen, and Heng Tao Shen. Discovery of convoys in trajectory databases. PVLDB, 1:1068-1080, 2008.
Panos Kalnis, Nikos Mamoulis, and Spiridon Bakiras. On discovering moving clusters in spatio-temporal data. In Advances in Spatial and Temporal Databases, volume 3633 of LNCS, pages 364-381. Springer, 2005.
Patrick Laube, Marc van Kreveld, and Stephan Imfeld. Finding REMO - detecting relative motion patterns in geospatial lifelines. In Developments in Spatial Data Handling, pages 201-215. Springer, 2005.
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Salman Parsa. A deterministic O(m log m) time algorithm for the Reeb graph. In Proc. 28th ACM Symposium on Computational Geometry, pages 269-276, 2012.
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From Proximity to Utility: A Voronoi Partition of Pareto Optima
We present an extension of Voronoi diagrams where not only the distance to the site is taken into account when considering which site the client is going to use, but additional attributes (i.e., prices or weights) are also considered. A cell in this diagram is then the loci of all clients that consider the same set of sites to be relevant. In particular, the precise site a client might use from this candidate set depends on parameters that might change between usages, and the candidate set lists all of the relevant sites. The resulting diagram is significantly more expressive than Voronoi diagrams, but naturally has the drawback that its complexity, even in the plane, might be quite high. Nevertheless, we show that if the attributes of the sites are drawn from the same distribution (note that the locations are fixed), then the expected complexity of the candidate diagram is near linear. To this end, we derive several new technical results, which are of independent interest.
Voronoi diagrams
expected complexity
backward analysis
Pareto optima
candidate diagram
Clarkson-Shor technique
689-703
Regular Paper
Hsien-Chih
Chang
Hsien-Chih Chang
Sariel
Har-Peled
Sariel Har-Peled
Benjamin
Raichel
Benjamin Raichel
10.4230/LIPIcs.SOCG.2015.689
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 Sympos. Principles Database Syst.\CNFPODS, pages 115-126, 2013.
P. K. Agarwal, S. Har-Peled, H. Kaplan, and M. Sharir. Union of random minkowski sums and network vulnerability analysis. Discrete Comput. Geom., 52(3):551-582, 2014.
P. K. Agarwal, J. Matoušek, and O. Schwarzkopf. Computing many faces in arrangements of lines and segments. SIAM J. Comput., 27(2):491-505, 1998.
F. Aurenhammer, R. Klein, and D.-T. Lee. Voronoi Diagrams and Delaunay Triangulations. World Scientific, 2013.
F. Aurenhammer and O. Schwarzkopf. A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams. Internat. J. Comput. Geom. Appl., pages 363-381, 1992.
Z.-D. Bai, L. Devroye, H.-K. Hwang, and T.-H. Tsai. Maxima in hypercubes. Random Struct. Alg., 27(3):290-309, 2005.
I. Bárány and M. Reitzner. On the variance of random polytopes. Adv. Math., 225(4):1986-2001, 2010.
I. Bárány and M. Reitzner. Poisson polytopes. Annals. Prob., 38(4):1507-1531, 2010.
M. de Berg, O. Cheong, M. van Kreveld, and M. H. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd edition, 2008.
S. Börzsönyi, D. Kossmann, and K. Stocker. The skyline operator. In Proc. 17th IEEE Int. Conf. Data Eng., pages 421-430, 2001.
H.-C. Chang, S. Har-Peled, and B. Raichel. From proximity to utility: A Voronoi partition of Pareto optima. CoRR, abs/1404.3403, 2014.
B. Chazelle and J. Friedman. A deterministic view of random sampling and its use in geometry. Combinatorica, 10(3):229-249, 1990.
K. L. Clarkson, K. Mehlhorn, and R. Seidel. Four results on randomized incremental constructions. Comput. Geom. Theory Appl., 3(4):185-212, 1993.
K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4:387-421, 1989.
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S. Har-Peled and B. Raichel. On the expected complexity of randomly weighted Voronoi diagrams. In Proc. 30th Annu. Sympos. Comput. Geom.\CNFSoCG, pages 232-241, 2014.
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H.-K. Hwang, T.-H. Tsai, and W.-M. Chen. Threshold phenomena in k-dominant skylines of random samples. SIAM J. Comput., 42(2):405-441, 2013.
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M. Sharir. The Clarkson-Shor technique revisited and extended. Comb., Prob. & Comput., 12(2):191-201, 2003.
M. Sharir and P. K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York, 1995.
W. Weil and J. A. Wieacker. Stochastic geometry. In P. M. Gruber and J. M. Wills, editors, Handbook of Convex Geometry, volume B, chapter 5.2, pages 1393-1438. North-Holland, 1993.
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Faster Deterministic Volume Estimation in the Oracle Model via Thin Lattice Coverings
We give a 2^{O(n)}(1+1/eps)^n time and poly(n)-space deterministic algorithm for computing a (1+eps)^n approximation to the volume of a general convex body K, which comes close to matching the (1+c/eps)^{n/2} lower bound for volume estimation in the oracle model by Barany and Furedi (STOC 1986, Proc. Amer. Math. Soc. 1988). This improves on the previous results of Dadush and Vempala (Proc. Nat'l Acad. Sci. 2013), which gave the above result only for symmetric bodies and achieved a dependence of 2^{O(n)}(1+log^{5/2}(1/eps)/eps^3)^n.
For our methods, we reduce the problem of volume estimation in K to counting lattice points in K subseteq R^n (via enumeration) for a specially constructed lattice L: a so-called thin covering of space with respect to K (more precisely, for which L + K = R^n and vol_n(K)/det(L) = 2^{O(n)}). The trade off between time and approximation ratio is achieved by scaling down the lattice.
As our main technical contribution, we give the first deterministic 2^{O(n)}-time and poly(n)-space construction of thin covering lattices for general convex bodies. This improves on a recent construction of Alon et al (STOC 2013) which requires exponential space and only works for symmetric bodies. For our construction, we combine the use of the M-ellipsoid from convex geometry (Milman, C.R. Math. Acad. Sci. Paris 1986) together with lattice sparsification and densification techniques (Dadush and Kun, SODA 2013; Rogers, J. London Math. Soc. 1950).
Deterministic Volume Estimation
Convex Geometry
Lattice Coverings of Space
Lattice Point Enumeration
704-718
Regular Paper
Daniel
Dadush
Daniel Dadush
10.4230/LIPIcs.SOCG.2015.704
N. Alon, A. Schraibman, T. Lee, and S. Vempala. The approximate rank of a matrix and its algorithmic applications. In STOC, 2013.
A. Andoni and P. Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. In FOCS, pages 459-468, 2006.
A. Becker, N. Gama, and A. Joux. Solving shortest and closest vector problems: The decomposition approach. Cryptology Eprint. Report 2013/685, 2013.
G. J. Butler. Simultaneous packing and covering in euclidean space. Proceedings of the London Mathematical Society, 25(3):721-735, 1972.
D. Dadush. Integer Programming, Lattice Algorithms, and Deterministic Volume Estimation. PhD thesis, Georgia Institute of Technology, 2012.
D. Dadush and G. Kun. Lattice sparsification and the approximate closest vector problem. In SODA, 2013.
D. Dadush, C. Peikert, and S. Vempala. Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In FOCS, 2011.
D. Dadush and S. Vempala. Near-optimal deterministic algorithms for volume computation via m-ellipsoids. Proceedings of the National Academy of Sciences, 2013.
M. E. Dyer, A. M. Frieze, and R. Kannan. A random polynomial-time algorithm for approximating the volume of convex bodies. J. ACM, 38(1):1-17, 1991. Preliminary version in STOC 1989.
U. Erez, S. Litsyn, and R. Zamir. Lattices which are good for (almost) everything. IEEE Transactions on Information Theory, 51(10):3401-3416, 2005.
Z. Füredi and I. Bárány. Computing the volume is difficult. In STOC, pages 442-447, New York, NY, USA, 1986. ACM.
Z. Füredi and I. Bárány. Approximation of the sphere by polytopes having few vertices. Proceedings of the AMS, 102(3), 1988.
M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, 1988.
P. M. Gruber. Convex and discrete geometry, volume 336. Springer Science & Business Media, 2007.
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G. Hanrot and D. Stehlé. Improved analysis of Kannan’s shortest lattice vector algorithm. In CRYPTO, pages 170-186, Berlin, Heidelberg, 2007. Springer-Verlag.
D. Micciancio and P. Voulgaris. A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations. SIAM Journal on Computing, 42(3):1364-1391, 2013. Preliminary version in STOC 2010.
V. D. Milman. Inégalités de Brunn-Minkowski inverse et applications at la theorie locales des espaces normes. C. R. Math. Acad. Sci. Paris, 302(1):25-28, 1986.
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C. A. Rogers and G. C. Shephard. The difference body of a convex body. Archiv der Mathematik, 8:220-233, 1957.
C. A. Rogers and C. Zong. Covering convex bodies by translates of convex bodies. Mathematika, 44:215-218, 6 1997.
D. B. Yudin and A. S. Nemirovski. Evaluation of the information complexity of mathematical programming problems (in russian). Ekonomika i Matematicheskie Metody, 13(2):3-45, 1976.
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Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings
We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (1992) and the optimal but randomized algorithm of Ramos (1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, (<= k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, epsilon-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (1991) and Chazelle (1993).
shallow cuttings
derandomization
halfspace range reporting
geometric data structures
719-732
Regular Paper
Timothy M.
Chan
Timothy M. Chan
Konstantinos
Tsakalidis
Konstantinos Tsakalidis
10.4230/LIPIcs.SOCG.2015.719
Peyman Afshani and Timothy M. Chan. Optimal halfspace range reporting in three dimensions. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '09, pages 180-186. SIAM, 2009.
Peyman Afshani, Timothy M. Chan, and Konstantinos Tsakalidis. Deterministic rectangle enclosure and offline dominance reporting on the RAM. In Proceedings of the Forty-First International Colloquium on Automata, Languages, and Programming, Part I, ICALP '14, pages 77-88, 2014.
Peyman Afshani and Konstantinos Tsakalidis. Optimal deterministic shallow cuttings for 3d dominance ranges. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 1389-1398. SIAM, 2014.
Pankaj K. Agarwal. Partitioning arrangements of lines I: An efficient deterministic algorithm. Discrete & Computational Geometry, 5(1):449-483, 1990.
Pankaj K. Agarwal. Intersection and Decomposition Algorithms for Planar Arrangements. Cambridge University Press, New York, NY, USA, 1991.
Pankaj K. Agarwal, Boris Aronov, Timothy M. Chan, and Micha Sharir. On levels in arrangements of lines, segments, planes, and triangles. Discrete & Computational Geometry, 19(3):315-331, 1998.
Gerth Stølting Brodal and Riko Jacob. Dynamic planar convex hull with optimal query time. In Proceedings of the Seventh Scandinavian Workshop on Algorithm Theory, SWAT '00, pages 57-70, 2000.
Gerth Stølting Brodal and Riko Jacob. Dynamic planar convex hull. In Proceedings of the Forty-Third Symposium on Foundations of Computer Science, FOCS '02, pages 617-626. IEEE, 2002. Current draft of full paper at https://pwgrp1.inf.ethz.ch/Current/DPCH/Journal/topdown.pdf.
Timothy M. Chan. Random sampling, halfspace range reporting, and construction of (≤ k)-levels in three dimensions. SIAM Journal on Computing, 30(2):561-575, 2000.
Timothy M. Chan. Low-dimensional linear programming with violations. SIAM Journal on Computing, 34(4):879-893, April 2005.
Timothy M. Chan. Three problems about dynamic convex hulls. International Journal of Computational Geometry & Applications, 22(04):341-364, 2012.
Timothy M. Chan, Kasper Green Larsen, and Mihai Pǎtraşcu. Orthogonal range searching on the RAM, revisited. In Proceedings of the Twenty-Seventh Symposium on Computational Geometry, SOCG '11, pages 1-10. ACM, 2011.
Timothy M. Chan and Mihai Pǎtraşcu. Transdichotomous results in computational geometry, I: point location in sublogarithmic time. SIAM J. Comput., 39(2):703-729, 2009.
Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete & Computational Geometry, 9(1):145-158, 1993.
Bernard Chazelle and Joel Friedman. A deterministic view of random sampling and its use in geometry. Combinatorica, 10(3):229-249, 1990.
Kenneth L. Clarkson. New applications of random sampling in computational geometry. Discrete & Computational Geometry, 2:195-222, 1987.
Martin E. Dyer. Linear time algorithms for two- and three-variable linear programs. SIAM Journal on Computing, 13(1):31-45, 1984.
Greg N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing, 16(6):1004-1022, 1987.
Michael T. Goodrich. Planar separators and parallel polygon triangulation. Journal of Computer and System Sciences, 51(3):374-389, 1995.
Sariel Har-Peled, Haim Kaplan, Micha Sharir, and Shakhar Smorodinsky. Epsilon-nets for halfspaces revisited. CoRR, abs/1410.3154, 2014.
David Haussler and Emo Welzl. ε-nets and simplex range queries. Discrete & Computational Geometry, 2(1):127-151, 1987.
Richard J. Lipton and Robert E. Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979.
Jiří Matoušek. Construction of ε-nets. Discrete & Computational Geometry, 5(1):427-448, 1990.
Jiří Matoušek. Cutting hyperplane arrangements. Discrete & Computational Geometry, 6(1):385-406, 1991.
Jiří Matoušek. Reporting points in halfspaces. Computational Geometry, 2(3):169-186, 1992.
Jiří Matoušek. On constants for cuttings in the plane. Discrete & Computational Geometry, 20(4):427-448, 1998.
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Nimrod Megiddo. Linear programming in linear time when the dimension is fixed. Journal of the ACM, 31(1):114-127, 1984.
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Edgar A. Ramos. Deterministic algorithms for 3-d diameter and some 2-d lower envelopes. In Proceedings of the Sixteenth Annual Symposium on Computational Geometry, SoCG '00, pages 290-299. ACM, 2000.
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A Simpler Linear-Time Algorithm for Intersecting Two Convex Polyhedra in Three Dimensions
Chazelle [FOCS'89] gave a linear-time algorithm to compute the intersection of two convex polyhedra in three dimensions. We present a simpler algorithm to do the same.
convex polyhedra
intersection
Dobkin–Kirkpatrick hierarchy
733-738
Regular Paper
Timothy M.
Chan
Timothy M. Chan
10.4230/LIPIcs.SOCG.2015.733
Nancy M. Amato, Michael T. Goodrich, and Edgar A. Ramos. A randomized algorithm for triangulating a simple polygon in linear time. Discrete and Computational Geometry, 26(2):245-265, 2001.
Timothy M. Chan. Deterministic algorithms for 2-d convex programming and 3-d online linear programming. Journal of Algorithms, 27(1):147-166, 1998.
Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete and Computational Geometry, 6:485-524, 1991.
Bernard Chazelle. An optimal algorithm for intersecting three-dimensional convex polyhedra. SIAM Journal on Computing, 21(4):671-696, 1992.
Kenneth L. Clarkson and Peter W. Shor. Application of random sampling in computational geometry, II. Discrete and Computational Geometry, 4:387-421, 1989.
David P. Dobkin and David G. Kirkpatrick. A linear algorithm for determining the separation of convex polyhedra. Journal of Algorithms, 6(3):381-392, 1985.
David P. Dobkin and David G. Kirkpatrick. Determining the separation of preprocessed polyhedra - A unified approach. In Proceedings of the 17th International Colloquium on Automata, Languages and Programming, pages 400-413, 1990.
Martin E. Dyer, Nimrod Megiddo, and Emo Welzl. Linear programming. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, chapter 45. CRC Press, second edition, 2004.
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David G. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12(1):28-35, 1983.
Andrew K. Martin. A simple primal algorithm for intersecting 3-polyhedra in linear time. Master’s thesis, Department of Computer Science, University of British Columbia, 1991. https://circle.ubc.ca/handle/2429/30114 or http://www.cs.ubc.ca/cgi-bin/tr/1991/TR-91-16.
Ketan Mulmuley. Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, Englewood Cliffs, NJ, 1993.
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Michael Ian Shamos and Dan Hoey. Closest-point problems. In Proceedings of the 16th Annual Symposium on Foundations of Computer Science, pages 151-162, 1975.
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Approximability of the Discrete Fréchet Distance
The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds.
In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399.
This raises the question of how well we can approximate the Fréchet distance (of two given d-dimensional point sequences of length n) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be 2^Theta(n). Moreover, we design an alpha-approximation algorithm that runs in time O(n log n + n^2 / alpha), for any alpha in [1, n]. Hence, an n^epsilon-approximation of the Fréchet distance can be computed in strongly subquadratic time, for any epsilon > 0.
Fréchet distance
approximation
lower bounds
Strong Exponential Time Hypothesis
739-753
Regular Paper
Karl
Bringmann
Karl Bringmann
Wolfgang
Mulzer
Wolfgang Mulzer
10.4230/LIPIcs.SOCG.2015.739
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proc. 55th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 434-443, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Proc. 41st Internat. Colloq. Automata Lang. Program. (ICALP), volume 8572 of LNCS, pages 39-51, 2014.
Amir Abboud, Ryan Williams, and Huacheng Yu. More applications of the polynomial method to algorithm design. In Proc. 26th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 218-230, 2015.
Pankaj K. Agarwal, Rinat Ben Avraham, Haim Kaplan, and Micha Sharir. Computing the discrete Fréchet distance in subquadratic time. SIAM J. Comput., 43(2):429-449, 2014.
Helmut Alt. Personal communication. 2012.
Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5(1-2):78-99, 1995.
Karl Bringmann. Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless SETH fails. In Proc. 55th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 661-670, 2014.
Karl Bringmann and Marvin Künnemann. Improved approximation for Fréchet distance on c-packed curves matching conditional lower bounds. arXiv:1408.1340, 2014.
Kevin Buchin, Maike Buchin, Wouter Meulemans, and Wolfgang Mulzer. Four soviets walk the dog - with an application to Alt’s conjecture. In Proc. 25th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 1399-1413, 2014.
Kevin Buchin, Maike Buchin, Rolf van Leusden, Wouter Meulemans, and Wolfgang Mulzer. Computing the Fréchet distance with a retractable leash. In Proc. 21st Annu. European Sympos. Algorithms (ESA), pages 241-252, 2013.
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.
Thomas Eiter and Heikki Mannila. Computing Discrete Fréchet Distance. Technical Report CD-TR 94/64, Christian Doppler Laboratory, 1994.
Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Comput. Geom. Theory Appl., 5(3):165-185, 1995.
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Allan Grønlund and Seth Pettie. Threesomes, degenerates, and love triangles. In Proc. 55th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 621-630, 2014.
Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. System Sci., 62(2):367-375, 2001.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity. J. Comput. System Sci., 63(4):512-530, 2001.
Mihai Pătraşcu and Ryan Williams. On the possibility of faster SAT algorithms. In Proc. 21st Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 1065-1075, 2010.
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Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoret. Comput. Sci., 348(2):357-365, 2005.
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The Hardness of Approximation of Euclidean k-Means
The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R^d, and the goal is to choose k center points in R^d so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general k and a (1+c)-approximation which runs in time poly(n) exp(k/c). At the other extreme, the only known computational complexity result for this problem is NP-hardness [Aloise et al.'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in R^d can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all k, d.
In this paper we provide the first hardness of approximation for the Euclidean k-means problem. Concretely, we show that there exists a constant c > 0 such that it is NP-hard to approximate the k-means objective to within a factor of (1+c). We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform G, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph H, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.
Euclidean k-means
Hardness of Approximation
Vertex Cover
754-767
Regular Paper
Pranjal
Awasthi
Pranjal Awasthi
Moses
Charikar
Moses Charikar
Ravishankar
Krishnaswamy
Ravishankar Krishnaswamy
Ali Kemal
Sinop
Ali Kemal Sinop
10.4230/LIPIcs.SOCG.2015.754
Daniel Aloise, Amit Deshpande, Pierre Hansen, and Preyas Popat. NP-hardness of Euclidean sum-of-squares clustering. Machine Learning, 75(2):245-248, 2009.
Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ron M. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Transactions on Information Theory, 38(2):509-516, 1992.
Noga Alon and Joel Spencer. The Probabilistic Method. John Wiley, 1992.
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 the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 106-113, 1998.
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Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM J. Comput., 33(3):544-562, 2004.
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Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals of Mathematics, 162(1):439-485, 2005.
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Dan Feldman, Morteza Monemizadeh, and Christian Sohler. A PTAS for k-means clustering based on weak coresets. In Proceedings of the 23rd ACM Symposium on Computational Geometry, Gyeongju, South Korea, June 6-8, 2007, pages 11-18, 2007.
Venkatesan Guruswami and Piotr Indyk. Embeddings and non-approximability of geometric problems. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA., pages 537-538, 2003.
Dorit S. Hochbaum and David B. Shmoys. A unified approach to approximation algorithms for bottleneck problems. J. ACM, 33(3):533-550, 1986.
Kamal Jain, Mohammad Mahdian, and Amin Saberi. A new greedy approach for facility location problems. In Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 731-740, 2002.
Kamal Jain and Vijay V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM, 48(2):274-296, 2001.
Tapas Kanungo, David M. Mount, Nathan S. Netanyahu, Christine D. Piatko, Ruth Silverman, and Angela Y. Wu. A local search approximation algorithm for k-means clustering. Comput. Geom., 28(2-3):89-112, 2004.
Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. Journal of Computer and System Sciences, 74(3):335-349, 2008.
Stavros G. Kolliopoulos and Satish Rao. A nearly linear-time approximation scheme for the Euclidean k-median problem. SIAM J. Comput., 37(3):757-782, 2007.
Guy Kortsarz, Michael Langberg, and Zeev Nutov. Approximating maximum subgraphs without short cycles. SIAM J. Discrete Math., 24(1):255-269, 2010.
Amit Kumar and Ravindran Kannan. Clustering with spectral norm and the k-means algorithm. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 299-308, 2010.
Amit Kumar, Yogish Sabharwal, and Sandeep Sen. A simple linear time (1+́ε)-approximation algorithm for k-means clustering in any dimensions. In 45th Symposium on Foundations of Computer Science (FOCS 2004), 17-19 October 2004, Rome, Italy, Proceedings, pages 454-462, 2004.
Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 901-910, 2013.
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Meena Mahajan, Prajakta Nimbhorkar, and Kasturi R. Varadarajan. The planar k-means problem is NP-hard. Theor. Comput. Sci., 442:13-21, 2012.
Jiri Matoušek. On approximate geometric k-clustering. Discrete and Computational Geometry, 24(1), 2000.
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Rafail Ostrovsky, Yuval Rabani, Leonard J. Schulman, and Chaitanya Swamy. The effectiveness of lloyd-type methods for the k-means problem. J. ACM, 59(6):28, 2012.
Xindong Wu, Vipin Kumar, J. Ross Quinlan, Joydeep Ghosh, Qiang Yang, Hiroshi Motoda, Geoffrey J. McLachlan, Angus F. M. Ng, Bing Liu, Philip S. Yu, Zhi-Hua Zhou, Michael Steinbach, David J. Hand, and Dan Steinberg. Top 10 algorithms in data mining. Knowl. Inf. Syst., 14(1):1-37, 2008.
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A Fire Fighter’s Problem
Suppose that a circular fire spreads in the plane at unit speed. A fire fighter can build a barrier at speed v > 1. How large must v be to ensure that the fire can be contained, and how should the fire fighter proceed? We provide two results. First, we analyze the natural strategy where the fighter keeps building a barrier along the frontier of the expanding fire. We prove that this approach contains the fire if v > v_c = 2.6144... holds. Second, we show that any "spiralling" strategy must have speed v > 1.618, the golden ratio, in order to succeed.
Motion Planning
Dynamic Environments
Spiralling strategies
Lower and upper bounds
768-780
Regular Paper
Rolf
Klein
Rolf Klein
Elmar
Langetepe
Elmar Langetepe
Christos
Levcopoulos
Christos Levcopoulos
10.4230/LIPIcs.SOCG.2015.768
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximate Geometric MST Range Queries
Range searching is a widely-used method in computational geometry for efficiently accessing local regions of a large data set. Typically, range searching involves either counting or reporting the points lying within a given query region, but it is often desirable to compute statistics that better describe the structure of the point set lying within the region, not just the count.
In this paper we consider the geometric minimum spanning tree (MST) problem in the context of range searching where approximation is allowed. We are given a set P of n points in R^d. The objective is to preprocess P so that given an admissible query region Q, it is possible to efficiently approximate the weight of the minimum spanning tree of the subset of P lying within Q. There are two natural sources of approximation error, first by treating Q as a fuzzy object and second by approximating the MST weight itself. To model this, we assume that we are given two positive real approximation parameters eps_q and eps_w. Following the typical practice in approximate range searching, the range is expressed as two shapes Q^- and Q^+, where Q^- is contained in Q which is contained in Q^+, and their boundaries are separated by a distance of at least eps_q diam(Q). Points within Q^- must be included and points external to Q^+ cannot be included. A weight W is a valid answer to the query if there exist subsets P' and P'' of P, such that Q^- is contained in P' which is contained in P'' which is contained in Q^+ and wt(MST(P')) <= W <= (1+eps_w) wt(MST(P'')).
In this paper, we present an efficient data structure for answering such queries. Our approach uses simple data structures based on quadtrees, and it can be applied whenever Q^- and Q^+ are compact sets of constant combinatorial complexity. It uses space O(n), and it answers queries in time O(log n + 1/(eps_q eps_w)^{d + O(1)}). The O(1) term is a small constant independent of dimension, and the hidden constant factor in the overall running time depends on d, but not on eps_q or eps_w. Preprocessing requires knowledge of eps_w, but not eps_q.
Geometric data structures
Minimum spanning trees
Range searching
Approximation algorithms
781-795
Regular Paper
Sunil
Arya
Sunil Arya
David M.
Mount
David M. Mount
Eunhui
Park
Eunhui Park
10.4230/LIPIcs.SOCG.2015.781
P. K. Agarwal, L. Arge, S. Govindarajan, J. Yanga, and K. Yi. Efficient external memory structures for range-aggregate queries. Comput. Geom. Theory Appl., 46:358-370, 2013.
A. Andoni, A. Nikolov, K. Onak, and G. Yaroslavtsev. Parallel algorithms for geometric graph problems. In Proc. 46th Annu. ACM Sympos. Theory Comput., pages 574-583, 2014.
S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. Assoc. Comput. Mach., 45:753-782, 1998.
S. Arya and T. M. Chan. Better ε-dependencies for offline approximate nearest neighbor search, euclidean minimum spanning trees, and ε-kernels. In Proc. 30th Annu. Sympos. Comput. Geom., pages 416-425, 2014.
S. Arya and D. M. Mount. Approximate range searching. Comput. Geom. Theory Appl., 17:135-163, 2000.
S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman, and A. Wu. An optimal algorithm for approximate nearest neighbor searching. J. Assoc. Comput. Mach., 45:891-923, 1998.
M. J. Bannister, W. E. Devanny, M. T. Goodrich, J. A. Simons, and L. Trott. Windows into geometric events: Data structures for time-windowed querying of temporal point sets. In Proc. 26th Canad. Conf. Comput. Geom., 2014.
M. J. Bannister, C. DuBois, D. Eppstein, and P. Smyth. Windows into relational events: data structures for contiguous subsequences of edges. In Proc. 24th Annu. ACM-SIAM Sympos. Discrete Algorithms, pages 856-864, 2013. (arXiv:1209.5791).
P. Brass, C. Knauer, C.-S. Shin, M. Smid, and I. Vigan. Range-aggregate queries for geometric extent problems. In Proc. 19th Computing: The Australasian Theory Symposium (CATS), pages 3-10, 2013.
P. B. Callahan and S. R. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. Assoc. Comput. Mach., 42:67-90, 1995.
P. B. Callahan and S. R. Kosaraju. Faster algorithms for some geometric graph problems in higher dimensions. In Proc. Eighth Annu. ACM-SIAM Sympos. Discrete Algorithms, pages 291-300, 1997.
A. Czumaj, F. Ergün, L. Fortnow, A. Magen, I. Newman, R. Rubinfeld, and C. Sohler. Approximating the weight of the Euclidean minimum spanning tree in sublinear time. SIAM J. Comput., 35:91-109, 2005.
A. Czumaj and C. Sohler. Estimating the weight of metric minimum spanning trees in sublinear time. SIAM J. Comput., 39:904-922, 2009.
G. Frahling, P. Indyk, and C. Sohler. Sampling in dynamic data streams and applications. Internat. J. Comput. Geom. Appl., 18:3-28, 2008.
Y. Nekrich and M. Smid. Approximating range-aggregate queries using coresets. In Proc. 22nd Canad. Conf. Comput. Geom., pages 253-256, 2010.
D. Papadias, Y. Tao, K. Mouratidis, and K. Hui. Aggregate nearest neighbor queries in spatial databases. ACM Transactions on Database Systems (TODS), 30:529-576, 2005.
E. Park and D. M. Mount. Output-sensitive well-separated pair decompositions for dynamic point sets. In Proc. 21st Internat. Conf. on Advances in Geographic Information Systems, pages 364-373, 2013. (doi: 10.1145/2525314.2525364).
J. Shan, D. Zhang, and B. Salzberg. On spatial-range closest-pair query. In T. Hadzilacos, Y. Manolopoulos, J. Roddick, and Y. Theodoridis, editors, Advances in Spatial and Temporal Databases, volume 2750 of Lecture Notes in Computer Science, pages 252-269. Springer, Berlin, 2003.
Y. Tao and D. Papadias. Range aggregate processing in spatial databases. IEEE Transactions on Knowledge and Data Engineering (TKDE), 16:1555-1570, 2004.
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Maintaining Contour Trees of Dynamic Terrains
We study the problem of maintaining the contour tree T of a terrain Sigma, represented as a triangulated xy-monotone surface, as the heights of its vertices vary continuously with time. We characterize the combinatorial changes in T and how they relate to topological changes in Sigma. We present a kinetic data structure (KDS) for maintaining T efficiently. It maintains certificates that fail, i.e., an event occurs, only when the heights of two adjacent vertices become equal or two saddle vertices appear on the same contour. Assuming that the heights of two vertices of Sigma become equal only O(1) times and these instances can be computed in O(1) time, the KDS processes O(kappa + n) events, where n is the number of vertices in Sigma and kappa is the number of events at which the combinatorial structure of T changes, and processes each event in O(log n) time. The KDS can be extended to maintain an augmented contour tree and a join/split tree.
Contour tree
dynamic terrain
kinetic data structure
796-811
Regular Paper
Pankaj K.
Agarwal
Pankaj K. Agarwal
Thomas
Mølhave
Thomas Mølhave
Morten
Revsbæk
Morten Revsbæk
Issam
Safa
Issam Safa
Yusu
Wang
Yusu Wang
Jungwoo
Yang
Jungwoo Yang
10.4230/LIPIcs.SOCG.2015.796
P. K. Agarwal, L. Arge, T. M. Murali, Kasturi R. Varadarajan, and J. S. Vitter. I/O-efficient algorithms for contour-line extraction and planar graph blocking. In Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, pages 117-126, 1998.
P. K. Agarwal and M. Sharir. Davenport-Schinzel sequences and their geometric applications. In Jörg-Rüdiger Sack and Jorge Urrutia, editors, Handbook of Computational Geometry, pages 1-47. Elsevier Science Publishers, 2000.
Pankaj K. Agarwal, Lars Arge, and Ke Yi. I/O-efficient batched union-find and its applications to terrain analysis. In Proc. 22nd Annu. Sympos. Comput. Geom., pages 167-176, 2006.
Lars Arge, Morten Revsbæk, and Norbert Zeh. I/O-efficient computation of water flow across a terrain. In Proc. 26th Annu. Sympos. Comput. Geom., pages 403-412, 2010.
J. Basch, L. J. Guibas, and J. Hershberger. Data structures for mobile data. J. Algorithms, 31(1):1-28, 1999.
K.G. Bemis, D. Silver, P.A. Rona, and C. Feng. Case study: a methodology for plume visualization with application to real-time acquisition and navigation. In Proc. IEEE Conf. Visualization, pages 481-494, 2000.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd edition, 2008.
Vasco Brattka and Peter Hertling. Feasible real random access machines. J. Complexity, 14(4):490-526, December 1998.
Hamish Carr, Jack Snoeyink, and Ulrike Axen. Computing contour trees in all dimensions. Comput. Geom., 24(2):75-94, 2003.
Hamish Carr, Jack Snoeyink, and Michiel van de Panne. Flexible isosurfaces: Simplifying and displaying scalar topology using the contour tree. Comput. Geom., 43(1):42-58, 2010.
A. Danner, T. Mølhave, K. Yi, P. K. Agarwal, L. Arge, and H. Mitásová. Terrastream: From elevation data to watershed hierarchies. In Proc. ACM Sympos. Advances in Geographic Information Systems, page 28, 2007.
Herbert Edelsbrunner, John Harer, Ajith Mascarenhas, Valerio Pascucci, and Jack Snoeyink. Time-varying Reeb graphs for continuous space-time data. Comput. Geom., 41(3):149-166, 2008.
L. Guibas. Modeling motion. In J. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, pages 1117-1134. Chapman and Hall/CRC, 2nd edition, 2004.
Leonidas J. Guibas. Kinetic data structures - a state of the art report. In Proc. Workshop Algorithmic Found. Robot., pages 191-209, 1998.
N. Max, R. Crawfis, and D. Williams. Visualization for climate modeling. IEEE Comput. Graphics and Appl., 13:34-40, 1993.
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B. S. Sohn and Bajaj C. L. Time-varying contour topology. IEEE Transactions on Visualization and Computer Graphics, 12(1):14-25, 2006.
A. Szymczak. Subdomain aware contour trees and contour evolution in time-dependent scalar fields. In Proc. Conf. Shape Model. and Appl., pages 136-144, 2005.
S. P. Tarasov and M. N. Vyalyi. Construction of contour trees in 3D in O(n log n) steps. In Proc. 14th Annu. Sympos. Comput. Geom., pages 68-75, 1998.
M. van Kreveld, R. van Oostrum, C. Bajaj, V. Pascucci, and D. Schikore. Contour trees and small seed sets for isosurface traversal. In Proc. 13th Annu. Sympos. Comput. Geom., pages 212-220, 1997.
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Hyperorthogonal Well-Folded Hilbert Curves
R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes - smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor Omega(2^{d/2}) larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.
space-filling curve
Hilbert curve
multi-dimensional
range query
R-tree
812-826
Regular Paper
Arie
Bos
Arie Bos
Herman J.
Haverkort
Herman J. Haverkort
10.4230/LIPIcs.SOCG.2015.812
J. Alber and R. Niedermeier. On multidimensional curves with Hilbert property. Theory of Computing Systems, 33(4):295-312, 2000.
L. Arge, M. de Berg, H. Haverkort, and K. Yi. The Priority R-tree: a practically efficient and worst-case optimal R-tree. ACM Tr. Algorithms, 4(1):9, 2008.
M. Bader. Space-filling curves: an introduction with applications in scientific computing. Springer, 2013.
A. R. Butz. Alternative algorithm for Hilbert’s space-filling curve. IEEE Trans. Comp., 20(4):424-426, 1971.
H. Haverkort. An inventory of three-dimensional Hilbert space-filling curves. CoRR, abs/1109.2323, 2011.
H. Haverkort. Harmonious Hilbert curves and other extradimensional space-filling curves. CoRR, abs/1211.0175, 2012.
H. Haverkort and F. van Walderveen. Locality and bounding-box quality of two-dimensional space-filling curves. Computational Geometry, 43(2):131-147, 2010.
H. Haverkort and F. van Walderveen. Four-dimensional Hilbert curves for R-trees. ACM J. Experimental Algorithmics, 16:3.4, 2011.
D. Hilbert. Über die stetige Abbildung einer Linie auf ein Flächenstück. Math. Ann., 38(3):459-460, 1891.
I. Kamel and C. Faloutsos. On packing R-trees. In Conf. on Information and Knowledge Management, pages 490-499, 1993.
K. V. R. Kanth and A. K. Singh. Optimal dynamic range searching in non-replicating index structures. In Int. Conf. Database Theory, LNCS 154, pages 257-276, 1999.
Y. Manolopoulos, A. Nanopoulos, A. N. Papadopoulos, and Y. Theodoridis. R-trees: Theory and Applications. Springer, 2005.
D. Moore. Fast Hilbert curve generation, sorting, and range queries. http://web.archive.org/web/www.caam.rice.edu/~dougm/twiddle/Hilbert/, 2000, retrieved 23 March 2015.
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G. Peano. Sur une courbe, qui remplit toute une aire plane. Math. Ann., 36(1):157-160, 1890.
H. Sagan. Space-Filling Curves. Universitext. Springer, 1994.
S. Sasburg. Approximating average and worst-case quality measure values for d-dimensional space-filling curves. Master’s thesis, Eindhoven University of Technology, 2011.
J.-M. Wierum. Definition of a new circular space-filling curve: βΩ-indexing. Technical Report TR-001-02, Paderborn Center for Parallel Computing PC², 2002.
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Topological Analysis of Scalar Fields with Outliers
Given a real-valued function f defined over a manifold M embedded in R^d, we are interested in recovering structural information about f from the sole information of its values on a finite sample P. Existing methods provide approximation to the persistence diagram of f when geometric noise and functional noise are bounded. However, they fail in the presence of aberrant values, also called outliers, both in theory and practice.
We propose a new algorithm that deals with outliers. We handle aberrant functional values with a method inspired from the k-nearest neighbors regression and the local median filtering, while the geometric outliers are handled using the distance to a measure. Combined with topological results on nested filtrations, our algorithm performs robust topological analysis of scalar fields in a wider range of noise models than handled by current methods. We provide theoretical guarantees and experimental results on the quality of our approximation of the sampled scalar field.
Persistent Homology
Topological Data Analysis
Scalar Field Analysis
Nested Rips Filtration
Distance to a Measure
827-841
Regular Paper
Mickaël
Buchet
Mickaël Buchet
Frédéric
Chazal
Frédéric Chazal
Tamal K.
Dey
Tamal K. Dey
Fengtao
Fan
Fengtao Fan
Steve Y.
Oudot
Steve Y. Oudot
Yusu
Wang
Yusu Wang
10.4230/LIPIcs.SOCG.2015.827
M. Buchet, F. Chazal, T. K. Dey, F. Fan, S. Y. Oudot, and Y. Wang. Topological analysis of scalar fields with outliers. arXiv preprint arXiv:1412.1680, 2014.
M. Buchet, F. Chazal, S. Oudot, and D. R. Sheehy. Efficient and robust persistent homology for measures. In Proceedings of the 26th ACM-SIAM symposium on Discrete algorithms. SIAM, 2015.
F. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Oudot. Proximity of persistence modules and their diagrams. In Proc. 25th ACM Sympos. on Comput. Geom., pages 237-246, 2009.
F. Chazal, D. Cohen-Steiner, and Q. Mérigot. Geometric inference for probability measures. Foundations of Computational Mathematics, 11(6):733-751, 2011.
F. Chazal, V. de Silva, M. Glisse, and S. Oudot. The structure and stability of persistence modules, 2013. arXiv:1207.3674.
F. Chazal, L. J. Guibas, S. Y. Oudot, and P. Skraba. Scalar field analysis over point cloud data. Discrete & Computational Geometry, 46(4):743-775, 2011.
F. Chazal and S. Y. Oudot. Towards persistence-based reconstruction in euclidean spaces. In Proceedings of the twenty-fourth annual symposium on Computational geometry, pages 232-241. ACM, 2008.
D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37(1):103-120, 2007.
T. K. Dey, J. Sun, and Y. Wang. Approximating cycles in a shortest basis of the first homology group from point data. Inverse Problems, 27(12):124004, 2011.
H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. Amer. Math. Soc., Providence, Rhode Island, 2009.
H. Federer. Curvature measures. Transactions of the American Mathematical Society, pages 418-491, 1959.
L. Guibas, D. Morozov, and Q. Mérigot. Witnessed k-distance. Discrete & Computational Geometry, 49(1):22-45, 2013.
L. Györfi. A distribution-free theory of nonparametric regression. Springer, 2002.
J. Kloke and G. Carlsson. Topological de-noising: Strengthening the topological signal. arXiv preprint arXiv:0910.5947, 2009.
S. Kpotufe. k-nn regression adapts to local intrinsic dimension. arXiv preprint arXiv:1110.4300, 2011.
A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete & Computational Geometry, 33(2):249-274, 2005.
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On Computability and Triviality of Well Groups
The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1.
Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set.
For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact.
For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.
nonlinear equations
robustness
well groups
computation
homotopy theory
842-856
Regular Paper
Peter
Franek
Peter Franek
Marek
Krcál
Marek Krcál
10.4230/LIPIcs.SOCG.2015.842
G. E. Alefeld, F. A. Potra, and Z. Shen. On the existence theorems of kantorovich, moore and miranda. Technical Report 01/04, Institut für Wissenschaftliches Rechnen und Mathematische Modellbildung, 2001.
A. Ben-Tal, L.E. Ghaoui, and A. Nemirovski. Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, 2009.
P. Bendich, H. Edelsbrunner, D. Morozov, and A. Patel. Homology and robustness of level and interlevel sets. Homology, Homotopy and Applications, 15(1):51-72, 2013.
G. Carlsson. Topology and data. Bull. Amer. Math. Soc. (N.S.), 46(2):255-308, 2009.
F. Chazal, A. Patel, and P. Škraba. Computing the robustness of roots. Applied Mathematics Letters, 25(11):1725 - 1728, November 2012.
P. Collins. Computability and representations of the zero set. Electron. Notes Theor. Comput. Sci., 221:37-43, December 2008.
H. Edelsbrunner and J. L. Harer. Computational topology. American Mathematical Society, Providence, RI, 2010.
H. Edelsbrunner, D. Morozov, and A. Patel. Quantifying transversality by measuring the robustness of intersections. Foundations of Computational Mathematics, 11(3):345-361, 2011.
Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. Quantifying transversality by measuring the robustness of intersections. Foundations of Computational Mathematics, 11(3):345-361, 2011.
P. Franek and M. Krčál. Robust satisfiability of systems of equations. In Proc. Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), 2014. Extended version accepted to Journal of ACM. Preprint in arXiv:1402.0858.
P. Franek and M. Krčál. On computability and triviality of well groups, 2015. Preprint arXiv:1501.03641v2.
P. Franek, S. Ratschan, and P. Zgliczynski. Quasi-decidability of a fragment of the analytic first-order theory of real numbers, 2012. Preprint in arXiv:1309.6280.
A. Frommer and B. Lang. Existence tests for solutions of nonlinear equations using Borsuk’s theorem. SIAM Journal on Numerical Analysis, 43(3):1348-1361, 2005.
F. Goudail and P. Réfrégier. Statistical Image Processing Techniques for Noisy Images: An Application-Oriented Approach. Kluwer Academic / Plenum Publishers, 2004.
A. Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2001.
N.J. Higham. Accuracy and Stability of Numerical Algorithms: Second Edition. Society for Industrial and Applied Mathematics, 2002.
R. B. Kearfott. On existence and uniqueness verification for non-smooth functions. Reliable Computing, 8(4):267-282, 2002.
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P. Franek, M. Krčál. Cohomotopy groups capture robust properties of zero sets. Manuscript in preparation, 2014.
V. V. Prasolov. Elements of Homology Theory. Graduate Studies in Mathematics. American Mathematical Society, 2007.
P. Škraba, B. Wang, Ch. Guoning, and P. Rosen. 2D vector field simplification based on robustness, 2014. to appear in IEEE Pacific Visualization (PacificVis).
M. Čadek, M. Krčál, J. Matoušek, F. Sergeraert, L. Vokř\'ιnek, and U. Wagner. Computing all maps into a sphere. J. ACM, 61(3):17:1-17:44, June 2014.
A.H. Wallace. Algebraic Topology: Homology and Cohomology. Dover Books on Mathematics Series. Dover Publications, 2007.
J.H.C. Whitehead. On the theory of obstructions. Annals of Mathematics, pages 68-84, 1951.
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Geometric Inference on Kernel Density Estimates
We show that geometric inference of a point cloud can be calculated by examining its kernel density estimate with a Gaussian kernel. This allows one to consider kernel density estimates, which are robust to spatial noise, subsampling, and approximate computation in comparison to raw point sets. This is achieved by examining the sublevel sets of the kernel distance, which isomorphically map to superlevel sets of the kernel density estimate. We prove new properties about the kernel distance, demonstrating stability results and allowing it to inherit reconstruction results from recent advances in distance-based topological reconstruction. Moreover, we provide an algorithm to estimate its topology using weighted Vietoris-Rips complexes.
topological data analysis
kernel density estimate
kernel distance
857-871
Regular Paper
Jeff M.
Phillips
Jeff M. Phillips
Bei
Wang
Bei Wang
Yan
Zheng
Yan Zheng
10.4230/LIPIcs.SOCG.2015.857
N. Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical Society, 68:337-404, 1950.
Sivaraman Balakrishnan, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Aarti Singh, and Larry Wasserman. Statistical inference for persistent homology. Technical report, ArXiv:1303.7117, March 2013.
James Biagioni and Jakob Eriksson. Map inference in the face of noise and disparity. In ACM SIGSPATIAL GIS, 2012.
Gérard Biau, Frédéric Chazal, David Cohen-Steiner, Luc Devroye, and Carlos Rodriguez. A weighted k-nearest neighbor density estimate for geometric inference. Electronic Journal of Statistics, 5:204-237, 2011.
Omer Bobrowski, Sayan Mukherjee, and Jonathan E. Taylor. Topological consistency via kernel estimation. Technical report, arXiv:1407.5272, 2014.
Peter Bubenik. Statistical topological data analysis using persistence landscapes. Jounral of Machine Learning Research, 2014.
Mickael Buchet, Frederic Chazal, Steve Y. Oudot, and Donald R. Sheehy. Efficient and robust persistent homology for measures. In SODA, 2015.
Frédéric Chazal and David Cohen-Steiner. Geometric inference. Tessellations in the Sciences, 2012.
Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J. Guibas, and Steve Y. Oudot. Proximity of persistence modules and their diagrams. In SOCG, 2009.
Frédéric Chazal, David Cohen-Steiner, and André Lieutier. Normal cone approximation and offset shape isotopy. CGTA, 42:566-581, 2009.
Frédéric Chazal, David Cohen-Steiner, and André Lieutier. A sampling theory for compact sets in Euclidean space. DCG, 41(3):461-479, 2009.
Frédéric Chazal, David Cohen-Steiner, and Quentin Mérigot. Geometric inference for probability measures. FOCM, 11(6):733-751, 2011.
Frederic Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules. arXiv:1207.3674, 2013.
Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Bertrand Michel, Alessandro Rinaldo, and Larry Wasserman. Robust topolical inference: Distance-to-a-measure and kernel distance. Technical report, arXiv:1412.7197, 2014.
Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Aarti Singh, and Larry Wasserman. On the bootstrap for persistence diagrams and landscapes. Modeling and Analysis of Information Systems, 20:96-105, 2013.
Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman. Stochastic convergence of persistence landscapes. In SOCG, 2014.
Frédéric Chazal and André Lieutier. Weak feature size and persistent homology: computing homology of solids in Rⁿ from noisy data samples. In SOCG, pages 255-262, 2005.
Frédéric Chazal and André Lieutier. Topology guaranteeing manifold reconstruction using distance function to noisy data. In SOCG, 2006.
Frédéric Chazal and Steve Oudot. Towards persistence-based reconstruction in euclidean spaces. In SOCG, 2008.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. DCG, 37:103-120, 2007.
Luc Devroye and László Györfi. Nonparametric Density Estimation: The L₁ View. Wiley, 1984.
Luc Devroye and Gábor Lugosi. Combinatorial Methods in Density Estimation. Springer-Verlag, 2001.
Herbert Edelsbrunner. The union of balls and its dual shape. In SOCG, 1993.
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Herbert Edelsbrunner, Brittany Terese Fasy, and Günter Rote. Add isotropic Gaussian kernels at own risk: More and more resiliant modes in higher dimensions. In SOCG, 2012.
Herbert Edelsbrunner and John Harer. Persistent homology. Contemporary Mathematics, 453:257-282, 2008.
Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Society, Providence, RI, USA, 2010.
Ahmed Elgammal, Ramani Duraiswami, David Harwood, and Larry S. Davis. Background and foreground modeling using nonparametric kernel density estimation for visual surveillance. Proc. IEEE, 90:1151-1163, 2002.
Brittany Terese Fasy, Jisu Kim, Fabrizio Lecci, and Clément Maria. Introduction to the R package TDA. Technical report, arXiV:1411.1830, 2014.
Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh. Statistical inference for persistent homology: Confidence sets for persistence diagrams. In The Annals of Statistics, volume 42, pages 2301-2339, 2014.
H. Federer. Curvature measures. Transactions of the American Mathematical Society, 93:418-491, 1959.
Mingchen Gao, Chao Chen, Shaoting Zhang, Zhen Qian, Dimitris Metaxas, and Leon Axel. Segmenting the papillary muscles and the trabeculae from high resolution cardiac CT through restoration of topological handles. In Proceedings International Conference on Information Processing in Medical Imaging, 2013.
Joan Glaunès. Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l'anatomie numérique. PhD thesis, Université Paris 13, 2005.
Karsten Grove. Critical point theory for distance functions. Proceedings of Symposia in Pure Mathematics, 54:357-385, 1993.
Leonidas Guibas, Quentin Mérigot, and Dmitriy Morozov. Witnessed k-distance. In SOCG, 2011.
Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002.
Matrial Hein and Olivier Bousquet. Hilbertian metrics and positive definite kernels on probability measures. In Proceedings 10th International Workshop on Artificial Intelligence and Statistics, 2005.
Sarang Joshi, Raj Varma Kommaraju, Jeff M. Phillips, and Suresh Venkatasubramanian. Comparing distributions and shapes using the kernel distance. In SOCG, 2011.
John M. Lee. Introduction to smooth manifolds. Springer, 2003.
Jie Liang, Herbert Edelsbrunner, Ping Fu, Pamidighantam V. Sudharkar, and Shankar Subramanian. Analytic shape computation of macromolecues: I. molecular area and volume through alpha shape. Proteins: Structure, Function, and Genetics, 33:1-17, 1998.
André Lieutier. Any open bounded subset of ℝⁿ has the same homotopy type as its medial axis. Computer-Aided Design, 36:1029-1046, 2004.
Quentin Mérigot. Geometric structure detection in point clouds. PhD thesis, Université de Nice Sophia-Antipolis, 2010.
Yuriy Mileyko, Sayan Mukherjee, and John Harer. Probability measures on the space of persistence diagrams. Inverse Problems, 27(12), 2011.
A. Müller. Integral probability metrics and their generating classes of functions. Advances in Applied Probability, 29(2):429-443, 1997.
Jeff M. Phillips. ε-samples for kernels. SODA, 2013.
Jeff M. Phillips and Suresh Venkatasubramanian. A gentle introduction to the kernel distance. arXiv:1103.1625, March 2011.
Jeff M. Phillips, Bei Wang, and Yan Zheng. Geometric inference on kernel density estimates. In arXiv:1307.7760, 2015.
Florian T. Pokorny, Carl Henrik, Hedvig Kjellström, and Danica Kragic. Persistent homology for learning densities with bounded support. In Neural Informations Processing Systems, 2012.
Charles A. Price, Olga Symonova, Yuriy Mileyko, Troy Hilley, and Joshua W. Weitz. Leaf gui: Segmenting and analyzing the structure of leaf veins and areoles. Plant Physiology, 155:236-245, 2011.
David W. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, 1992.
Donald R. Sheehy. A multicover nerve for geometric inference. CCCG, 2012.
Bernard W. Silverman. Using kernel density esimates to inversitigate multimodality. J. R. Sratistical Society B, 43:97-99, 1981.
Bernard W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC, 1986.
Bharath K. Sriperumbudur, Arthur Gretton, Kenji Fukumizu, Bernhard Schölkopf, and Gert R. G. Lanckriet. Hilbert space embeddings and metrics on probability measures. JMLR, 11:1517-1561, 2010.
Kathryn Turner, Yuriy Mileyko, Sayan Mukherjee, and John Harer. Fréchet means for distributions of persistence diagrams. DCG, 2014.
Cédric Villani. Topics in Optimal Transportation. American Mathematical Society, 2003.
Grace Wahba. Support vector machines, reproducing kernel Hilbert spaces, and randomization. In Advances in Kernel Methods - Support Vector Learning, pages 69-88. The MIT Press, 1999.
Yan Zheng, Jeffrey Jestes, Jeff M. Phillips, and Feifei Li. Quality and efficiency in kernel density estimates for large data. In SIGMOD, 2012.
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Modeling Real-World Data Sets (Invited Talk)
Traditionally, the performance of algorithms is evaluated using worst-case analysis. For a number of problems, this type of analysis gives overly pessimistic results: Worst-case inputs are rather artificial and do not occur in practical applications. In this lecture we review some alternative analysis approaches leading to more realistic and robust performance evaluations.
Specifically, we focus on the approach of modeling real-world data sets. We report on two studies performed by the author for the problems of self-organizing search and paging. In these settings real data sets exhibit locality of reference. We devise mathematical models capturing locality. Furthermore, we present combined theoretical and experimental analyses in which the theoretically proven and experimentally observed performance guarantees match up to very small relative errors.
Worst-case analysis
real data sets
locality of reference
paging
self-organizing lists
872-872
Invited Talk
Susanne
Albers
Susanne Albers
10.4230/LIPIcs.SOCG.2015.872
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