eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
0
0
10.4230/LIPIcs.SoCG.2017
article
LIPIcs, Volume 77, SoCG'17, Complete Volume
Aronov, Boris
Katz, Matthew J.
LIPIcs, Volume 77, SoCG'17, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017/LIPIcs.SoCG.2017.pdf
Analysis of Algorithms and Problem Complexity, Nonnumerical Algorithms and Problems – Geometrical problems and computations, Discrete Mathematics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
0:i
0:xviii
10.4230/LIPIcs.SoCG.2017.0
article
Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers, Sponsors
Aronov, Boris
Katz, Matthew J.
Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers, Sponsors
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.0/LIPIcs.SoCG.2017.0.pdf
Front Matter
Table of Contents
Foreword
Conference Organization
External Reviewers
Sponsors
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
1:1
1:1
10.4230/LIPIcs.SoCG.2017.1
article
The Geometry and Topology of Crystals: From Sphere-Packing to Tiling, Nets, and Knots (Invited Talk)
Robins, Vanessa
Crystal structures have inspired developments in geometry since the Ancient Greeks conceived of Platonic solids after observing tetrahedral, cubical and octahedral mineral forms in their local environment. The internal structure of crystals became accessible with the development of x-ray diffraction techniques just over 100 years ago, and a key step in developing this method was understanding the arrangement of atoms in the simplest crystals as close-packings of spheres. Determining a crystal structure via x-ray diffraction unavoidably requires prior models, and this has led to the intense study of sphere packing, atom-bond networks, and arrangements of polyhedra by crystallographers investigating ever more complex compounds. In the 21st century, chemists are exploring the possibilities of coordination polymers, a wide class of crystalline materials that self-assemble from metal cations and organic ligands into periodic framework materials. Longer organic ligands mean these compounds can form multi-component interwoven network structures where the "edges" are no longer constrained to join nearest-neighbour "nodes" as in simpler atom-bond networks. The challenge for geometers is to devise algorithms for enumerating relevant structures and to devise invariants that will distinguish between different modes of interweaving. This talk will survey various methods from computational geometry and topology that are currently used to describe crystalline structures and outline research directions to address some of the open questions suggested above.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.1/LIPIcs.SoCG.2017.1.pdf
Mathematical crystallography
Combinatorial tiling theory
Graphs and surfaces in the 3-torus
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
2:1
2:1
10.4230/LIPIcs.SoCG.2017.2
article
The Algebraic Revolution in Combinatorial and Computational Geometry: State of the Art (Invited Talk)
Sharir, Micha
For the past 10 years, combinatorial geometry (and to some extent, computational geometry too) has gone through a dramatic revolution, due to the infusion of techniques from algebraic geometry and algebra that have proven effective in solving a variety of hard problems that were thought to be unreachable with more traditional techniques. The new era has begun with two groundbreaking papers of Guth and Katz, the second of which has (almost completely) solved the distinct distances problem of Erdos, open since 1946.
In this talk I will survey some of the progress that has been made since then, including a variety of problems on distinct and repeated distances and other configurations, on incidences between points and lines, curves, and surfaces in two, three, and higher dimensions, on polynomials vanishing on Cartesian products with applications, on cycle elimination for lines and triangles in three dimensions, on range searching with semialgebraic sets, and I will most certainly run out of time while doing so.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.2/LIPIcs.SoCG.2017.2.pdf
Combinatorial Geometry
Incidences
Polynomial method
Algebraic Geometry
Distances
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
3:1
3:15
10.4230/LIPIcs.SoCG.2017.3
article
Irrational Guards are Sometimes Needed
Abrahamsen, Mikkel
Adamaszek, Anna
Miltzow, Tillmann
In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon so that the guards together can see the whole polygon. We say that a guard at position x sees a point y if the line segment xy is contained in the polygon.
Despite an extensive study of the art gallery problem, it remained an open question whether there are polygons given by integer coordinates that require guard positions with irrational coordinates in any optimal solution. We give a positive answer to this question by constructing a monotone polygon with integer coordinates that can be guarded by three guards only when we allow to place the guards at points with irrational coordinates. Otherwise, four guards are needed. By extending this example, we show that for every n, there is a polygon which can be guarded by 3n guards with irrational coordinates but needs 4n guards if the coordinates have to be rational. Subsequently, we show that there are rectilinear polygons given by integer coordinates that require guards with irrational coordinates in any optimal solution.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.3/LIPIcs.SoCG.2017.3.pdf
art gallery problem
computational geometry
irrational numbers
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
4:1
4:15
10.4230/LIPIcs.SoCG.2017.4
article
Minimum Perimeter-Sum Partitions in the Plane
Abrahamsen, Mikkel
de Berg, Mark
Buchin, Kevin
Mehr, Mehran
Mehrabi, Ali D.
Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.4/LIPIcs.SoCG.2017.4.pdf
Computational geometry
clustering
minimum-perimeter partition
convex hull
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
5:1
5:16
10.4230/LIPIcs.SoCG.2017.5
article
Range-Clustering Queries
Abrahamsen, Mikkel
de Berg, Mark
Buchin, Kevin
Mehr, Mehran
Mehrabi, Ali D.
In a geometric k-clustering problem the goal is to partition a set of points in R^d into k subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering queries on a point set S: given a query box Q and an integer k > 2, compute an optimal k-clustering for the subset of S inside Q. We obtain the following results.
* We present a general method to compute a (1+epsilon)-approximation to a range-clustering query, where epsilon>0 is a parameter that can be specified as part of the query. Our method applies to a large class of clustering problems, including k-center clustering in any Lp-metric and a variant of k-center clustering where the goal is to minimize the sum (instead of maximum) of the cluster sizes.
* We extend our method to deal with capacitated k-clustering problems, where each of the clusters should not contain more than a given number of points.
* For the special cases of rectilinear k-center clustering in R^1, and in R^2 for k = 2 or 3, we present data structures that answer range-clustering queries exactly.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.5/LIPIcs.SoCG.2017.5.pdf
Geometric data structures
clustering
k-center problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
6:1
6:16
10.4230/LIPIcs.SoCG.2017.6
article
Best Laid Plans of Lions and Men
Abrahamsen, Mikkel
Holm, Jacob
Rotenberg, Eva
Wulff-Nilsen, Christian
We answer the following question dating back to J.E. Littlewood (1885-1977): Can two lions catch a man in a bounded area with rectifiable lakes? The lions and the man are all assumed to be points moving with at most unit speed. That the lakes are rectifiable means that their boundaries are finitely long. This requirement is to avoid pathological examples where the man survives forever because any path to the lions is infinitely long. We show that the answer to the question is not always "yes", by giving an example of a region R in the plane where the man has a strategy to survive forever. R is a polygonal region with holes and the exterior and interior boundaries are pairwise disjoint, simple polygons. Our construction is the first truly two-dimensional example where the man can survive.
Next, we consider the following game played on the entire plane instead of a bounded area: There is any finite number of unit speed lions and one fast man who can run with speed 1+epsilon for some value epsilon>0. Can the man always survive? We answer the question in the affirmative for any constant epsilon>0.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.6/LIPIcs.SoCG.2017.6.pdf
Lion and man game
Pursuit evasion game
Winning strategy
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
7:1
7:16
10.4230/LIPIcs.SoCG.2017.7
article
Faster Algorithms for the Geometric Transportation Problem
Agarwal, Pankaj K.
Fox, Kyle
Panigrahi, Debmalya
Varadarajan, Kasturi R.
Xiao, Allen
Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric:
* For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost.
* For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point.
* An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.7/LIPIcs.SoCG.2017.7.pdf
transportation map
earth mover's distance
shape matching
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
8:1
8:16
10.4230/LIPIcs.SoCG.2017.8
article
A Superlinear Lower Bound on the Number of 5-Holes
Aichholzer, Oswin
Balko, Martin
Hackl, Thomas
Kyncl, Jan
Parada, Irene
Scheucher, Manfred
Valtr, Pavel
Vogtenhuber, Birgit
Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h_5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position.
Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h_5(n) have been of order Omega(n) and O(n^2), respectively. We show that h_5(n) = Omega(n(log n)^(4/5)), obtaining the first superlinear lower bound on h_5(n).
The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.8/LIPIcs.SoCG.2017.8.pdf
Erdös-Szekeres type problem
k-hole
empty k-gon
empty pentagon
planar point set
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
9:1
9:16
10.4230/LIPIcs.SoCG.2017.9
article
A Universal Slope Set for 1-Bend Planar Drawings
Angelini, Patrizio
Bekos, Michael A.
Liotta, Giuseppe
Montecchiani, Fabrizio
We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.9/LIPIcs.SoCG.2017.9.pdf
Slope number
1-bend drawings
planar graphs
angular resolution
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
10:1
10:15
10.4230/LIPIcs.SoCG.2017.10
article
Near-Optimal epsilon-Kernel Construction and Related Problems
Arya, Sunil
da Fonseca, Guilherme D.
Mount, David M.
The computation of (i) eps-kernels, (ii) approximate diameter, and (iii) approximate bichromatic closest pair are fundamental problems in geometric approximation. In each case the input is a set of points in d-dimensional space for a constant d and an approximation parameter eps > 0. In this paper, we describe new algorithms for these problems, achieving significant improvements to the exponent of the eps-dependency in their running times, from roughly d to d/2 for the first two problems and from roughly d/3 to d/4 for problem (iii).
These results are all based on an efficient decomposition of a convex body using a hierarchy of Macbeath regions, and contrast to previous solutions that decomposed the space using quadtrees and grids. By further application of these techniques, we also show that it is possible to obtain near-optimal preprocessing time for the most efficient data structures for (iv) approximate nearest neighbor searching, (v) directional width queries, and (vi) polytope membership queries.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.10/LIPIcs.SoCG.2017.10.pdf
Approximation
diameter
kernel
coreset
nearest neighbor
polytope membership
bichromatic closest pair
Macbeath regions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
11:1
11:15
10.4230/LIPIcs.SoCG.2017.11
article
Exact Algorithms for Terrain Guarding
Ashok, Pradeesha
Fomin, Fedor V.
Kolay, Sudeshna
Saurabh, Saket
Zehavi, Meirav
Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size on T that guards all of the points on T. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an n^O(sqrt{k})-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.11/LIPIcs.SoCG.2017.11.pdf
Terrain Guarding
Art Gallery
Exponential-Time Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
12:1
12:16
10.4230/LIPIcs.SoCG.2017.12
article
Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences
Balko, Martin
Cibulka, Josef
Valtr, Pavel
Let d and k be integers with 1 <= k <= d-1. Let Lambda be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in the intersection of Lambda with K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n * ... * n grid is at least Omega(n^(d(d-k)/(d-1)-epsilon)) and at most O(n^(d(d-k)/(d-1))), where epsilon > 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach. We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover the intersection of Lambda with K.
We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer. For d > =3 and epsilon in (0,1), we show that there is an integer r=r(d,epsilon) such that for all positive integers n, m the following statement is true. There is a set of n points in R^d and an arrangement of m hyperplanes in R^d with no K_(r,r) in their incidence graph and with at least Omega((mn)^(1-(2d+3)/((d+2)(d+3)) - epsilon)) incidences if d is odd and Omega((mn)^(1-(2d^2+d-2)/((d+2)(d^2+2d-2)) - epsilon)) incidences if d is even.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.12/LIPIcs.SoCG.2017.12.pdf
lattice point
covering
linear subspace
point-hyperplane incidence
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
13:1
13:15
10.4230/LIPIcs.SoCG.2017.13
article
Subquadratic Algorithms for Algebraic Generalizations of 3SUM
Barba, Luis
Cardinal, Jean
Iacono, John
Langerman, Stefan
Ooms, Aurélien
Solomon, Noam
The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n^{11/6}) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. Finally, we shed light on the General Position Testing (GPT) problem: "Given n points in the plane, do three of them lie on a line?", a key problem in computational geometry.
We prove that there exist bounded-degree algebraic decision trees of depth O(n^{12/7+e}) that solve 3POL, and that 3POL can be solved in O(n^2 (log log n)^{3/2} / (log n)^{1/2}) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in subquadratic time when the input points lie on o((log n)^{1/6}/(log log n)^{1/2}) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.13/LIPIcs.SoCG.2017.13.pdf
3SUM
subquadratic algorithms
general position testing
range searching
dominance reporting
polynomial curves
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
14:1
14:16
10.4230/LIPIcs.SoCG.2017.14
article
Towards a Topology-Shape-Metrics Framework for Ortho-Radial Drawings
Barth, Lukas
Niedermann, Benjamin
Rutter, Ignaz
Wolf, Matthias
Ortho-Radial drawings are a generalization of orthogonal drawings to grids that are formed by concentric circles and straight-line spokes emanating from the circles' center. Such drawings have applications in schematic graph layouts, e.g., for metro maps and destination maps.
A plane graph is a planar graph with a fixed planar embedding. We give a combinatorial characterization of the plane graphs that admit a planar ortho-radial drawing without bends. Previously, such a characterization was only known for paths, cycles, and theta graphs, and in the special case of rectangular drawings for cubic graphs, where the contour of each face is required to be a rectangle.
The characterization is expressed in terms of an ortho-radial representation that, similar to Tamassia's orthogonal representations for orthogonal drawings describes such a drawing combinatorially in terms of angles around vertices and bends on the edges. In this sense our characterization can be seen as a first step towards generalizing the Topology-Shape-Metrics framework of Tamassia to ortho-radial drawings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.14/LIPIcs.SoCG.2017.14.pdf
Graph Drawing
Ortho-Radial Drawings
Combinatorial Characterization
Bend Minimization
Topology-Shape-Metrics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
15:1
15:15
10.4230/LIPIcs.SoCG.2017.15
article
On the Number of Ordinary Lines Determined by Sets in Complex Space
Basit, Abdul
Dvir, Zeev
Saraf, Shubhangi
Wolf, Charles
Kelly's theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.15/LIPIcs.SoCG.2017.15.pdf
Incidences
Combinatorial Geometry
Designs
Polynomial Method
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
16:1
16:16
10.4230/LIPIcs.SoCG.2017.16
article
On Optimal 2- and 3-Planar Graphs
Bekos, Michael A.
Kaufmann, Michael
Raftopoulou, Chrysanthi N.
A graph is k-planar if it can be drawn in the plane such that no edge is crossed more than k times. While for k=1, optimal 1-planar graphs, i.e., those with n vertices and exactly 4n-8 edges, have been completely characterized, this has not been the case for k > 1. For k=2,3 and 4, upper bounds on the edge density have been developed for the case of simple graphs by Pach and Tóth, Pach et al. and Ackerman, which have been used to improve the well-known "Crossing Lemma". Recently, we proved that these bounds also apply to non-simple 2- and 3-planar graphs without homotopic parallel edges and self-loops.
In this paper, we completely characterize optimal 2- and 3-planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.16/LIPIcs.SoCG.2017.16.pdf
topological graphs
optimal k-planar graphs
characterization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
17:1
17:15
10.4230/LIPIcs.SoCG.2017.17
article
Reachability in a Planar Subdivision with Direction Constraints
Binham, Daniel
Manhaes de Castro, Pedro Machado
Vigneron, Antoine
Given a planar subdivision with n vertices, each face having a cone of possible directions of travel, our goal is to decide which vertices of the subdivision can be reached from a given starting point s. We give an O(n log n)-time algorithm for this problem, as well as an Omega(n log n) lower bound in the algebraic computation tree model. We prove that the generalization where two cones of directions per face are allowed is NP-hard.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.17/LIPIcs.SoCG.2017.17.pdf
Design and analysis of geometric algorithms
Path planning
Reachability
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
18:1
18:16
10.4230/LIPIcs.SoCG.2017.18
article
Fine-Grained Complexity of Coloring Unit Disks and Balls
Biró, Csaba
Bonnet, Édouard
Marx, Dániel
Miltzow, Tillmann
Rzazewski, Pawel
On planar graphs, many classic algorithmic problems enjoy a certain "square root phenomenon" and can be solved significantly faster than what is known to be possible on general graphs: for example, Independent Set, 3-Coloring, Hamiltonian Cycle, Dominating Set can be solved in time 2^O(sqrt{n}) on an n-vertex planar graph, while no 2^o(n) algorithms exist for general graphs, assuming the Exponential Time Hypothesis (ETH). The square root in the exponent seems to be best possible for planar graphs: assuming the ETH, the running time for these problems cannot be improved to 2^o(sqrt{n}). In some cases, a similar speedup can be obtained for 2-dimensional geometric problems, for example, there are 2^O(sqrt{n}log n) time algorithms for Independent Set on unit disk graphs or for TSP on 2-dimensional point sets.
In this paper, we explore whether such a speedup is possible for geometric coloring problems. On the one hand, geometric objects can behave similarly to planar graphs: 3-Coloring can be solved in time 2^O(sqrt{n}) on the intersection graph of n unit disks in the plane and, assuming the ETH, there is no such algorithm with running time 2^o(sqrt{n}). On the other hand, if the number L of colors is part of the input, then no such speedup is possible: Coloring the intersection graph of n unit disks with L colors cannot be solved in time 2^o(n), assuming the ETH. More precisely, we exhibit a smooth increase of complexity as the number L of colors increases: If we restrict the number of colors to L=Theta(n^alpha) for some 0<=alpha<=1, then the problem of coloring the intersection graph of n unit disks with L colors
* can be solved in time exp(O(n^{{1+alpha}/2}log n))=exp( O(sqrt{nL}log n)), and
* cannot be solved in time exp(o(n^{{1+alpha}/2}))=exp(o(sqrt{nL})), unless the ETH fails.
More generally, we consider the problem of coloring d-dimensional unit balls in the Euclidean space and obtain analogous results showing that the problem
* can be solved in time exp(O(n^{{d-1+alpha}/d}log n))=exp(O(n^{1-1/d}L^{1/d}log n)), and
* cannot be solved in time exp(n^{{d-1+alpha}/d-epsilon})= exp (O(n^{1-1/d-epsilon}L^{1/d})) for any epsilon>0, unless the ETH fails.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.18/LIPIcs.SoCG.2017.18.pdf
unit disk graphs
unit ball graphs
coloring
exact algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
19:1
19:16
10.4230/LIPIcs.SoCG.2017.19
article
Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams
Boissonnat, Jean-Daniel
Rouxel-Labbé, Mael
Wintraecken, Mathijs
The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R^2 and on surfaces embedded in R^3 as detailed in our experimental companion paper.
In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Omega equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Omega under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.19/LIPIcs.SoCG.2017.19.pdf
Riemannian Geometry
Voronoi diagram
Delaunay triangulation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
20:1
20:15
10.4230/LIPIcs.SoCG.2017.20
article
An Approximation Algorithm for the Art Gallery Problem
Bonnet, Édouard
Miltzow, Tillmann
Given a simple polygon P on n vertices, two points x, y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum-size set S such that every point in P is visible from a point in S. The set S is referred to as guards. Assuming integer coordinates and a specific general position on the vertices of P, we present the first O(log OPT)-approximation algorithm for the point guard problem. This algorithm combines ideas in papers of Efrat and Har-Peled and Deshpande et al. We also point out a mistake in the latter.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.20/LIPIcs.SoCG.2017.20.pdf
computational geometry
art gallery
approximation algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
21:1
21:15
10.4230/LIPIcs.SoCG.2017.21
article
Self-Approaching Paths in Simple Polygons
Bose, Prosenjit
Kostitsyna, Irina
Langerman, Stefan
We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, |ac| >= |bc|. We analyze the properties, and present a characterization of shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points inside a polygon can be forced to follow a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find a self-approaching path inside a polygon connecting two points under a model of computation which assumes that we can calculate involute curves of high order.
Lastly, we provide an algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.21/LIPIcs.SoCG.2017.21.pdf
self-approaching path
simple polygon
shortest path
involute curve
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
22:1
22:15
10.4230/LIPIcs.SoCG.2017.22
article
Maximum Volume Subset Selection for Anchored Boxes
Bringmann, Karl
Cabello, Sergio
Emmerich, Michael T. M.
Let B be a set of n axis-parallel boxes in d-dimensions such that each box has a corner at the origin and the other corner in the positive quadrant, and let k be a positive integer. We study the problem of selecting k boxes in B that maximize the volume of the union of the selected boxes. The research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known algorithms in any dimension d>2 enumerate all size-k subsets. We show that:
* The problem is NP-hard already in 3 dimensions.
* In 3 dimensions, we break the enumeration of all size-k subsets, by providing an n^O(sqrt(k)) algorithm.
* For any constant dimension d, we give an efficient polynomial-time approximation scheme.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.22/LIPIcs.SoCG.2017.22.pdf
geometric optimization
subset selection
hypervolume indicator
Klee’s 23 measure problem
boxes
NP-hardness
PTAS
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
23:1
23:16
10.4230/LIPIcs.SoCG.2017.23
article
Declutter and Resample: Towards Parameter Free Denoising
Buchet, Mickael
Dey, Tamal K.
Wang, Jiayuan
Wang, Yusu
In many data analysis applications the following scenario is commonplace: we are given a point set that is supposed to sample a hidden ground truth K in a metric space, but it got corrupted with noise so that some of the data points lie far away from K creating outliers also termed as ambient noise. One of the main goals of denoising algorithms is to eliminate such noise so that the curated data lie within a bounded Hausdorff distance of K. Popular denoising approaches such as deconvolution and thresholding often require the user to set several parameters and/or to choose an appropriate noise model while guaranteeing only asymptotic convergence. Our goal is to lighten this burden as much as possible while ensuring theoretical guarantees in all cases.
Specifically, first, we propose a simple denoising algorithm that requires only a single parameter but provides a theoretical guarantee on the quality of the output on general input points. We argue that this single parameter cannot be avoided. We next present a simple algorithm that avoids even this parameter by paying for it with a slight strengthening of the sampling condition on the input points which is not unrealistic. We also provide some preliminary empirical evidence that our algorithms
are effective in practice.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.23/LIPIcs.SoCG.2017.23.pdf
denoising
parameter free
k-distance,compact sets
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
24:1
24:15
10.4230/LIPIcs.SoCG.2017.24
article
Ham Sandwich is Equivalent to Borsuk-Ulam
C. S., Karthik
Saha, Arpan
The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in R^n can always be simultaneously bisected by an (n-1)-dimensional hyperplane.
In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and the Ham Sandwich theorem. The main technical result we show towards establishing the equivalence is the following: For every odd polynomial restricted to the hypersphere f:S^n->R, there exists a compact set A in R^{n+1}, such that for every x in S^n we have f(x)=vol(A cap H^+) - vol(A cap H^-), where H is the oriented hyperplane containing the origin with x as the normal. A noteworthy aspect of the proof of the above result is the use of hyperspherical harmonics.
Finally, using the above result we prove that there exist constants n_0, epsilon_0>0 such that for every n>= n_0 and epsilon <= epsilon_0/sqrt{48n}, any query algorithm to find an epsilon-bisecting (n-1)-dimensional hyperplane of n compact set in [-n^4.51,n^4.51]^n, even with success probability 2^-Omega(n), requires 2^Omega(n) queries.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.24/LIPIcs.SoCG.2017.24.pdf
Ham Sandwich theorem
Borsuk-Ulam theorem
Query Complexity
Hyperspherical Harmonics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
25:1
25:15
10.4230/LIPIcs.SoCG.2017.25
article
Local Equivalence and Intrinsic Metrics between Reeb Graphs
Carrière, Mathieu
Oudot, Steve
As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as functional distortion or interleaving that are provably more discriminative than the so-called bottleneck distance, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough locally, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the intrinsic metrics induced by these distances, which turn out to be all globally equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.25/LIPIcs.SoCG.2017.25.pdf
Reeb Graphs
Extended Persistence
Induced Metrics
Topological Data Analysis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
26:1
26:15
10.4230/LIPIcs.SoCG.2017.26
article
Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry
Chan, Timothy M.
We apply the polynomial method - specifically, Chebyshev polynomials - to obtain a number of new results on geometric approximation algorithms in low constant dimensions. For example, we give an algorithm for constructing epsilon-kernels (coresets for approximate width and approximate convex hull) in close to optimal time O(n + (1/epsilon)^{(d-1)/2}), up to a small near-(1/epsilon)^{3/2} factor, for any d-dimensional n-point set. We obtain an improved data structure for Euclidean *approximate nearest neighbor search* with close to O(n log n + (1/epsilon)^{d/4} n) preprocessing time and O((1/epsilon)^{d/4} log n) query time. We obtain improved approximation algorithms for discrete Voronoi diagrams, diameter, and bichromatic closest pair in the L_s-metric for any even integer constant s >= 2. The techniques are general and may have further applications.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.26/LIPIcs.SoCG.2017.26.pdf
diameter
coresets
approximate nearest neighbor search
the polynomial method
streaming
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
27:1
27:15
10.4230/LIPIcs.SoCG.2017.27
article
Orthogonal Range Searching in Moderate Dimensions: k-d Trees and Range Trees Strike Back
Chan, Timothy M.
We revisit the orthogonal range searching problem and the exact l_infinity nearest neighbor searching problem for a static set of n points when the dimension d is moderately large. We give the first data structure with near linear space that achieves truly sublinear query time when the dimension is any constant multiple of log n. Specifically, the preprocessing time and space are O(n^{1+delta}) for any constant delta>0, and the expected query time is n^{1-1/O(c log c)} for d = c log n. The data structure is simple and is based on a new "augmented, randomized, lopsided" variant of k-d trees. It matches (in fact, slightly improves) the performance of previous combinatorial algorithms that work only in the case of offline queries [Impagliazzo, Lovett, Paturi, and Schneider (2014) and Chan (SODA'15)]. It leads to slightly faster combinatorial algorithms for all-pairs shortest paths in general real-weighted graphs and rectangular Boolean matrix multiplication.
In the offline case, we show that the problem can be reduced to the Boolean orthogonal vectors problem and thus admits an n^{2-1/O(log c)}-time non-combinatorial algorithm [Abboud, Williams, and Yu (SODA'15)]. This reduction is also simple and is based on range trees.
Finally, we use a similar approach to obtain a small improvement to Indyk's data structure [FOCS'98] for approximate l_infinity nearest neighbor search when d = c log n.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.27/LIPIcs.SoCG.2017.27.pdf
computational geometry
data structures
range searching
nearest neighbor searching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
28:1
28:13
10.4230/LIPIcs.SoCG.2017.28
article
Dynamic Orthogonal Range Searching on the RAM, Revisited
Chan, Timothy M.
Tsakalidis, Konstantinos
We study a longstanding problem in computational geometry: 2-d dynamic orthogonal range reporting. We present a new data structure achieving O(log n / log log n + k) optimal query time and O(log^{2/3+o(1)}n) update time (amortized) in the word RAM model, where n is the number of data points and k is the output size. This is the first improvement in over 10 years of Mortensen's previous result [SIAM J. Comput., 2006], which has O(log^{7/8+epsilon}n) update time for an arbitrarily small constant epsilon.
In the case of 3-sided queries, our update time reduces to O(log^{1/2+epsilon}n), improving Wilkinson's previous bound [ESA 2014] of O(log^{2/3+epsilon}n).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.28/LIPIcs.SoCG.2017.28.pdf
dynamic data structures
range searching
computational geometry
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
29:1
29:15
10.4230/LIPIcs.SoCG.2017.29
article
On Bend-Minimized Orthogonal Drawings of Planar 3-Graphs
Chang, Yi-Jun
Yen, Hsu-Chun
An orthogonal drawing of a graph is a planar drawing where each edge is drawn as a sequence of horizontal and vertical line segments. Finding a bend-minimized orthogonal drawing of a planar graph of maximum degree 4 is NP-hard. The problem becomes tractable for planar graphs of maximum degree 3, and the fastest known algorithm takes O(n^5 log n) time. Whether a faster algorithm exists has been a long-standing open problem in graph drawing. In this paper we present an algorithm that takes only O~(n^{17/7}) time, which is a significant improvement over the previous state of the art.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.29/LIPIcs.SoCG.2017.29.pdf
Bend minimization
graph drawing
orthogonal drawing
planar graph
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
30:1
30:15
10.4230/LIPIcs.SoCG.2017.30
article
Adaptive Planar Point Location
Cheng, Siu-Wing
Lau, Man-Kit
We present a self-adjusting point location structure for convex subdivisions. Let n be the number of vertices in a convex subdivision S. Our structure for S uses O(n) space and processes any online query sequence sigma in O(n + OPT) time, where OPT is the minimum time required by any linear decision tree for answering point location queries in S to process sigma. The O(n + OPT) time bound includes the preprocessing time. Our result is a two-dimensional analog of the static optimality property of splay trees. For connected subdivisions, we achieve a processing time of O(|sigma| log log n + n + OPT).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.30/LIPIcs.SoCG.2017.30.pdf
point location
planar subdivision
static optimality
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
31:1
31:15
10.4230/LIPIcs.SoCG.2017.31
article
High Dimensional Consistent Digital Segments
Chiu, Man-Kwun
Korman, Matias
We consider the problem of digitalizing Euclidean line segments from R^d to Z^d. Christ {et al.} (DCG, 2012) showed how to construct a set of {consistent digital segments} (CDS) for d=2: a collection of segments connecting any two points in Z^2 that satisfies the natural extension of the Euclidean axioms to Z^d. In this paper we study the construction of CDSs in higher dimensions.
We show that any total order can be used to create a set of {consistent digital rays} CDR in Z^d (a set of rays emanating from a fixed point p that satisfies the extension of the Euclidean axioms). We fully characterize for which total orders the construction holds and study their Hausdorff distance, which in particular positively answers the question posed by Christ {et al.}.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.31/LIPIcs.SoCG.2017.31.pdf
Consistent Digital Line Segments
Digital Geometry
Computer Vision
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
32:1
32:16
10.4230/LIPIcs.SoCG.2017.32
article
TSP With Locational Uncertainty: The Adversarial Model
Citovsky, Gui
Mayer, Tyler
Mitchell, Joseph S. B.
In this paper we study a natural special case of the Traveling Salesman Problem (TSP) with point-locational-uncertainty which we will call the adversarial TSP problem (ATSP). Given a metric space (X, d) and a set of subsets R = {R_1, R_2, ... , R_n} : R_i subseteq X, the goal is to devise an ordering of the regions, sigma_R, that the tour will visit such that when a single point is chosen from each region, the induced tour over those points in the ordering prescribed by sigma_R is as short as possible. Unlike the classical locational-uncertainty-TSP problem, which focuses on minimizing the expected length of such a tour when the point within each region is chosen according to some probability distribution, here, we focus on the adversarial model in which once the choice of sigma_R is announced, an adversary selects a point from each region in order to make the resulting tour as long as possible. In other words, we consider an offline problem in which the goal is to determine an ordering of the regions R that is optimal with respect to the ``worst'' point possible within each region being chosen by an adversary, who knows the chosen ordering. We give a 3-approximation when R is a set of arbitrary regions/sets of points in a metric space. We show how geometry leads to improved constant factor approximations when regions are parallel line segments of the same lengths, and a polynomial-time approximation scheme (PTAS) for the important special case in which R is a set of disjoint unit disks in the plane.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.32/LIPIcs.SoCG.2017.32.pdf
traveling salesperson problem
TSP with neighborhoods
approximation algorithms
uncertainty
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
33:1
33:16
10.4230/LIPIcs.SoCG.2017.33
article
On Planar Greedy Drawings of 3-Connected Planar Graphs
Da Lozzo, Giordano
D'Angelo, Anthony
Frati, Fabrizio
A graph drawing is greedy if, for every ordered pair of vertices (x,y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed.
In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra.
In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra's result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.33/LIPIcs.SoCG.2017.33.pdf
Greedy drawings
3-connectivity
planar graphs
convex drawings
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
34:1
34:16
10.4230/LIPIcs.SoCG.2017.34
article
Origamizer: A Practical Algorithm for Folding Any Polyhedron
Demaine, Erik D.
Tachi, Tomohiro
It was established at SoCG'99 that every polyhedral complex can be folded from a sufficiently large square of paper, but the known algorithms are extremely impractical, wasting most of the material and making folds through many layers of paper. At a deeper level, these foldings get the topology wrong, introducing many gaps (boundaries) in the surface, which results in flimsy foldings in practice. We develop a new algorithm designed specifically for the practical folding of real paper into complicated polyhedral models. We prove that the algorithm correctly folds any oriented polyhedral manifold, plus an arbitrarily small amount of additional structure on one side of the surface (so for closed manifolds, inside the model). This algorithm is the first to attain the watertight property: for a specified cutting of the manifold into a topological disk with boundary, the folding maps the boundary of the paper to within epsilon of the specified boundary of the surface (in Fréchet distance). Our foldings also have the geometric feature that every convex face is folded seamlessly, i.e., as one unfolded convex polygon of the piece of paper. This work provides the theoretical underpinnings for Origamizer, freely available software written by the second author, which has enabled practical folding of many complex polyhedral models such as the Stanford bunny.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.34/LIPIcs.SoCG.2017.34.pdf
origami
folding
polyhedra
Voronoi diagram
computational geometry
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
35:1
35:15
10.4230/LIPIcs.SoCG.2017.35
article
Computing the Geometric Intersection Number of Curves
Despré, Vincent
Lazarus, Francis
The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most l on a combinatorial surface of complexity n we describe simple algorithms to (1) compute the geometric intersection number of c in O(n+ l^2) time, (2) construct a curve homotopic to c that realizes this geometric intersection number in O(n+l^4) time, (3) decide if the geometric intersection number of c is zero, i.e. if c is homotopic to a simple curve, in O(n+l log^2 l) time.
To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a O(n+g^2l^2) time complexity on a genus g surface without boundary. No polynomial time algorithm was known for problem (2). Interestingly, our solution to problem (3) is the first quasi-linear algorithm since the problem was raised by Poincare more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most l in O(n+ l^2) time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.35/LIPIcs.SoCG.2017.35.pdf
computational topology
curves on surfaces
combinatorial geodesic
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
36:1
36:16
10.4230/LIPIcs.SoCG.2017.36
article
Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers
Dey, Tamal K.
Mémoli, Facundo
Wang, Yusu
Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called nerve of a cover of the domain.
In this paper, we aim to analyze the topological information encoded in these structures in order to provide better understanding of these structures and facilitate their practical usage.
More specifically, we show that the one-dimensional homology of the nerve complex N(U) of a path-connected cover U of a domain X cannot be richer than that of the domain X itself. Intuitively, this result means that no new H_1-homology class can be "created" under a natural map from X to the nerve complex N(U). Equipping X with a pseudometric d, we further refine this result and characterize the classes of H_1(X) that may survive in the nerve complex using the notion of size of the covering elements in U. These fundamental results about nerve complexes then lead to an analysis of the H_1-homology of Reeb spaces, mappers and multiscale mappers.
The analysis of H_1-homology groups unfortunately does not extend to higher dimensions. Nevertheless, by using a map-induced metric, establishing a Gromov-Hausdorff convergence result between mappers and the domain, and interleaving relevant modules, we can still analyze the persistent homology groups of (multiscale) mappers to establish a connection to Reeb spaces.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.36/LIPIcs.SoCG.2017.36.pdf
Topology
Nerves
Mapper
Multiscale Mapper
Reeb Spaces
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
37:1
37:16
10.4230/LIPIcs.SoCG.2017.37
article
Locality-Sensitive Hashing of Curves
Driemel, Anne
Silvestri, Francesco
We study data structures for storing a set of polygonal curves in R^d such that, given a query curve, we can efficiently retrieve similar curves from the set, where similarity is measured using the discrete Fréchet distance or the dynamic time warping distance. To this end we devise the first locality-sensitive hashing schemes for these distance measures. A major challenge is posed by the fact that these distance measures internally optimize the alignment between the curves. We give solutions for different types of alignments including constrained and unconstrained versions. For unconstrained alignments, we improve over a result by Indyk [SoCG 2002] for short curves. Let n be the number of input curves and let m be the maximum complexity of a curve in the input. In the particular case where m <= (a/(4d)) log n, for some fixed a>0, our solutions imply an approximate near-neighbor data structure for the discrete Fréchet distance that uses space in O(n^(1+a) log n) and achieves query time in O(n^a log^2 n) and constant approximation factor. Furthermore, our solutions provide a trade-off between approximation quality and computational performance: for any parameter k in [m], we can give a data structure that uses space in O(2^(2k) m^(k-1) n log n + nm), answers queries in O( 2^(2k) m^(k) log n) time and achieves approximation factor in O(m/k).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.37/LIPIcs.SoCG.2017.37.pdf
Locality-Sensitive Hashing
Frechet distance
Dynamic Time Warping
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
38:1
38:15
10.4230/LIPIcs.SoCG.2017.38
article
Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning
Dutta, Kunal
Ghosh, Arijit
Jartoux, Bruno
Mustafa, Nabil H.
The packing lemma of Haussler states that given a set system (X,R) with bounded VC dimension, if every pair of sets in R have large symmetric difference, then R cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity.
In this paper we present several new results and applications related to packings:
* an optimal lower bound for shallow packings,
* improved bounds on Mnets, providing a combinatorial analogue to Macbeath regions in convex geometry,
* we observe that Mnets provide a general, more powerful framework from which the state-of-the-art unweighted epsilon-net results follow immediately, and
* simplifying and generalizing one of the main technical tools in [Fox et al. , J. of the EMS, to appear].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.38/LIPIcs.SoCG.2017.38.pdf
Epsilon-nets
Haussler's packing lemma
Mnets
shallow-cell complexity
shallow packing lemma
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
39:1
39:16
10.4230/LIPIcs.SoCG.2017.39
article
Topological Data Analysis with Bregman Divergences
Edelsbrunner, Herbert
Wagner, Hubert
We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.39/LIPIcs.SoCG.2017.39.pdf
Topological data analysis
Bregman divergences
persistent homology
proximity complexes
algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
40:1
40:15
10.4230/LIPIcs.SoCG.2017.40
article
Finding Small Hitting Sets in Infinite Range Spaces of Bounded VC-Dimension
Elbassioni, Khaled
We consider the problem of finding a small hitting set in an infinite range space F=(Q,R) of bounded VC-dimension. We show that, under reasonably general assumptions, the infinite-dimensional convex relaxation can be solved (approximately) efficiently by multiplicative weight updates. As a consequence, we get an algorithm that finds, for any delta>0, a set of size O(s_F(z^*_F)) that hits (1-delta)-fraction of R (with respect to a given measure) in time proportional to log(1/delta), where s_F(1/epsilon) is the size of the smallest epsilon-net the range space admits, and z^*_F is the value of the fractional optimal solution. This exponentially improves upon previous results which achieve the same approximation guarantees with running time proportional to poly(1/delta). Our assumptions hold, for instance, in the case when the range space represents the visibility regions of a polygon in the plane, giving thus a deterministic polynomial-time O(log z^*_F)-approximation algorithm for guarding (1-delta)-fraction of the area of any given simple polygon, with running time proportional to polylog(1/delta).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.40/LIPIcs.SoCG.2017.40.pdf
VC-dimension
approximation algorithms
fractional covering
multiplicative weights update
art gallery problem
polyhedral separators
geometric cove
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
41:1
41:15
10.4230/LIPIcs.SoCG.2017.41
article
A Nearly Quadratic Bound for the Decision Tree Complexity of k-SUM
Ezra, Esther
Sharir, Micha
We show that the k-SUM problem can be solved by a linear decision tree of depth O(n^2 log^2 n),improving the recent bound O(n^3 log^3 n) of Cardinal et al. Our bound depends linearly on k, and allows us to conclude that the number of linear queries required to decide the n-dimensional Knapsack or SubsetSum problems is only O(n^3 log n), improving the currently best known bounds by a factor of n. Our algorithm extends to the RAM model, showing that the k-SUM problem can be solved in expected polynomial time, for any fixed k, with the above bound on the number of linear queries. Our approach relies on a new point-location mechanism, exploiting "Epsilon-cuttings" that are based on vertical decompositions in hyperplane arrangements in high dimensions.
A major side result of the analysis in this paper is a sharper bound on the complexity of the vertical decomposition of such an arrangement (in terms of its dependence on the dimension). We hope that this study will reveal further structural properties of vertical decompositions in hyperplane arrangements.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.41/LIPIcs.SoCG.2017.41.pdf
k-SUM and k-LDT
linear decision tree
hyperplane arrangements
point-location
vertical decompositions
Epsilon-cuttings
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
42:1
42:16
10.4230/LIPIcs.SoCG.2017.42
article
Computing the Fréchet Gap Distance
Fan, Chenglin
Raichel, Benjamin
Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Frechet distance is one of the most well studied similarity measures. Informally, the Fréchet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study a variant called the Fréchet gap distance. In the man and dog analogy, the Fréchet gap distance minimizes the difference of the longest and smallest leash lengths used over the entire walk. This measure in some ways better captures our intuitive notions of curve similarity, for example giving distance zero to translated copies of the same curve.
The Fréchet gap distance was originally introduced by Filtser and Katz (2015) in the context of the discrete Fréchet distance. Here we study the continuous version, which presents a number of additional challenges not present in discrete case. In particular, the continuous nature makes bounding and searching over the critical events a rather difficult task.
For this problem we give an O(n^5 log(n)) time exact algorithm and a more efficient O(n^2 log(n) + (n^2/epsilon) log(1/epsilon)) time (1+epsilon)-approximation algorithm, where n is the total number of vertices of the input curves. Note that for (small enough) constant epsilon and ignoring logarithmic factors, our approximation has quadratic running time, matching the lower bound, assuming SETH (Bringmann 2014), for approximating the standard Fréchet distance for general curves.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.42/LIPIcs.SoCG.2017.42.pdf
Frechet Distance
Approximation
Polygonal Curves
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
43:1
43:15
10.4230/LIPIcs.SoCG.2017.43
article
Erdös-Hajnal Conjecture for Graphs with Bounded VC-Dimension
Fox, Jacob
Pach, János
Suk, Andrew
The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e^{(log n)^{1 - o(1)}}. The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, e^{c sqrt{log n}}, due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erdos-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e^{Omega(log n)}. Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties.
Our main tool is a partitioning result found by Lovasz-Szegedy and Alon-Fischer-Newman, which is called the "ultra-strong regularity lemma" for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/epsilon)^{O(d)}, improving the original bound of (1/epsilon)^{O(d^2)} in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(n^k)-time algorithm for finding a partition meeting the requirements in the k-uniform setting.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.43/LIPIcs.SoCG.2017.43.pdf
VC-dimension
Ramsey theory
regularity lemma
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
44:1
44:15
10.4230/LIPIcs.SoCG.2017.44
article
Implementing Delaunay Triangulations of the Bolza Surface
Iordanov, Iordan
Teillaud, Monique
The CGAL library offers software packages to compute Delaunay triangulations of the (flat) torus of genus one in two and three dimensions. To the best of our knowledge, there is no available software for the simplest possible extension, i.e., the Bolza surface, a hyperbolic manifold homeomorphic to a torus of genus two.
In this paper, we present an implementation based on the theoretical results and the incremental algorithm proposed last year at SoCG by Bogdanov, Teillaud, and Vegter. We describe the representation of the triangulation, we detail the different steps of the algorithm, we study predicates, and report experimental results.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.44/LIPIcs.SoCG.2017.44.pdf
hyperbolic surface
Fuchsian group
arithmetic issues
Dehn's algorithm
CGAL
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
45:1
45:16
10.4230/LIPIcs.SoCG.2017.45
article
Lower Bounds for Differential Privacy from Gaussian Width
Kattis, Assimakis
Nikolov, Aleksandar
We study the optimal sample complexity of a given workload of linear queries under the constraints of differential privacy. The sample complexity of a query answering mechanism under error parameter alpha is the smallest n such that the mechanism answers the workload with error at most alpha on any database of size n. Following a line of research started by Hardt and Talwar [STOC 2010], we analyze sample complexity using the tools of asymptotic convex geometry. We study the sensitivity polytope, a natural convex body associated with a query workload that quantifies how query answers can change between neighboring databases. This is the information that, roughly speaking, is protected by a differentially private algorithm, and, for this reason, we expect that a "bigger" sensitivity polytope implies larger sample complexity. Our results identify the mean Gaussian width as an appropriate measure of the size of the polytope, and show sample complexity lower bounds in terms of this quantity. Our lower bounds completely characterize the workloads for which the Gaussian noise mechanism is optimal up to constants as those having asymptotically maximal Gaussian width.
Our techniques also yield an alternative proof of Pisier's Volume Number Theorem which also suggests an approach to improving the parameters of the theorem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.45/LIPIcs.SoCG.2017.45.pdf
differential privacy
convex geometry
lower bounds
sample complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
46:1
46:15
10.4230/LIPIcs.SoCG.2017.46
article
Constrained Triangulations, Volumes of Polytopes, and Unit Equations
Kerber, Michael
Tichy, Robert
Weitzer, Mario
Given a polytope P in R^d and a subset U of its vertices, is there a triangulation of P using d-simplices that all contain U? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of P. Our proof relates triangulations of P to triangulations of its "shadow", a projection to a lower-dimensional space determined by U. In particular, we obtain a formula relating the volume of P with the volume of its shadow. This leads to an exact formula for the volume of a polytope arising in the theory of unit equations.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.46/LIPIcs.SoCG.2017.46.pdf
constrained triangulations
simplotopes
volumes of polytopes
projections of polytopes
unit equations
S-integers
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
47:1
47:15
10.4230/LIPIcs.SoCG.2017.47
article
Proper Coloring of Geometric Hypergraphs
Keszegh, Balázs
Pálvölgyi, Dömötör
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.47/LIPIcs.SoCG.2017.47.pdf
discrete geometry
decomposition of multiple coverings
geometric hypergraph coloring
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
48:1
48:16
10.4230/LIPIcs.SoCG.2017.48
article
Computing Representative Networks for Braided Rivers
Kleinhans, Maarten
van Kreveld, Marc
Ophelders, Tim
Sonke, Willem
Speckmann, Bettina
Verbeek, Kevin
Drainage networks on terrains have been studied extensively from an algorithmic perspective. However, in drainage networks water flow cannot bifurcate and hence they do not model braided rivers (multiple channels which split and join, separated by sediment bars). We initiate the algorithmic study of braided rivers by employing the descending quasi Morse-Smale complex on the river bed (a polyhedral terrain), and extending it with a certain ordering of bars from the one river bank to the other. This allows us to compute a graph that models a representative channel network, consisting of lowest paths. To ensure that channels in this network are sufficiently different we define a sand function that represents the volume of sediment separating them. We show that in general the problem of computing a maximum network of non-crossing channels which are delta-different from each other (as measured by the sand function) is NP-hard. However, using our ordering between the river banks, we can compute a maximum delta-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.48/LIPIcs.SoCG.2017.48.pdf
braided rivers
Morse-Smale complex
persistence
network extraction
polyhedral terrain
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
49:1
49:15
10.4230/LIPIcs.SoCG.2017.49
article
A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations
Lubiw, Anna
Masárová, Zuzana
Wagner, Uli
Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n^7) on the length of the flip sequence.
Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.49/LIPIcs.SoCG.2017.49.pdf
triangulations
reconfiguration
flip
constrained triangulations
Delaunay triangulation
shellability
piecewise linear balls
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
50:1
50:16
10.4230/LIPIcs.SoCG.2017.50
article
A Spectral Gap Precludes Low-Dimensional Embeddings
Naor, Assaf
We prove that if an n-vertex O(1)-expander embeds with average distortion D into a finite dimensional normed space X, then necessarily the dimension of X is at least n^{c/D} for some universal constant c>0. This is sharp up to the value of the constant c, and it improves over the previously best-known estimate dim(X)> c(log n)^2/D^2 of Linial, London and Rabinovich, strengthens a theorem of Matousek, and answers a question of Andoni, Nikolov, Razenshteyn and Waingarten.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.50/LIPIcs.SoCG.2017.50.pdf
Metric embeddings
dimensionality reduction
expander graphs
nonlinear spectral gaps
nearest neighbor search
complex interpolation
Markov type.
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
51:1
51:15
10.4230/LIPIcs.SoCG.2017.51
article
Dynamic Geodesic Convex Hulls in Dynamic Simple Polygons
Oh, Eunjin
Ahn, Hee-Kap
We consider the geodesic convex hulls of points in a simple polygonal region in the presence of non-crossing line segments (barriers) that subdivide the region into simply connected faces. We present an algorithm together with data structures for maintaining the geodesic convex hull of points in each face in a sublinear update time under the fully-dynamic setting where both input points and barriers change by insertions and deletions. The algorithm processes a mixed update sequence of insertions and deletions of points and barriers. Each update takes O(n^2/3 log^2 n) time with high probability, where n is the total number of the points and barriers at the moment. Our data structures support basic queries on the geodesic convex hull, each of which takes O(polylog n) time. In addition, we present an algorithm together with data structures for geodesic triangle counting queries under the fully-dynamic setting. With high probability, each update takes O(n^2/3 log n) time, and each query takes O(n^2/3 log n) time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.51/LIPIcs.SoCG.2017.51.pdf
Dynamic geodesic convex hull
dynamic simple polygons
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
52:1
52:15
10.4230/LIPIcs.SoCG.2017.52
article
Voronoi Diagrams for a Moderate-Sized Point-Set in a Simple Polygon
Oh, Eunjin
Ahn, Hee-Kap
Given a set of sites in a simple polygon, a geodesic Voronoi diagram partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones for m < n/polylog n. Moreover, the algorithms for the nearest-point and farthest-point Voronoi diagrams are optimal for m < n/polylog n. This partially answers a question posed by Mitchell in the Handbook of Computational Geometry.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.52/LIPIcs.SoCG.2017.52.pdf
Simple polygons
Voronoi diagrams
geodesic distance
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
53:1
53:16
10.4230/LIPIcs.SoCG.2017.53
article
A Quest to Unravel the Metric Structure Behind Perturbed Networks
Parthasarathy, Srinivasan
Sivakoff, David
Tian, Minghao
Wang, Yusu
Graphs and network data are ubiquitous across a wide spectrum of scientific and application domains. Often in practice, an input graph can be considered as an observed snapshot of a (potentially
continuous) hidden domain or process. Subsequent analysis, processing, and inferences are then performed on this observed graph. In this paper we advocate the perspective that an observed graph is often a noisy version of some discretized 1-skeleton of a hidden domain, and specifically we will consider the following natural network model: We assume that there is a true graph G^* which is a certain proximity graph for points sampled from a hidden domain X; while the observed graph G is an Erdos-Renyi type perturbed version of G^*.
Our network model is related to, and slightly generalizes, the much-celebrated small-world network model originally proposed by Watts and Strogatz. However, the main question we aim to answer is orthogonal to the usual studies of network models (which often focuses on characterizing / predicting behaviors and properties of real-world networks). Specifically, we aim to recover the metric structure of G^* (which reflects that of the hidden space X as we will show) from the observed graph G. Our main result is that a simple filtering process based on the Jaccard index can recover this metric within a multiplicative factor of 2 under our network model. Our work makes one step towards the general question of inferring structure of a hidden space from its observed noisy graph representation. In addition, our results also provide a theoretical understanding for Jaccard-Index-based denoising approaches.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.53/LIPIcs.SoCG.2017.53.pdf
metric structure
Erdös-Rényi perturbation
graphs
doubling measure
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
54:1
54:16
10.4230/LIPIcs.SoCG.2017.54
article
From Crossing-Free Graphs on Wheel Sets to Embracing Simplices and Polytopes with Few Vertices
Pilz, Alexander
Welzl, Emo
Wettstein, Manuel
A set P = H cup {w} of n+1 points in the plane is called a wheel set if all points but w are extreme. We show that for the purpose of counting crossing-free geometric graphs on P, it suffices to know the so-called frequency vector of P. While there are roughly 2^n distinct order types that correspond to wheel sets, the number of frequency vectors is only about 2^{n/2}.
We give simple formulas in terms of the frequency vector for the number of crossing-free spanning cycles, matchings, w-embracing triangles, and many more. Based on these formulas, the corresponding numbers of graphs can be computed efficiently.
Also in higher dimensions, wheel sets turn out to be a suitable model to approach the problem of computing the simplicial depth of a point w in a set H, i.e., the number of simplices spanned by H that contain w. While the concept of frequency vectors does not generalize easily, we show how to apply similar methods in higher dimensions. The result is an O(n^{d-1}) time algorithm for computing the simplicial depth of a point w in a set H of n d-dimensional points, improving on the previously best bound of O(n^d log n).
Configurations equivalent to wheel sets have already been used by Perles for counting the faces of high-dimensional polytopes with few vertices via the Gale dual. Based on that we can compute the number of facets of the convex hull of n=d+k points in general position in R^d in time O(n^max(omega,k-2)) where omega = 2.373, even though the asymptotic number of facets may be as large as n^k.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.54/LIPIcs.SoCG.2017.54.pdf
Geometric Graph
Wheel Set
Simplicial Depth
Gale Transform
Polytope
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
55:1
55:15
10.4230/LIPIcs.SoCG.2017.55
article
Approximate Range Counting Revisited
Rahul, Saladi
We study range-searching for colored objects, where one has to count (approximately) the number of colors present in a query range. The problems studied mostly involve orthogonal range-searching in two and three dimensions, and the dual setting of rectangle stabbing by points. We present optimal and near-optimal solutions for these problems. Most of the results are obtained via reductions to the approximate uncolored version, and improved data-structures for them. An additional contribution of this work is the introduction of nested shallow cuttings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.55/LIPIcs.SoCG.2017.55.pdf
orthogonal range searching
rectangle stabbing
colors
approximate count
geometric data structures
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
56:1
56:15
10.4230/LIPIcs.SoCG.2017.56
article
Coloring Curves That Cross a Fixed Curve
Rok, Alexandre
Walczak, Bartosz
We prove that for every integer t greater than or equal to 1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is chi-bounded. This is essentially the strongest chi-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers k > 1 and t > 0, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has O(n log n) edges.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.56/LIPIcs.SoCG.2017.56.pdf
String graphs
chi-boundedness
k-quasi-planar graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
57:1
57:16
10.4230/LIPIcs.SoCG.2017.57
article
Barcodes of Towers and a Streaming Algorithm for Persistent Homology
Kerber, Michael
Schreiber, Hannah
A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and can be efficiently computed by a streaming algorithm, both in theory and in practice. Furthermore, we show that our approach can be combined with a streaming algorithm to compute the barcode of the tower via matrix reduction. The space complexity of the algorithm does not depend on the length of the tower, but the maximal size of any subcomplex within the tower. Experimental evaluations show that our approach can efficiently handle towers with billions of complexes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.57/LIPIcs.SoCG.2017.57.pdf
Persistent Homology
Topological Data Analysis
Matrix reduction
Streaming algorithms
Simplicial Approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
58:1
58:16
10.4230/LIPIcs.SoCG.2017.58
article
Algorithmic Interpretations of Fractal Dimension
Sidiropoulos, Anastasios
Sridhar, Vijay
We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces which agrees with standard notions used to empirically estimate the fractal dimension of various sets. We define the fractal dimension of some metric space to be the infimum delta>0, such that for any eps>0, for any ball B of radius r >= 2eps, and for any eps-net N, we have |B cap N|=O((r/eps)^delta).
Using this definition we obtain faster algorithms for a plethora of classical problems on sets of low fractal dimension in Euclidean space. Our results apply to exact and fixed-parameter algorithms, approximation schemes, and spanner constructions. Interestingly, the dependence of the performance of these algorithms on the fractal dimension nearly matches the currently best-known dependence on the standard Euclidean dimension. Thus, when the fractal dimension is strictly smaller than the ambient dimension, our results yield improved solutions in all of these settings.
We remark that our definition of fractal definition is equivalent up to constant factors to the well-studied notion of doubling dimension.
However, in the problems that we consider, the dimension appears in the exponent of the running time, and doubling dimension is not precise enough for capturing the best possible such exponent for subsets of Euclidean space. Thus our work is orthogonal to previous results on spaces of low doubling dimension; while algorithms on spaces of low doubling dimension seek to extend results from the case of low dimensional Euclidean spaces to more general metric spaces, our goal is to obtain faster algorithms for special pointsets in Euclidean space.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.58/LIPIcs.SoCG.2017.58.pdf
fractal dimension
exact algorithms
fixed parameter tractability
approximation schemes
spanners
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
59:1
59:15
10.4230/LIPIcs.SoCG.2017.59
article
Disjointness Graphs of Segments
Pach, János
Tardos, Gábor
Tóth, Géza
The disjointness graph G=G(S) of a set of segments S in R^d, d>1 is a graph whose vertex set is S and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies chi(G)<=omega(G)^4+omega(G)^3 where omega(G) denotes the clique number of G. It follows, that S has at least cn^{1/5} pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments.
We show that computing omega(G) and chi(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of G in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (omega(G)=2), but whose chromatic numbers are arbitrarily large.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.59/LIPIcs.SoCG.2017.59.pdf
disjointness graph
chromatic number
clique number
chi-bounded
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
60:1
60:16
10.4230/LIPIcs.SoCG.2017.60
article
Bicriteria Rectilinear Shortest Paths among Rectilinear Obstacles in the Plane
Wang, Haitao
Given a rectilinear domain P of h pairwise-disjoint rectilinear obstacles with a total of n vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points s and t in P. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when h is relatively small. For example, for computing a minimum-link shortest s-t path, the previous algorithm runs in O(n log^{3/2} n) time while the time of our new algorithm is O(n + h log^{3/2} h).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.60/LIPIcs.SoCG.2017.60.pdf
rectilinear paths
shortest paths
minimum-link paths
bicriteria paths
rectilinear polygons
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
61:1
61:16
10.4230/LIPIcs.SoCG.2017.61
article
Quickest Visibility Queries in Polygonal Domains
Wang, Haitao
Let s be a point in a polygonal domain P of h-1 holes and n vertices. We consider the following quickest visibility query problem. Given a query point q in P, the goal is to find a shortest path in P to move from s to see q as quickly as possible. Previously, Arkin et al. (SoCG 2015) built a data structure of size O(n^2 2^alpha(n) log n) that can answer each query in O(K log^2 n) time, where alpha(n) is the inverse Ackermann function and K is the size of the visibility polygon of q in P (and K can be Theta(n) in the worst case). In this paper, we present a new data structure of size O(n log h + h^2) that can answer each query in O(h log h log n) time. Our result improves the previous work when h is relatively small. In particular, if h is a constant, then our result even matches the best result for the simple polygon case (i.e., h = 1), which is optimal. As a by-product, we also have a new algorithm for the following shortest-path-to-segment query problem. Given a query line segment tau in P, the query seeks a shortest path from s to all points of tau. Previously, Arkin et al. gave a data structure of size O(n^2 2^alpha(n) log n) that can answer each query in O(log^2 n) time, and another data structure of size O(n^3 log n) with O(log n) query time. We present a data structure of size O(n) with query time O(h log n/h), which favors small values of h and is optimal when h = O(1).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.61/LIPIcs.SoCG.2017.61.pdf
shortest paths
visibility
quickest visibility queries
shortest path to segments
polygons with holes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
62:1
62:5
10.4230/LIPIcs.SoCG.2017.62
article
Zapping Zika with a Mosquito-Managing Drone: Computing Optimal Flight Patterns with Minimum Turn Cost (Multimedia Contribution)
Becker, Aaron T.
Debboun, Mustapha
Fekete, Sándor P.
Krupke, Dominik
Nguyen, An
We present results arising from the problem of sweeping a mosquito-infested area with an Un-manned Aerial Vehicle (UAV) equipped with an electrified metal grid. This is related to the Traveling Salesman Problem, the Lawn Mower Problem and, most closely, Milling with TurnCost. Planning a good trajectory can be reduced to considering penalty and budget variants of covering a grid graph with minimum turn cost. On the theoretical side, we show the solution of a problem from The Open Problems Project that had been open for more than 15 years, and hint at approximation algorithms. On the practical side, we describe an exact method based on Integer Programming that is able to compute provably optimal instances with over 500 pixels. These solutions are actually used for practical trajectories, as demonstrated in the video.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.62/LIPIcs.SoCG.2017.62.pdf
Covering tours
turn cost
complexity
exact optimization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
63:1
63:5
10.4230/LIPIcs.SoCG.2017.63
article
Ruler of the Plane - Games of Geometry (Multimedia Contribution)
Beekhuis, Sander
Buchin, Kevin
Castermans, Thom
Hurks, Thom
Sonke, Willem
Ruler of the Plane is a set of games illustrating concepts from combinatorial and computational geometry. The games are based on the art gallery problem, ham-sandwich cuts, the Voronoi game, and geometric network connectivity problems like the Euclidean minimum spanning tree and traveling salesperson problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.63/LIPIcs.SoCG.2017.63.pdf
art gallery problem
ham-sandwich cuts
Voronoi game
traveling sales-person problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
64:1
64:5
10.4230/LIPIcs.SoCG.2017.64
article
Folding Free-Space Diagrams: Computing the Fréchet Distance between 1-Dimensional Curves (Multimedia Contribution)
Buchin, Kevin
Chun, Jinhee
Löffler, Maarten
Markovic, Aleksandar
Meulemans, Wouter
Okamoto, Yoshio
Shiitada, Taichi
By folding the free-space diagram for efficient preprocessing, we show that the Frechet distance between 1D curves can be computed in O(nk log n) time, assuming one curve has ply k.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.64/LIPIcs.SoCG.2017.64.pdf
Frechet distance
ply
k-packed curves
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
65:1
65:4
10.4230/LIPIcs.SoCG.2017.65
article
Cardiac Trabeculae Segmentation: an Application of Computational Topology (Multimedia Contribution)
Chen, Chao
Metaxas, Dimitris
Wang, Yusu
Wu, Pengxiang
In this video, we present a research project on cardiac trabeculae segmentation. Trabeculae are fine muscle columns within human ventricles whose both ends are attached to the wall. Extracting these structures are very challenging even with state-of-the-art image segmentation techniques. We observed that these structures form natural topological handles. Based on such observation, we developed a topological approach, which employs advanced computational topology methods and achieve high quality segmentation results.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.65/LIPIcs.SoCG.2017.65.pdf
image segmentation
trabeculae
persistent homology
homology localization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
66:1
66:5
10.4230/LIPIcs.SoCG.2017.66
article
MatchTheNet - An Educational Game on 3-Dimensional Polytopes (Multimedia Contribution)
Joswig, Michael
Loho, Georg
Lorenz, Benjamin
Raber, Rico
We present an interactive game which challenges a single player to match 3-dimensional polytopes to their planar nets. It is open source, and it runs in standard web browsers.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.66/LIPIcs.SoCG.2017.66.pdf
three-dimensional convex polytopes
unfoldings
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2017-06-20
77
67:1
67:4
10.4230/LIPIcs.SoCG.2017.67
article
On Balls in a Hilbert Polygonal Geometry (Multimedia Contribution)
Nielsen, Frank
Shao, Laetitia
Hilbert geometry is a metric geometry that extends the hyperbolic Cayley-Klein geometry. In this video, we explain the shape of balls and their properties in a convex polygonal Hilbert geometry. First, we study the combinatorial properties of Hilbert balls, showing that the shapes of Hilbert polygonal balls depend both on the center location and on the complexity of the Hilbert domain but not on their radii. We give an explicit description of the Hilbert ball for any given center and radius. We then study the intersection of two Hilbert balls. In particular, we consider the cases of empty intersection and internal/external tangencies.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol077-socg2017/LIPIcs.SoCG.2017.67/LIPIcs.SoCG.2017.67.pdf
Projective geometry
Hilbert geometry
balls