Volume
LIPIcs, Volume 224
38th International Symposium on Computational Geometry (SoCG 2022)
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Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG)
Event
SoCG 2022, June 7-10, 2022, Berlin, GermanyEditors
Publication Details
- published at: 2022-06-01
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-227-3
- DBLP: db/conf/compgeom/compgeom2022
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Document
Complete Volume
LIPIcs, Volume 224, SoCG 2022, Complete Volume
Abstract
LIPIcs, Volume 224, SoCG 2022, Complete Volume
Cite as
38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 1-1110, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@Proceedings{goaoc_et_al:LIPIcs.SoCG.2022,
title = {{LIPIcs, Volume 224, SoCG 2022, Complete Volume}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {1--1110},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022},
URN = {urn:nbn:de:0030-drops-160075},
doi = {10.4230/LIPIcs.SoCG.2022},
annote = {Keywords: LIPIcs, Volume 224, SoCG 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2022.0,
author = {Goaoc, Xavier and Kerber, Michael},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {0:i--0:xx},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.0},
URN = {urn:nbn:de:0030-drops-160087},
doi = {10.4230/LIPIcs.SoCG.2022.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Tiling with Squares and Packing Dominos in Polynomial Time
Abstract
A polyomino is a polygonal region with axis-parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container P. We give polynomial-time algorithms for deciding if P can be tiled with k× k squares for any fixed k which can be part of the input (that is, deciding if P is the union of a set of non-overlapping k× k squares) and for packing P with a maximum number of non-overlapping and axis-parallel 2× 1 dominos, allowing rotations by 90^∘. As packing is more general than tiling, the latter algorithm can also be used to decide if P can be tiled by 2× 1 dominos.
These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of 2× 2 squares is known to be NP-hard [J. Algorithms 1990]. For our three problems there are known pseudo-polynomial-time algorithms, that is, algorithms with running times polynomial in the area or perimeter of P. However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial-time algorithms for the problems. Concretely, we give a simple O(nlog n)-time algorithm for tiling with squares, where n is the number of corners of P. We then give a more involved algorithm that reduces the problems of packing and tiling with dominos to finding a maximum and perfect matching in a graph with O(n³) vertices. This leads to algorithms with running times O(n³(log³ n)/(log²log n)) and O(n³(log² n)/(log log n)), respectively.
Cite as
Anders Aamand, Mikkel Abrahamsen, Thomas Ahle, and Peter M. R. Rasmussen. Tiling with Squares and Packing Dominos in Polynomial Time. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 1:1-1:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{aamand_et_al:LIPIcs.SoCG.2022.1,
author = {Aamand, Anders and Abrahamsen, Mikkel and Ahle, Thomas and Rasmussen, Peter M. R.},
title = {{Tiling with Squares and Packing Dominos in Polynomial Time}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {1:1--1:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.1},
URN = {urn:nbn:de:0030-drops-160096},
doi = {10.4230/LIPIcs.SoCG.2022.1},
annote = {Keywords: packing, tiling, polyominos}
}
Document
On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem
Abstract
We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s.
When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case.
We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well.
The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics.
Cite as
Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.2,
author = {Afshani, Peyman and de Berg, Mark and Buchin, Kevin and Gao, Jie and L\"{o}ffler, Maarten and Nayyeri, Amir and Raichel, Benjamin and Sarkar, Rik and Wang, Haotian and Yang, Hao-Tsung},
title = {{On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {2:1--2:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.2},
URN = {urn:nbn:de:0030-drops-160109},
doi = {10.4230/LIPIcs.SoCG.2022.2},
annote = {Keywords: Approximation, Motion Planning, Scheduling}
}
Document
On Semialgebraic Range Reporting
Abstract
Semialgebraic range searching, arguably the most general version of range searching, is a fundamental problem in computational geometry. In the problem, we are to preprocess a set of points in ℝ^D such that the subset of points inside a semialgebraic region described by a constant number of polynomial inequalities of degree Δ can be found efficiently.
Relatively recently, several major advances were made on this problem. Using algebraic techniques, "near-linear space" data structures [Agarwal et al., 2013; Matoušek and Patáková, 2015] with almost optimal query time of Q(n) = O(n^{1-1/D+o(1)}) were obtained. For "fast query" data structures (i.e., when Q(n) = n^{o(1)}), it was conjectured that a similar improvement is possible, i.e., it is possible to achieve space S(n) = O(n^{D+o(1)}). The conjecture was refuted very recently by Afshani and Cheng [Afshani and Cheng, 2021]. In the plane, i.e., D = 2, they proved that S(n) = Ω(n^{Δ+1 - o(1)}/Q(n)^{(Δ+3)Δ/2}) which shows Ω(n^{Δ+1-o(1)}) space is needed for Q(n) = n^{o(1)}. While this refutes the conjecture, it still leaves a number of unresolved issues: the lower bound only works in 2D and for fast queries, and neither the exponent of n or Q(n) seem to be tight even for D = 2, as the best known upper bounds have S(n) = O(n^{m+o(1)}/Q(n)^{(m-1)D/(D-1)}) where m = binom(D+Δ,D)-1 = Ω(Δ^D) is the maximum number of parameters to define a monic degree-Δ D-variate polynomial, for any constant dimension D and degree Δ.
In this paper, we resolve two of the issues: we prove a lower bound in D-dimensions, for constant D, and show that when the query time is n^{o(1)}+O(k), the space usage is Ω(n^{m-o(1)}), which almost matches the Õ(n^{m}) upper bound and essentially closes the problem for the fast-query case, as far as the exponent of n is considered in the pointer machine model. When considering the exponent of Q(n), we show that the analysis in [Afshani and Cheng, 2021] is tight for D = 2, by presenting matching upper bounds for uniform random point sets. This shows either the existing upper bounds can be improved or to obtain better lower bounds a new fundamentally different input set needs to be constructed.
Cite as
Peyman Afshani and Pingan Cheng. On Semialgebraic Range Reporting. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.3,
author = {Afshani, Peyman and Cheng, Pingan},
title = {{On Semialgebraic Range Reporting}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {3:1--3:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.3},
URN = {urn:nbn:de:0030-drops-160117},
doi = {10.4230/LIPIcs.SoCG.2022.3},
annote = {Keywords: Computational Geometry, Range Searching, Data Structures and Algorithms, Lower Bounds}
}
Document
Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems
Abstract
Let 𝒯 be a set of n planar semi-algebraic regions in ℝ³ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess 𝒯 into a data structure so that for a query object γ, which is also a plate, we can quickly answer various intersection queries, such as detecting whether γ intersects any plate of 𝒯, reporting all the plates intersected by γ, or counting them. We focus on two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree algebraic arcs in ℝ³ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in ℝ³. These interesting special cases form the building blocks for the general case.
By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we obtain a variety of results with different storage and query-time bounds, depending on the complexity of the input and query objects. For example, if 𝒯 is a set of plates and the query objects are arcs, we obtain a data structure that uses O^*(n^{4/3}) storage (where the O^*(⋅) notation hides subpolynomial factors) and answers an intersection query in O^*(n^{2/3}) time. Alternatively, by increasing the storage to O^*(n^{3/2}), the query time can be decreased to O^*(n^{ρ}), where ρ = (2t-3)/3(t-1) < 2/3 and t ≥ 3 is the number of parameters needed to represent the query arcs.
Cite as
Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2022.4,
author = {Agarwal, Pankaj K. and Aronov, Boris and Ezra, Esther and Katz, Matthew J. and Sharir, Micha},
title = {{Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {4:1--4:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.4},
URN = {urn:nbn:de:0030-drops-160126},
doi = {10.4230/LIPIcs.SoCG.2022.4},
annote = {Keywords: Intersection searching, Semi-algebraic range searching, Point-enclosure queries, Ray-shooting queries, Polynomial partitions, Cylindrical algebraic decomposition, Multi-level partition trees, Collision detection}
}
Document
Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs
Abstract
Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized twisted drawings. A simple drawing is generalized twisted if there is a point O such that every ray emanating from O crosses every edge of the drawing at most once and there is a ray emanating from O which crosses every edge exactly once.
Via this new class of simple drawings, we show that every simple drawing of the complete graph with n vertices contains Ω(n^{1/2}) pairwise disjoint edges and a plane path of length Ω((log n)/(log log n)). Both results improve over previously known best lower bounds. On the way we show several structural results about and properties of generalized twisted drawings. We further present different characterizations of generalized twisted drawings, which might be of independent interest.
Cite as
Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger. Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2022.5,
author = {Aichholzer, Oswin and Garc{\'\i}a, Alfredo and Tejel, Javier and Vogtenhuber, Birgit and Weinberger, Alexandra},
title = {{Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {5:1--5:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.5},
URN = {urn:nbn:de:0030-drops-160136},
doi = {10.4230/LIPIcs.SoCG.2022.5},
annote = {Keywords: Simple drawings, simple topological graphs, disjoint edges, plane matching, plane path}
}
Document
Edge Partitions of Complete Geometric Graphs
Abstract
In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs).
Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars.
Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.
Cite as
Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, and Birgit Vogtenhuber. Edge Partitions of Complete Geometric Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2022.6,
author = {Aichholzer, Oswin and Obenaus, Johannes and Orthaber, Joachim and Paul, Rosna and Schnider, Patrick and Steiner, Raphael and Taubner, Tim and Vogtenhuber, Birgit},
title = {{Edge Partitions of Complete Geometric Graphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {6:1--6:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.6},
URN = {urn:nbn:de:0030-drops-160141},
doi = {10.4230/LIPIcs.SoCG.2022.6},
annote = {Keywords: edge partition, complete geometric graph, plane spanning tree, wheel set}
}
Document
Minimum-Error Triangulations for Sea Surface Reconstruction
Abstract
We apply state-of-the-art computational geometry methods to the problem of reconstructing a time-varying sea surface from tide gauge records. Our work builds on a recent article by Nitzke et al. (Computers & Geosciences, 157:104920, 2021) who have suggested to learn a triangulation D of a given set of tide gauge stations. The objective is to minimize the misfit of the piecewise linear surface induced by D to a reference surface that has been acquired with satellite altimetry. The authors restricted their search to k-order Delaunay (k-OD) triangulations and used an integer linear program in order to solve the resulting optimization problem.
In geometric terms, the input to our problem consists of two sets of points in ℝ² with elevations: a set 𝒮 that is to be triangulated, and a set ℛ of reference points. Intuitively, we define the error of a triangulation as the average vertical distance of a point in ℛ to the triangulated surface that is obtained by interpolating elevations of 𝒮 linearly in each triangle. Our goal is to find the triangulation of 𝒮 that has minimum error with respect to ℛ.
In our work, we prove that the minimum-error triangulation problem is NP-hard and cannot be approximated within any multiplicative factor in polynomial time unless P = NP. At the same time we show that the problem instances that occur in our application (considering sea level data from several hundreds of tide gauge stations worldwide) can be solved relatively fast using dynamic programming when restricted to k-OD triangulations for k ≤ 7. In particular, instances for which the number of connected components of the so-called k-OD fixed-edge graph is small can be solved within few seconds.
Cite as
Anna Arutyunova, Anne Driemel, Jan-Henrik Haunert, Herman Haverkort, Jürgen Kusche, Elmar Langetepe, Philip Mayer, Petra Mutzel, and Heiko Röglin. Minimum-Error Triangulations for Sea Surface Reconstruction. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{arutyunova_et_al:LIPIcs.SoCG.2022.7,
author = {Arutyunova, Anna and Driemel, Anne and Haunert, Jan-Henrik and Haverkort, Herman and Kusche, J\"{u}rgen and Langetepe, Elmar and Mayer, Philip and Mutzel, Petra and R\"{o}glin, Heiko},
title = {{Minimum-Error Triangulations for Sea Surface Reconstruction}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {7:1--7:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.7},
URN = {urn:nbn:de:0030-drops-160155},
doi = {10.4230/LIPIcs.SoCG.2022.7},
annote = {Keywords: Minimum-Error Triangulation, k-Order Delaunay Triangulations, Data dependent Triangulations, Sea Surface Reconstruction, fixed-Edge Graph}
}
Document
Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization
Abstract
In this paper, we focus on one particular instance of the shape reconstruction problem, in which the shape we wish to reconstruct is an orientable smooth submanifold of the Euclidean space. Assuming we have as input a simplicial complex K that approximates the submanifold (such as the Čech complex or the Rips complex), we recast the reconstruction problem as a 𝓁₁-norm minimization problem in which the optimization variable is a chain of K. Providing that K satisfies certain reasonable conditions, we prove that the considered minimization problem has a unique solution which triangulates the submanifold and coincides with the flat Delaunay complex introduced and studied in a companion paper [D. Attali and A. Lieutier, 2022]. Since the objective is a weighted 𝓁₁-norm and the contraints are linear, the triangulation process can thus be implemented by linear programming.
Cite as
Dominique Attali and André Lieutier. Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{attali_et_al:LIPIcs.SoCG.2022.8,
author = {Attali, Dominique and Lieutier, Andr\'{e}},
title = {{Delaunay-Like Triangulation of Smooth Orientable Submanifolds by 𝓁₁-Norm Minimization}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {8:1--8:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.8},
URN = {urn:nbn:de:0030-drops-160162},
doi = {10.4230/LIPIcs.SoCG.2022.8},
annote = {Keywords: manifold reconstruction, Delaunay complex, triangulation, sampling conditions, optimization, 𝓁₁-norm minimization, simplicial complex, chain, fundamental class}
}
Document
Tighter Bounds for Reconstruction from ε-Samples
Abstract
We show that reconstructing a curve in ℝ^d for d ≥ 2 from a 0.66-sample is always possible using an algorithm similar to the classical NN-Crust algorithm. Previously, this was only known to be possible for 0.47-samples in ℝ² and 1/3-samples in ℝ^d for d ≥ 3. In addition, we show that there is not always a unique way to reconstruct a curve from a 0.72-sample; this was previously only known for 1-samples. We also extend this non-uniqueness result to hypersurfaces in all higher dimensions.
Cite as
Håvard Bakke Bjerkevik. Tighter Bounds for Reconstruction from ε-Samples. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bakkebjerkevik:LIPIcs.SoCG.2022.9,
author = {Bakke Bjerkevik, H\r{a}vard},
title = {{Tighter Bounds for Reconstruction from \epsilon-Samples}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {9:1--9:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.9},
URN = {urn:nbn:de:0030-drops-160170},
doi = {10.4230/LIPIcs.SoCG.2022.9},
annote = {Keywords: Curve reconstruction, surface reconstruction, \epsilon-sampling}
}
Document
Erdős-Szekeres-Type Problems in the Real Projective Plane
Abstract
We consider point sets in the real projective plane ℝ𝒫² and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős-Szekeres-type problems.
We provide asymptotically tight bounds for a variant of the Erdős-Szekeres theorem about point sets in convex position in ℝ𝒫², which was initiated by Harborth and Möller in 1994. The notion of convex position in ℝ𝒫² agrees with the definition of convex sets introduced by Steinitz in 1913.
For k ≥ 3, an (affine) k-hole in a finite set S ⊆ ℝ² is a set of k points from S in convex position with no point of S in the interior of their convex hull. After introducing a new notion of k-holes for points sets from ℝ𝒫², called projective k-holes, we find arbitrarily large finite sets of points from ℝ𝒫² with no projective 8-holes, providing an analogue of a classical result by Horton from 1983. We also prove that they contain only quadratically many projective k-holes for k ≤ 7. On the other hand, we show that the number of k-holes can be substantially larger in ℝ𝒫² than in ℝ² by constructing, for every k ∈ {3,… ,6}, sets of n points from ℝ² ⊂ ℝ𝒫² with Ω(n^{3-3/5k}) projective k-holes and only O(n²) affine k-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in ℝ𝒫² and about some algorithmic aspects.
The study of extremal problems about point sets in ℝ𝒫² opens a new area of research, which we support by posing several open problems.
Cite as
Martin Balko, Manfred Scheucher, and Pavel Valtr. Erdős-Szekeres-Type Problems in the Real Projective Plane. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{balko_et_al:LIPIcs.SoCG.2022.10,
author = {Balko, Martin and Scheucher, Manfred and Valtr, Pavel},
title = {{Erd\H{o}s-Szekeres-Type Problems in the Real Projective Plane}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {10:1--10:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.10},
URN = {urn:nbn:de:0030-drops-160182},
doi = {10.4230/LIPIcs.SoCG.2022.10},
annote = {Keywords: real projective plane, point set, convex position, k-gon, k-hole, Erd\H{o}s-Szekeres theorem, Horton set, random point set}
}
Document
True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs
Abstract
We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set 𝒟 of n unit disks inducing a unit-disk graph G_𝒟 and a number p ∈ [n], one can partition 𝒟 into p subsets 𝒟₁,… ,𝒟_p such that for every i ∈ [p] and every 𝒟' ⊆ 𝒟_i, the graph obtained from G_𝒟 by contracting all edges between the vertices in 𝒟_i $1𝒟' admits a tree decomposition in which each bag consists of O(p+|𝒟'|) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved very recently by Marx et al. [SODA'22] and Bandyapadhyay et al. [SODA'22].
By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem (CDT) for unit-disk graphs, resolving an open question in the work Panolan et al. [SODA'19]. On the algorithmic side, we obtain a new FPT algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in 2^{O(√k log k)} ⋅ n^{O(1)} time, where k denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA'22] (which works more generally for disk graphs) and is almost optimal, as the problem cannot be solved in 2^{o(√k)} ⋅ n^{O(1)} time assuming the ETH.
Cite as
Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, and Jie Xue. True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2022.11,
author = {Bandyapadhyay, Sayan and Lochet, William and Lokshtanov, Daniel and Saurabh, Saket and Xue, Jie},
title = {{True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {11:1--11:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.11},
URN = {urn:nbn:de:0030-drops-160190},
doi = {10.4230/LIPIcs.SoCG.2022.11},
annote = {Keywords: unit-disk graphs, tree decomposition, contraction decomposition, bipartization}
}
Document
Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds
Abstract
We consider the unlabeled motion-planning problem of m unit-disc robots moving in a simple polygonal workspace of n edges. The goal is to find a motion plan that moves the robots to a given set of m target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end.
If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time O(n log n + mn + m²) if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance 4 apart, any two target positions are at least distance 4 apart, and any pair of a start and a target positions is at least distance 3 apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.
Cite as
Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot. Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{banyassady_et_al:LIPIcs.SoCG.2022.12,
author = {Banyassady, Bahareh and de Berg, Mark and Bringmann, Karl and Buchin, Kevin and Fernau, Henning and Halperin, Dan and Kostitsyna, Irina and Okamoto, Yoshio and Slot, Stijn},
title = {{Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {12:1--12:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.12},
URN = {urn:nbn:de:0030-drops-160203},
doi = {10.4230/LIPIcs.SoCG.2022.12},
annote = {Keywords: motion planning, computational geometry, simple polygon}
}
Document
Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures
Abstract
It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in [Yair Bartal et al., 2019] (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has 1+ε average distortion for embedding into k-dimensional Euclidean space, where k = O(1/ε²), and for more general q-norms of distortion, k = O(max{1/ε²,q/ε}), whereas tight lower bounds were established only for large values of q via reduction to the worst case.
In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any 1 ≤ q ≤ O((log (2ε² n))/ε) and ε ≥ 1/(√n), where n is the size of the subset of Euclidean space.
Our results also imply that the JL method is optimal for various distortion measures commonly used in practice, such as stress, energy and relative error. We prove that if any of these measures is bounded by ε then k = Ω(1/ε²), for any ε ≥ 1/(√n), matching the upper bounds of [Yair Bartal et al., 2019] and extending their tightness results for the full range moment analysis.
Our results may indicate that the JL dimensionality reduction method should be considered more often in practical applications, and the bounds we provide for its quality should be served as a measure for comparison when evaluating the performance of other methods and heuristics.
Cite as
Yair Bartal, Ora Nova Fandina, and Kasper Green Larsen. Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bartal_et_al:LIPIcs.SoCG.2022.13,
author = {Bartal, Yair and Fandina, Ora Nova and Larsen, Kasper Green},
title = {{Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {13:1--13:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.13},
URN = {urn:nbn:de:0030-drops-160219},
doi = {10.4230/LIPIcs.SoCG.2022.13},
annote = {Keywords: average distortion, practical dimensionality reduction, JL transform}
}
Document
Quasi-Universality of Reeb Graph Distances
Abstract
We establish bi-Lipschitz bounds certifying quasi-universality (universality up to a constant factor) for various distances between Reeb graphs: the interleaving distance, the functional distortion distance, and the functional contortion distance. The definition of the latter distance is a novel contribution, and for the special case of contour trees we also prove strict universality of this distance. Furthermore, we prove that for the special case of merge trees the functional contortion distance coincides with the interleaving distance, yielding universality of all four distances in this case.
Cite as
Ulrich Bauer, Håvard Bakke Bjerkevik, and Benedikt Fluhr. Quasi-Universality of Reeb Graph Distances. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bauer_et_al:LIPIcs.SoCG.2022.14,
author = {Bauer, Ulrich and Bjerkevik, H\r{a}vard Bakke and Fluhr, Benedikt},
title = {{Quasi-Universality of Reeb Graph Distances}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {14:1--14:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.14},
URN = {urn:nbn:de:0030-drops-160221},
doi = {10.4230/LIPIcs.SoCG.2022.14},
annote = {Keywords: Reeb graphs, contour trees, merge trees, distances, universality, interleaving distance, functional distortion distance, functional contortion distance}
}
Document
Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations
Abstract
Motivated by computational aspects of persistent homology for Vietoris–Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris–Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We consider the notion of geodesic defect to extend this result to general metric spaces in a way that is also compatible with the filtration. We further show that for finite tree metrics the Vietoris–Rips complexes collapse to their corresponding subforests. We relate our result to modern computational methods by showing that these collapses are induced by the apparent pairs gradient, which is used as an algorithmic optimization in Ripser, explaining its particularly strong performance on tree-like metric data.
Cite as
Ulrich Bauer and Fabian Roll. Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bauer_et_al:LIPIcs.SoCG.2022.15,
author = {Bauer, Ulrich and Roll, Fabian},
title = {{Gromov Hyperbolicity, Geodesic Defect, and Apparent Pairs in Vietoris-Rips Filtrations}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {15:1--15:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.15},
URN = {urn:nbn:de:0030-drops-160237},
doi = {10.4230/LIPIcs.SoCG.2022.15},
annote = {Keywords: Vietoris–Rips complexes, persistent homology, discrete Morse theory, apparent pairs, hyperbolicity, geodesic defect, Ripser}
}
Document
Acute Tours in the Plane
Abstract
We confirm the following conjecture of Fekete and Woeginger from 1997: for any sufficiently large even number n, every set of n points in the plane can be connected by a spanning tour (Hamiltonian cycle) consisting of straight-line edges such that the angle between any two consecutive edges is at most π/2. Our proof is constructive and suggests a simple O(nlog n)-time algorithm for finding such a tour. The previous best-known upper bound on the angle is 2π/3, and it is due to Dumitrescu, Pach and Tóth (2009).
Cite as
Ahmad Biniaz. Acute Tours in the Plane. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 16:1-16:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{biniaz:LIPIcs.SoCG.2022.16,
author = {Biniaz, Ahmad},
title = {{Acute Tours in the Plane}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {16:1--16:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.16},
URN = {urn:nbn:de:0030-drops-160240},
doi = {10.4230/LIPIcs.SoCG.2022.16},
annote = {Keywords: planar points, acute tour, Hamiltonian cycle, equitable partition}
}
Document
ETH-Tight Algorithms for Finding Surfaces in Simplicial Complexes of Bounded Treewidth
Abstract
Given a simplicial complex with n simplices, we consider the Connected Subsurface Recognition (c-SR) problem of finding a subcomplex that is homeomorphic to a given connected surface with a fixed boundary. We also study the related Sum-of-Genus Subsurface Recognition (SoG) problem, where we instead search for a surface whose boundary, number of connected components, and total genus are given. For both of these problems, we give parameterized algorithms with respect to the treewidth k of the Hasse diagram that run in 2^{O(k log k)}n^{O(1)} time. For the SoG problem, we also prove that our algorithm is optimal assuming the exponential-time hypothesis. In fact, we prove the stronger result that our algorithm is ETH-tight even without restriction on the total genus.
Cite as
Mitchell Black, Nello Blaser, Amir Nayyeri, and Erlend Raa Vågset. ETH-Tight Algorithms for Finding Surfaces in Simplicial Complexes of Bounded Treewidth. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{black_et_al:LIPIcs.SoCG.2022.17,
author = {Black, Mitchell and Blaser, Nello and Nayyeri, Amir and V\r{a}gset, Erlend Raa},
title = {{ETH-Tight Algorithms for Finding Surfaces in Simplicial Complexes of Bounded Treewidth}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {17:1--17:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.17},
URN = {urn:nbn:de:0030-drops-160253},
doi = {10.4230/LIPIcs.SoCG.2022.17},
annote = {Keywords: Computational Geometry, Surface Recognition, Treewidth, Hasse Diagram, Simplicial Complexes, Low-Dimensional Topology, Parameterized Complexity, Computational Complexity}
}
Document
Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes
Abstract
The combinatorial diameter diam(P) of a polytope P is the maximum shortest path distance between any pair of vertices. In this paper, we provide upper and lower bounds on the combinatorial diameter of a random "spherical" polytope, which is tight to within one factor of dimension when the number of inequalities is large compared to the dimension. More precisely, for an n-dimensional polytope P defined by the intersection of m i.i.d. half-spaces whose normals are chosen uniformly from the sphere, we show that diam(P) is Ω(n m^{1/(n-1)}) and O(n² m^{1/(n-1)} + n⁵ 4ⁿ) with high probability when m ≥ 2^{Ω(n)}.
For the upper bound, we first prove that the number of vertices in any fixed two dimensional projection sharply concentrates around its expectation when m is large, where we rely on the Θ(n² m^{1/(n-1)}) bound on the expectation due to Borgwardt [Math. Oper. Res., 1999]. To obtain the diameter upper bound, we stitch these "shadows paths" together over a suitable net using worst-case diameter bounds to connect vertices to the nearest shadow. For the lower bound, we first reduce to lower bounding the diameter of the dual polytope P^∘, corresponding to a random convex hull, by showing the relation diam(P) ≥ (n-1)(diam(P^∘)-2). We then prove that the shortest path between any "nearly" antipodal pair vertices of P^∘ has length Ω(m^{1/(n-1)}).
Cite as
Gilles Bonnet, Daniel Dadush, Uri Grupel, Sophie Huiberts, and Galyna Livshyts. Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bonnet_et_al:LIPIcs.SoCG.2022.18,
author = {Bonnet, Gilles and Dadush, Daniel and Grupel, Uri and Huiberts, Sophie and Livshyts, Galyna},
title = {{Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {18:1--18:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.18},
URN = {urn:nbn:de:0030-drops-160269},
doi = {10.4230/LIPIcs.SoCG.2022.18},
annote = {Keywords: Random Polytopes, Combinatorial Diameter, Hirsch Conjecture}
}
Document
Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions
Abstract
In this paper we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode encodes the rank invariant as a ℤ-linear combination of rank invariants of indicator modules supported on segments in the poset. It can also be enriched to encode the generalized rank invariant as a ℤ-linear combination of generalized rank invariants in fixed classes of interval modules. In the paper we develop the theory behind these rank decompositions, showing under what conditions they exist and are unique - so the signed barcode is canonically defined. We also illustrate the contribution of the signed barcode to the exploration of multi-parameter persistence modules through a practical example.
Cite as
Magnus Bakke Botnan, Steffen Oppermann, and Steve Oudot. Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{botnan_et_al:LIPIcs.SoCG.2022.19,
author = {Botnan, Magnus Bakke and Oppermann, Steffen and Oudot, Steve},
title = {{Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {19:1--19:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.19},
URN = {urn:nbn:de:0030-drops-160276},
doi = {10.4230/LIPIcs.SoCG.2022.19},
annote = {Keywords: Topological data analysis, multi-parameter persistent homology}
}
Document
Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves
Abstract
The Dynamic Time Warping (DTW) distance is a popular measure of similarity for a variety of sequence data. For comparing polygonal curves π, σ in ℝ^d, it provides a robust, outlier-insensitive alternative to the Fréchet distance. However, like the Fréchet distance, the DTW distance is not invariant under translations. Can we efficiently optimize the DTW distance of π and σ under arbitrary translations, to compare the curves' shape irrespective of their absolute location?
There are surprisingly few works in this direction, which may be due to its computational intricacy: For the Euclidean norm, this problem contains as a special case the geometric median problem, which provably admits no exact algebraic algorithm (that is, no algorithm using only addition, multiplication, and k-th roots). We thus investigate exact algorithms for non-Euclidean norms as well as approximation algorithms for the Euclidean norm.
For the L₁ norm in ℝ^d, we provide an 𝒪(n^{2(d+1)})-time algorithm, i.e., an exact polynomial-time algorithm for constant d. Here and below, n bounds the curves' complexities. For the Euclidean norm in ℝ², we show that a simple problem-specific insight leads to a (1+ε)-approximation in time 𝒪(n³/ε²). We then show how to obtain a subcubic 𝒪̃(n^{2.5}/ε²) time algorithm with significant new ideas; this time comes close to the well-known quadratic time barrier for computing DTW for fixed translations. Technically, the algorithm is obtained by speeding up repeated DTW distance estimations using a dynamic data structure for maintaining shortest paths in weighted planar digraphs. Crucially, we show how to traverse a candidate set of translations using space-filling curves in a way that incurs only few updates to the data structure.
We hope that our results will facilitate the use of DTW under translation both in theory and practice, and inspire similar algorithmic approaches for related geometric optimization problems.
Cite as
Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, Dániel Marx, and André Nusser. Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2022.20,
author = {Bringmann, Karl and Kisfaludi‑Bak, S\'{a}ndor and K\"{u}nnemann, Marvin and Marx, D\'{a}niel and Nusser, Andr\'{e}},
title = {{Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {20:1--20:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.20},
URN = {urn:nbn:de:0030-drops-160287},
doi = {10.4230/LIPIcs.SoCG.2022.20},
annote = {Keywords: Dynamic Time Warping, Sequence Similarity Measures}
}
Document
Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs
Abstract
We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in d-dimensional Euclidean space, such as balls, segments, or hypercubes, and whose edges correspond to pairs of intersecting shapes. The diameter of a graph is the largest distance realized by a pair of vertices in the graph.
Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since in the related class of planar graphs the diameter can be computed in 𝒪̃(n^{5/3}) time [Cabello 2019, Gawrychowski et al. 2021].
In this work we (conditionally) rule out sub-quadratic algorithms in several classes of intersection graphs, i.e., algorithms of running time 𝒪(n^{2-δ}) for some δ > 0. In particular, there are no sub-quadratic algorithms already for fat objects in small dimensions: unit balls in ℝ³ or congruent equilateral triangles in ℝ². For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic approximations already in ℝ². It seems that the hardness of approximation may also depend on dimensionality: for axis-parallel unit hypercubes in ℝ^{12}, distinguishing between diameter 2 and 3 needs quadratic time (ruling out (3/2-ε)- approximations), whereas for axis-parallel unit squares, we give an algorithm that distinguishes between diameter 2 and 3 in near-linear time.
Note that many of our lower bounds match the best known algorithms up to sub-polynomial factors. Ultimately, this fine-grained perspective may enable us to determine for which shapes we can have efficient algorithms and approximation schemes for diameter computation.
Cite as
Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, André Nusser, and Zahra Parsaeian. Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2022.21,
author = {Bringmann, Karl and Kisfaludi‑Bak, S\'{a}ndor and K\"{u}nnemann, Marvin and Nusser, Andr\'{e} and Parsaeian, Zahra},
title = {{Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {21:1--21:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.21},
URN = {urn:nbn:de:0030-drops-160294},
doi = {10.4230/LIPIcs.SoCG.2022.21},
annote = {Keywords: Hardness in P, Geometric Intersection Graph, Graph Diameter, Orthogonal Vectors, Hyperclique Detection}
}
Document
Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time
Abstract
Dynamic Time Warping is arguably the most popular similarity measure for time series, where we define a time series to be a one-dimensional polygonal curve. The drawback of Dynamic Time Warping is that it is sensitive to the sampling rate of the time series. The Fréchet distance is an alternative that has gained popularity, however, its drawback is that it is sensitive to outliers.
Continuous Dynamic Time Warping (CDTW) is a recently proposed alternative that does not exhibit the aforementioned drawbacks. CDTW combines the continuous nature of the Fréchet distance with the summation of Dynamic Time Warping, resulting in a similarity measure that is robust to sampling rate and to outliers. In a recent experimental work of Brankovic et al., it was demonstrated that clustering under CDTW avoids the unwanted artifacts that appear when clustering under Dynamic Time Warping and under the Fréchet distance. Despite its advantages, the major shortcoming of CDTW is that there is no exact algorithm for computing CDTW, in polynomial time or otherwise.
In this work, we present the first exact algorithm for computing CDTW of one-dimensional curves. Our algorithm runs in time 𝒪(n⁵) for a pair of one-dimensional curves, each with complexity at most n. In our algorithm, we propagate continuous functions in the dynamic program for CDTW, where the main difficulty lies in bounding the complexity of the functions. We believe that our result is an important first step towards CDTW becoming a practical similarity measure between curves.
Cite as
Kevin Buchin, André Nusser, and Sampson Wong. Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{buchin_et_al:LIPIcs.SoCG.2022.22,
author = {Buchin, Kevin and Nusser, Andr\'{e} and Wong, Sampson},
title = {{Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.22},
URN = {urn:nbn:de:0030-drops-160307},
doi = {10.4230/LIPIcs.SoCG.2022.22},
annote = {Keywords: Computational Geometry, Curve Similarity, Fr\'{e}chet distance, Dynamic Time Warping, Continuous Dynamic Time Warping}
}
Document
Long Plane Trees
Abstract
In the longest plane spanning tree problem, we are given a finite planar point set 𝒫, and our task is to find a plane (i.e., noncrossing) spanning tree T_OPT for 𝒫 with maximum total Euclidean edge length |T_OPT|. Despite more than two decades of research, it remains open if this problem is NP-hard. Thus, previous efforts have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates |T_OPT|. The approximate trees in these algorithms all have small unweighted diameter, typically three or four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms.
We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree T_ALG with diameter at most four and |T_ALG| ≥ 0.546 ⋅ |T_OPT|. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter d ≥ 3, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most d (compared to a longest plane tree without constraints).
Cite as
Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec. Long Plane Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{cabello_et_al:LIPIcs.SoCG.2022.23,
author = {Cabello, Sergio and Hoffmann, Michael and Klost, Katharina and Mulzer, Wolfgang and Tkadlec, Josef},
title = {{Long Plane Trees}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {23:1--23:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.23},
URN = {urn:nbn:de:0030-drops-160311},
doi = {10.4230/LIPIcs.SoCG.2022.23},
annote = {Keywords: geometric network design, spanning trees, plane straight-line graphs, approximation algorithms}
}
Document
The Universal 𝓁^p-Metric on Merge Trees
Abstract
Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an 𝓁^p-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the p-Wasserstein distance between the associated barcodes. For each p ∈ [1,∞], we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the p = ∞ case, this gives a novel proof of universality for the interleaving distance on merge trees.
Cite as
Robert Cardona, Justin Curry, Tung Lam, and Michael Lesnick. The Universal 𝓁^p-Metric on Merge Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{cardona_et_al:LIPIcs.SoCG.2022.24,
author = {Cardona, Robert and Curry, Justin and Lam, Tung and Lesnick, Michael},
title = {{The Universal 𝓁^p-Metric on Merge Trees}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {24:1--24:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.24},
URN = {urn:nbn:de:0030-drops-160325},
doi = {10.4230/LIPIcs.SoCG.2022.24},
annote = {Keywords: merge trees, hierarchical clustering, persistent homology, Wasserstein distances, interleavings}
}
Document
On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class
Abstract
Homology features of spaces which appear in applications, for instance 3D meshes, are among the most important topological properties of these objects. Given a non-trivial cycle in a homology class, we consider the problem of computing a representative in that homology class which is optimal. We study two measures of optimality, namely, the lexicographic order of cycles (the lex-optimal cycle) and the bottleneck norm (a bottleneck-optimal cycle). We give a simple algorithm for computing the lex-optimal cycle for a 1-homology class in a closed orientable surface. In contrast to this, our main result is that, in the case of 3-manifolds of size n² in the Euclidean 3-space, the problem of finding a bottleneck optimal cycle cannot be solved more efficiently than solving a system of linear equations with an n × n sparse matrix. From this reduction, we deduce several hardness results. Most notably, we show that for 3-manifolds given as a subset of the 3-space of size n², persistent homology computations are at least as hard as rank computation (for sparse matrices) while ordinary homology computations can be done in O(n² log n) time. This is the first such distinction between these two computations. Moreover, it follows that the same disparity exists between the height persistent homology computation and general sub-level set persistent homology computation for simplicial complexes in the 3-space.
Cite as
Erin Wolf Chambers, Salman Parsa, and Hannah Schreiber. On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chambers_et_al:LIPIcs.SoCG.2022.25,
author = {Chambers, Erin Wolf and Parsa, Salman and Schreiber, Hannah},
title = {{On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {25:1--25:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.25},
URN = {urn:nbn:de:0030-drops-160338},
doi = {10.4230/LIPIcs.SoCG.2022.25},
annote = {Keywords: computational topology, bottleneck optimal cycles, homology}
}
Document
Parameterized Algorithms for Upward Planarity
Abstract
We obtain new parameterized algorithms for the classical problem of determining whether a directed acyclic graph admits an upward planar drawing. Our results include a new fixed-parameter algorithm parameterized by the number of sources, an XP-algorithm parameterized by treewidth, and a fixed-parameter algorithm parameterized by treedepth. All three algorithms are obtained using a novel framework for the problem that combines SPQR tree-decompositions with parameterized techniques. Our approach unifies and pushes beyond previous tractability results for the problem on series-parallel digraphs, single-source digraphs and outerplanar digraphs.
Cite as
Steven Chaplick, Emilio Di Giacomo, Fabrizio Frati, Robert Ganian, Chrysanthi N. Raftopoulou, and Kirill Simonov. Parameterized Algorithms for Upward Planarity. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 26:1-26:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chaplick_et_al:LIPIcs.SoCG.2022.26,
author = {Chaplick, Steven and Di Giacomo, Emilio and Frati, Fabrizio and Ganian, Robert and Raftopoulou, Chrysanthi N. and Simonov, Kirill},
title = {{Parameterized Algorithms for Upward Planarity}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {26:1--26:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.26},
URN = {urn:nbn:de:0030-drops-160349},
doi = {10.4230/LIPIcs.SoCG.2022.26},
annote = {Keywords: Upward planarity, parameterized algorithms, SPQR trees, treewidth, treedepth}
}
Document
Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres
Abstract
A closed quasigeodesic on a convex polyhedron is a closed curve that is locally straight outside of the vertices, where it forms an angle at most π on both sides. While the existence of a simple closed quasigeodesic on a convex polyhedron has been proved by Pogorelov in 1949, finding a polynomial-time algorithm to compute such a simple closed quasigeodesic has been repeatedly posed as an open problem. Our first contribution is to propose an extended definition of quasigeodesics in the intrinsic setting of (not necessarily convex) polyhedral spheres, and to prove the existence of a weakly simple closed quasigeodesic in such a setting. Our proof does not proceed via an approximation by smooth surfaces, but relies on an adapation of the disk flow of Hass and Scott to the context of polyhedral surfaces. Our second result is to leverage this existence theorem to provide a finite algorithm to compute a weakly simple closed quasigeodesic on a polyhedral sphere. On a convex polyhedron, our algorithm computes a simple closed quasigeodesic, solving an open problem of Demaine, Hersterberg and Ku.
Cite as
Jean Chartier and Arnaud de Mesmay. Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chartier_et_al:LIPIcs.SoCG.2022.27,
author = {Chartier, Jean and de Mesmay, Arnaud},
title = {{Finding Weakly Simple Closed Quasigeodesics on Polyhedral Spheres}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {27:1--27:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.27},
URN = {urn:nbn:de:0030-drops-160350},
doi = {10.4230/LIPIcs.SoCG.2022.27},
annote = {Keywords: Quasigeodesic, polyhedron, curve-shortening process, disk flow, weakly simple}
}
Document
Tight Lower Bounds for Approximate & Exact k-Center in ℝ^d
Abstract
In the discrete k-Center problem, we are given a metric space (P,dist) where |P| = n and the goal is to select a set C ⊆ P of k centers which minimizes the maximum distance of a point in P from its nearest center. For any ε > 0, Agarwal and Procopiuc [SODA '98, Algorithmica '02] designed an (1+ε)-approximation algorithm for this problem in d-dimensional Euclidean space which runs in O(dn log k) + (k/ε)^{O (k^{1-1/d})}⋅ n^{O(1)} time. In this paper we show that their algorithm is essentially optimal: if for some d ≥ 2 and some computable function f, there is an f(k)⋅(1/ε)^{o (k^{1-1/d})} ⋅ n^{o (k^{1-1/d})} time algorithm for (1+ε)-approximating the discrete k-Center on n points in d-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails.
We obtain our lower bound by designing a gap reduction from a d-dimensional constraint satisfaction problem (CSP) to discrete d-dimensional k-Center. This reduction has the property that there is a fixed value ε (depending on the CSP) such that the optimal radius of k-Center instances corresponding to satisfiable and unsatisfiable instances of the CSP is < 1 and ≥ (1+ε) respectively. Our claimed lower bound on the running time for approximating discrete k-Center in d-dimensions then follows from the lower bound due to Marx and Sidiropoulos [SoCG '14] for checking the satisfiability of the aforementioned d-dimensional CSP.
As a byproduct of our reduction, we also obtain that the exact algorithm of Agarwal and Procopiuc [SODA '98, Algorithmica '02] which runs in n^{O (d⋅ k^{1-1/d})} time for discrete k-Center on n points in d-dimensional Euclidean space is asymptotically optimal. Formally, we show that if for some d ≥ 2 and some computable function f, there is an f(k)⋅n^{o (k^{1-1/d})} time exact algorithm for the discrete k-Center problem on n points in d-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails. Previously, such a lower bound was only known for d = 2 and was implicit in the work of Marx [IWPEC '06].
Cite as
Rajesh Chitnis and Nitin Saurabh. Tight Lower Bounds for Approximate & Exact k-Center in ℝ^d. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chitnis_et_al:LIPIcs.SoCG.2022.28,
author = {Chitnis, Rajesh and Saurabh, Nitin},
title = {{Tight Lower Bounds for Approximate \& Exact k-Center in \mathbb{R}^d}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {28:1--28:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.28},
URN = {urn:nbn:de:0030-drops-160365},
doi = {10.4230/LIPIcs.SoCG.2022.28},
annote = {Keywords: k-center, Euclidean space, Exponential Time Hypothesis (ETH), lower bound}
}
Document
Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) Without Flat Angles
Abstract
A foundational result in origami mathematics is Kawasaki and Justin’s simple, efficient characterization of flat foldability for unassigned single-vertex crease patterns (where each crease can fold mountain or valley) on flat material. This result was later generalized to cones of material, where the angles glued at the single vertex may not sum to 360^∘. Here we generalize these results to when the material forms a complex (instead of a manifold), and thus the angles are glued at the single vertex in the structure of an arbitrary planar graph (instead of a cycle). Like the earlier characterizations, we require all creases to fold mountain or valley, not remain unfolded flat; otherwise, the problem is known to be NP-complete (weakly for flat material and strongly for complexes). Equivalently, we efficiently characterize which combinatorially embedded planar graphs with prescribed edge lengths can fold flat, when all angles must be mountain or valley (not unfolded flat). Our algorithm runs in O(n log³ n) time, improving on the previous best algorithm of O(n² log n).
Cite as
Lily Chung, Erik D. Demaine, Dylan Hendrickson, and Victor Luo. Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) Without Flat Angles. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chung_et_al:LIPIcs.SoCG.2022.29,
author = {Chung, Lily and Demaine, Erik D. and Hendrickson, Dylan and Luo, Victor},
title = {{Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) Without Flat Angles}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {29:1--29:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.29},
URN = {urn:nbn:de:0030-drops-160371},
doi = {10.4230/LIPIcs.SoCG.2022.29},
annote = {Keywords: Graph drawing, folding, origami, polyhedral complex, algorithms}
}
Document
Hop-Spanners for Geometric Intersection Graphs
Abstract
A t-spanner of a graph G = (V,E) is a subgraph H = (V,E') that contains a uv-path of length at most t for every uv ∈ E. It is known that every n-vertex graph admits a (2k-1)-spanner with O(n^{1+1/k}) edges for k ≥ 1. This bound is the best possible for 1 ≤ k ≤ 9 and is conjectured to be optimal due to Erdős' girth conjecture.
We study t-spanners for t ∈ {2,3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O(nlog n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(nlog n) edges, and this bound is the best possible. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(nlog n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(nlog² n) edges.
Cite as
Jonathan B. Conroy and Csaba D. Tóth. Hop-Spanners for Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{conroy_et_al:LIPIcs.SoCG.2022.30,
author = {Conroy, Jonathan B. and T\'{o}th, Csaba D.},
title = {{Hop-Spanners for Geometric Intersection Graphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {30:1--30:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.30},
URN = {urn:nbn:de:0030-drops-160381},
doi = {10.4230/LIPIcs.SoCG.2022.30},
annote = {Keywords: geometric intersection graph, unit disk graph, hop-spanner}
}
Document
Persistent Cup-Length
Abstract
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser.
The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces.
In this paper, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.
Cite as
Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent Cup-Length. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{contessoto_et_al:LIPIcs.SoCG.2022.31,
author = {Contessoto, Marco and M\'{e}moli, Facundo and Stefanou, Anastasios and Zhou, Ling},
title = {{Persistent Cup-Length}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {31:1--31:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.31},
URN = {urn:nbn:de:0030-drops-160398},
doi = {10.4230/LIPIcs.SoCG.2022.31},
annote = {Keywords: cohomology, cup product, persistence, cup length, Gromov-Hausdorff distance}
}
Document
Three-Chromatic Geometric Hypergraphs
Abstract
We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.
Cite as
Gábor Damásdi and Dömötör Pálvölgyi. Three-Chromatic Geometric Hypergraphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 32:1-32:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{damasdi_et_al:LIPIcs.SoCG.2022.32,
author = {Dam\'{a}sdi, G\'{a}bor and P\'{a}lv\"{o}lgyi, D\"{o}m\"{o}t\"{o}r},
title = {{Three-Chromatic Geometric Hypergraphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {32:1--32:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.32},
URN = {urn:nbn:de:0030-drops-160401},
doi = {10.4230/LIPIcs.SoCG.2022.32},
annote = {Keywords: Discrete geometry, Geometric hypergraph coloring, Decomposition of multiple coverings}
}
Document
A Solution to Ringel’s Circle Problem
Abstract
We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales-Jewett theorem and which may be of independent interest.
Cite as
James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, and Bartosz Walczak. A Solution to Ringel’s Circle Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{davies_et_al:LIPIcs.SoCG.2022.33,
author = {Davies, James and Keller, Chaya and Kleist, Linda and Smorodinsky, Shakhar and Walczak, Bartosz},
title = {{A Solution to Ringel’s Circle Problem}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {33:1--33:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.33},
URN = {urn:nbn:de:0030-drops-160413},
doi = {10.4230/LIPIcs.SoCG.2022.33},
annote = {Keywords: circle arrangement, chromatic number, Gallai’s theorem, polynomial method}
}
Document
Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications
Abstract
The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval I of a 𝐙²-indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank over I by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing M to that path. If the bifiltration and I have at most t simplices and points respectively, this computation takes O(t^ω) time where ω ∈ [2,2.373) is the exponent of matrix multiplication. Among others, we apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M, determine whether M is interval decomposable and, if so, compute all intervals supporting its summands.
Cite as
Tamal K. Dey, Woojin Kim, and Facundo Mémoli. Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dey_et_al:LIPIcs.SoCG.2022.34,
author = {Dey, Tamal K. and Kim, Woojin and M\'{e}moli, Facundo},
title = {{Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {34:1--34:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.34},
URN = {urn:nbn:de:0030-drops-160420},
doi = {10.4230/LIPIcs.SoCG.2022.34},
annote = {Keywords: Multiparameter persistent homology, Zigzag persistent homology, Generalized Persistence Diagrams, M\"{o}bius inversion}
}
Document
Tracking Dynamical Features via Continuation and Persistence
Abstract
Multivector fields and combinatorial dynamical systems have recently become a subject of interest due to their potential for use in computational methods. In this paper, we develop a method to track an isolated invariant set - a salient feature of a combinatorial dynamical system - across a sequence of multivector fields. This goal is attained by placing the classical notion of the "continuation" of an isolated invariant set in the combinatorial setting. In particular, we give a "Tracking Protocol" that, when given a seed isolated invariant set, finds a canonical continuation of the seed across a sequence of multivector fields. In cases where it is not possible to continue, we show how to use zigzag persistence to track homological features associated with the isolated invariant sets. This construction permits viewing continuation as a special case of persistence.
Cite as
Tamal K. Dey, Michał Lipiński, Marian Mrozek, and Ryan Slechta. Tracking Dynamical Features via Continuation and Persistence. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dey_et_al:LIPIcs.SoCG.2022.35,
author = {Dey, Tamal K. and Lipi\'{n}ski, Micha{\l} and Mrozek, Marian and Slechta, Ryan},
title = {{Tracking Dynamical Features via Continuation and Persistence}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {35:1--35:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.35},
URN = {urn:nbn:de:0030-drops-160439},
doi = {10.4230/LIPIcs.SoCG.2022.35},
annote = {Keywords: combinatorial dynamical systems, continuation, index pair, Conley index, persistent homology}
}
Document
On the Discrete Fréchet Distance in a Graph
Abstract
The Fréchet distance is a well-studied similarity measure between curves that is widely used throughout computer science. Motivated by applications where curves stem from paths and walks on an underlying graph (such as a road network), we define and study the Fréchet distance for paths and walks on graphs. When provided with a distance oracle of G with O(1) query time, the classical quadratic-time dynamic program can compute the Fréchet distance between two walks P and Q in a graph G in O(|P|⋅|Q|) time. We show that there are situations where the graph structure helps with computing Fréchet distance: when the graph G is planar, we apply existing (approximate) distance oracles to compute a (1+ε)-approximation of the Fréchet distance between any shortest path P and any walk Q in O(|G|log|G|/√ε+|P|+|Q|/ε) time. We generalise this result to near-shortest paths, i.e. κ-straight paths, as we show how to compute a (1+ε)-approximation between a κ-straight path P and any walk Q in O(|G|log|G|/√ε+|P|+(κ|Q|)/ε) time. Our algorithmic results hold for both the strong and the weak discrete Fréchet distance over the shortest path metric in G.
Finally, we show that additional assumptions on the input, such as our assumption on path straightness, are indeed necessary to obtain truly subquadratic running time. We provide a conditional lower bound showing that the Fréchet distance, or even its 1.01-approximation, between arbitrary paths in a weighted planar graph cannot be computed in O((|P|⋅|Q|)^{1-δ}) time for any δ > 0 unless the Orthogonal Vector Hypothesis fails. For walks, this lower bound holds even when G is planar, unit-weight and has O(1) vertices.
Cite as
Anne Driemel, Ivor van der Hoog, and Eva Rotenberg. On the Discrete Fréchet Distance in a Graph. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{driemel_et_al:LIPIcs.SoCG.2022.36,
author = {Driemel, Anne and van der Hoog, Ivor and Rotenberg, Eva},
title = {{On the Discrete Fr\'{e}chet Distance in a Graph}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {36:1--36:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.36},
URN = {urn:nbn:de:0030-drops-160448},
doi = {10.4230/LIPIcs.SoCG.2022.36},
annote = {Keywords: Fr\'{e}chet, graphs, planar, complexity analysis}
}
Document
Computing a Link Diagram from Its Exterior
Abstract
A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.
Cite as
Nathan M. Dunfield, Malik Obeidin, and Cameron Gates Rudd. Computing a Link Diagram from Its Exterior. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 37:1-37:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dunfield_et_al:LIPIcs.SoCG.2022.37,
author = {Dunfield, Nathan M. and Obeidin, Malik and Rudd, Cameron Gates},
title = {{Computing a Link Diagram from Its Exterior}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {37:1--37:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.37},
URN = {urn:nbn:de:0030-drops-160457},
doi = {10.4230/LIPIcs.SoCG.2022.37},
annote = {Keywords: computational topology, low-dimensional topology, knot, knot exterior, knot diagram, link, link exterior, link diagram}
}
Document
On Comparable Box Dimension
Abstract
Two boxes in ℝ^d are comparable if one of them is a subset of a translation of the other one. The comparable box dimension of a graph G is the minimum integer d such that G can be represented as a touching graph of comparable axis-aligned boxes in ℝ^d. We show that proper minor-closed classes have bounded comparable box dimension and explore further properties of this notion.
Cite as
Zdeněk Dvořák, Daniel Gonçalves, Abhiruk Lahiri, Jane Tan, and Torsten Ueckerdt. On Comparable Box Dimension. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dvorak_et_al:LIPIcs.SoCG.2022.38,
author = {Dvo\v{r}\'{a}k, Zden\v{e}k and Gon\c{c}alves, Daniel and Lahiri, Abhiruk and Tan, Jane and Ueckerdt, Torsten},
title = {{On Comparable Box Dimension}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {38:1--38:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.38},
URN = {urn:nbn:de:0030-drops-160461},
doi = {10.4230/LIPIcs.SoCG.2022.38},
annote = {Keywords: geometric graphs, minor-closed graph classes, treewidth fragility}
}
Document
Weak Coloring Numbers of Intersection Graphs
Abstract
Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for a positive integer k, we seek a vertex ordering such that every vertex can (weakly respectively strongly) reach in k steps only few vertices that precede it in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in structural and algorithmic graph theory. Recently, Dvořák, McCarty, and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in ℝ^d, such as homothets of a compact convex object, or comparable axis-aligned boxes.
In this paper, we prove upper and lower bounds for the k-th weak coloring numbers of these classes of intersection graphs. As a consequence, we describe a natural graph class whose strong coloring numbers are polynomial in k, but the weak coloring numbers are exponential. We also observe a surprising difference in terms of the dependence of the weak coloring numbers on the dimension between touching graphs of balls (single-exponential) and hypercubes (double-exponential).
Cite as
Zdeněk Dvořák, Jakub Pekárek, Torsten Ueckerdt, and Yelena Yuditsky. Weak Coloring Numbers of Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dvorak_et_al:LIPIcs.SoCG.2022.39,
author = {Dvo\v{r}\'{a}k, Zden\v{e}k and Pek\'{a}rek, Jakub and Ueckerdt, Torsten and Yuditsky, Yelena},
title = {{Weak Coloring Numbers of Intersection Graphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {39:1--39:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.39},
URN = {urn:nbn:de:0030-drops-160477},
doi = {10.4230/LIPIcs.SoCG.2022.39},
annote = {Keywords: geometric intersection graphs, weak and strong coloring numbers}
}
Document
ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p
Abstract
Fix p ∈ [1,∞), K ∈ (0,∞) and a probability measure μ. We prove that for every n ∈ ℕ, ε ∈ (0,1) and x₁,…,x_n ∈ L_p(μ) with ‖max_{i ∈ {1,…,n}}|x_i|‖_{L_p(μ)} ≤ K, there exists d ≤ (32e² (2K)^{2p}log n)/ε² and vectors y₁,…, y_n ∈ 𝓁_p^d such that
∀i,j∈{1,…,n}, ‖x_i-x_j‖^p_{L_p(μ)}-ε ≤ ‖y_i-y_j‖_{𝓁_p^d}^p ≤ ‖x_i-x_j‖^p_{L_p(μ)}+ε.
Moreover, the argument implies the existence of a greedy algorithm which outputs {y_i}_{i = 1}ⁿ after receiving {x_i}_{i = 1}ⁿ as input. The proof relies on a derandomized version of Maurey’s empirical method (1981) combined with a combinatorial idea of Ball (1990) and a suitable change of measure. Motivated by the above embedding, we introduce the notion of ε-isometric dimension reduction of the unit ball B_E of a normed space (E,‖⋅‖_E) and we prove that B_{𝓁_p} does not admit ε-isometric dimension reduction by linear operators for any value of p≠2.
Cite as
Alexandros Eskenazis. ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{eskenazis:LIPIcs.SoCG.2022.40,
author = {Eskenazis, Alexandros},
title = {{\epsilon-Isometric Dimension Reduction for Incompressible Subsets of 𝓁\underlinep}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {40:1--40:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.40},
URN = {urn:nbn:de:0030-drops-160486},
doi = {10.4230/LIPIcs.SoCG.2022.40},
annote = {Keywords: Dimension reduction, \epsilon-isometric embedding, Maurey’s empirical method, change of measure}
}
Document
Short Topological Decompositions of Non-Orientable Surfaces
Abstract
We investigate short topological decompositions of non-orientable surfaces and provide algorithms to compute them. Our main result is a polynomial-time algorithm that for any graph embedded in a non-orientable surface computes a canonical non-orientable system of loops so that any loop from the canonical system intersects any edge of the graph in at most 30 points. The existence of such short canonical systems of loops was well known in the orientable case and an open problem in the non-orientable case. Our proof techniques combine recent work of Schaefer-Štefankovič with ideas coming from computational biology, specifically from the signed reversal distance algorithm of Hannenhalli-Pevzner. This result confirms a special case of a conjecture of Negami on the joint crossing number of two embeddable graphs. We also provide a correction for an argument of Negami bounding the joint crossing number of two non-orientable graph embeddings.
Cite as
Niloufar Fuladi, Alfredo Hubard, and Arnaud de Mesmay. Short Topological Decompositions of Non-Orientable Surfaces. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{fuladi_et_al:LIPIcs.SoCG.2022.41,
author = {Fuladi, Niloufar and Hubard, Alfredo and de Mesmay, Arnaud},
title = {{Short Topological Decompositions of Non-Orientable Surfaces}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {41:1--41:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.41},
URN = {urn:nbn:de:0030-drops-160492},
doi = {10.4230/LIPIcs.SoCG.2022.41},
annote = {Keywords: Computational topology, embedded graph, non-orientable surface, joint crossing number, canonical system of loop, surface decomposition}
}
Document
Robust Radical Sylvester-Gallai Theorem for Quadratics
Abstract
We prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials. More precisely, given a parameter 0 < δ ≤ 1 and a finite collection ℱ of irreducible and pairwise independent polynomials of degree at most 2, we say that ℱ is a (δ, 2)-radical Sylvester-Gallai configuration if for any polynomial F_i ∈ ℱ, there exist δ(|ℱ|-1) polynomials F_j such that |rad (F_i, F_j) ∩ ℱ| ≥ 3, that is, the radical of F_i, F_j contains a third polynomial in the set. We prove that any (δ, 2)-radical Sylvester-Gallai configuration ℱ must be of low dimension: that is dim span_ℂ{ℱ} = poly(1/δ).
Cite as
Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Robust Radical Sylvester-Gallai Theorem for Quadratics. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{garg_et_al:LIPIcs.SoCG.2022.42,
author = {Garg, Abhibhav and Oliveira, Rafael and Sengupta, Akash Kumar},
title = {{Robust Radical Sylvester-Gallai Theorem for Quadratics}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {42:1--42:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.42},
URN = {urn:nbn:de:0030-drops-160505},
doi = {10.4230/LIPIcs.SoCG.2022.42},
annote = {Keywords: Sylvester-Gallai theorem, arrangements of hypersurfaces, locally correctable codes, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra}
}
Document
Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials
Abstract
In this work we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if {𝒬} ⊂ ℂ[x₁.…,x_n] is a finite set, |{𝒬}| = m, of irreducible quadratic polynomials that satisfy the following condition
There is δ > 0 such that for every Q ∈ {𝒬} there are at least δ m polynomials P ∈ {𝒬} such that whenever Q and P vanish then so does a third polynomial in {𝒬}⧵{Q,P}.
then dim(span) = Poly(1/δ).
The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/δ) on the dimension (in the first work an upper bound of O(1/δ²) was given, which was improved to O(1/δ) in the second work).
Cite as
Shir Peleg and Amir Shpilka. Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{peleg_et_al:LIPIcs.SoCG.2022.43,
author = {Peleg, Shir and Shpilka, Amir},
title = {{Robust Sylvester-Gallai Type Theorem for Quadratic Polynomials}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {43:1--43:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.43},
URN = {urn:nbn:de:0030-drops-160515},
doi = {10.4230/LIPIcs.SoCG.2022.43},
annote = {Keywords: Sylvester-Gallai theorem, quadratic polynomials, Algebraic computation}
}
Document
Swap, Shift and Trim to Edge Collapse a Filtration
Abstract
Boissonnat and Pritam introduced an algorithm to reduce a filtration of flag (or clique) complexes, which can in particular speed up the computation of its persistent homology. They used so-called edge collapse to reduce the input flag filtration and their reduction method required only the 1-skeleton of the filtration. In this paper we revisit the use of edge collapse for efficient computation of persistent homology. We first give a simple and intuitive explanation of the principles underlying that algorithm. This in turn allows us to propose various extensions including a zigzag filtration simplification algorithm. We finally show some experiments to better understand how it behaves.
Cite as
Marc Glisse and Siddharth Pritam. Swap, Shift and Trim to Edge Collapse a Filtration. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{glisse_et_al:LIPIcs.SoCG.2022.44,
author = {Glisse, Marc and Pritam, Siddharth},
title = {{Swap, Shift and Trim to Edge Collapse a Filtration}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {44:1--44:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.44},
URN = {urn:nbn:de:0030-drops-160525},
doi = {10.4230/LIPIcs.SoCG.2022.44},
annote = {Keywords: edge collapse, flag complex, graph, persistent homology}
}
Document
Hardness and Approximation of Minimum Convex Partition
Abstract
We consider the Minimum Convex Partition problem: Given a set P of n points in the plane, draw a plane graph G on P, with positive minimum degree, such that G partitions the convex hull of P into a minimum number of convex faces. We show that Minimum Convex Partition is NP-hard, and we give several approximation algorithms, from an 𝒪(log OPT)-approximation running in 𝒪(n⁸)-time, where OPT denotes the minimum number of convex faces needed, to an 𝒪(√nlog n)-approximation algorithm running in 𝒪(n²)-time. We say that a point set is k-directed if the (straight) lines containing at least three points have up to k directions. We present an 𝒪(k)-approximation algorithm running in n^{𝒪(k)}-time. Those hardness and approximation results also holds for the Minimum Convex Tiling problem, defined similarly but allowing the use of Steiner points. The approximation results are obtained by relating the problem to the Covering Points with Non-Crossing Segments problem. We show that this problem is NP-hard, and present an FPT algorithm. This allows us to obtain a constant-approximation FPT algorithm for the Minimum Convex Partition Problem where the parameter is the number of faces.
Cite as
Nicolas Grelier. Hardness and Approximation of Minimum Convex Partition. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{grelier:LIPIcs.SoCG.2022.45,
author = {Grelier, Nicolas},
title = {{Hardness and Approximation of Minimum Convex Partition}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {45:1--45:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.45},
URN = {urn:nbn:de:0030-drops-160530},
doi = {10.4230/LIPIcs.SoCG.2022.45},
annote = {Keywords: degenerate point sets, point cover, non-crossing segments, approximation algorithm, complexity}
}
Document
Parameterised Partially-Predrawn Crossing Number
Abstract
Inspired by the increasingly popular research on extending partial graph drawings, we propose a new perspective on the traditional and arguably most important geometric graph parameter, the crossing number. Specifically, we define the partially predrawn crossing number to be the smallest number of crossings in any drawing of a graph, part of which is prescribed on the input (not counting the prescribed crossings). Our main result - an FPT-algorithm to compute the partially predrawn crossing number - combines advanced ideas from research on the classical crossing number and so called partial planarity in a very natural but intricate way. Not only do our techniques generalise the known FPT-algorithm by Grohe for computing the standard crossing number, they also allow us to substantially improve a number of recent parameterised results for various drawing extension problems.
Cite as
Thekla Hamm and Petr Hliněný. Parameterised Partially-Predrawn Crossing Number. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{hamm_et_al:LIPIcs.SoCG.2022.46,
author = {Hamm, Thekla and Hlin\v{e}n\'{y}, Petr},
title = {{Parameterised Partially-Predrawn Crossing Number}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {46:1--46:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.46},
URN = {urn:nbn:de:0030-drops-160547},
doi = {10.4230/LIPIcs.SoCG.2022.46},
annote = {Keywords: Crossing Number, Drawing Extension, Partial Planarity, Parameterised Complexity}
}
Document
Approximation Algorithms for Maximum Matchings in Geometric Intersection Graphs
Abstract
We present a (1-ε)-approximation algorithms for maximum cardinality matchings in disk intersection graphs - all with near linear running time. We also present an estimation algorithm that returns (1±ε)-approximation to the size of such matchings - this algorithm runs in linear time for unit disks, and O(n log n) for general disks (as long as the density is relatively small).
Cite as
Sariel Har-Peled and Everett Yang. Approximation Algorithms for Maximum Matchings in Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{harpeled_et_al:LIPIcs.SoCG.2022.47,
author = {Har-Peled, Sariel and Yang, Everett},
title = {{Approximation Algorithms for Maximum Matchings in Geometric Intersection Graphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {47:1--47:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.47},
URN = {urn:nbn:de:0030-drops-160555},
doi = {10.4230/LIPIcs.SoCG.2022.47},
annote = {Keywords: Matchings, disk intersection graphs, approximation algorithms}
}
Document
The Complexity of the Hausdorff Distance
Abstract
We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class ∀∃_<ℝ. This implies that the problem is NP-, co-NP-, ∃ℝ- and ∀ℝ-hard.
Cite as
Paul Jungeblut, Linda Kleist, and Tillmann Miltzow. The Complexity of the Hausdorff Distance. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{jungeblut_et_al:LIPIcs.SoCG.2022.48,
author = {Jungeblut, Paul and Kleist, Linda and Miltzow, Tillmann},
title = {{The Complexity of the Hausdorff Distance}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {48:1--48:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.48},
URN = {urn:nbn:de:0030-drops-160567},
doi = {10.4230/LIPIcs.SoCG.2022.48},
annote = {Keywords: Hausdorff Distance, Semi-Algebraic Set, Existential Theory of the Reals, Universal Existential Theory of the Reals, Complexity Theory}
}
Document
Dynamic Connectivity in Disk Graphs
Abstract
Let S ⊆ ℝ² be a set of n planar sites, such that each s ∈ S has an associated radius r_s > 0. Let 𝒟(S) be the disk intersection graph for S. It has vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii r_s, r_t intersect. Our goal is to design data structures that maintain the connectivity structure of 𝒟(S) as sites are inserted and/or deleted.
First, we consider unit disk graphs, i.e., r_s = 1, for all s ∈ S. We describe a data structure that has O(log² n) amortized update and O(log n/log log n) amortized query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ r_s ≤ Ψ, for a Ψ ≥ 1 known in advance. In the fully dynamic case, we achieve amortized update time O(Ψ λ₆(log n) log⁷ n) and query time O(log n/log log n), where λ_s(n) is the maximum length of a Davenport-Schinzel sequence of order s on n symbols. In the incremental case, where only insertions are allowed, we get logarithmic dependency on Ψ, with O(α(n)) query time and O(logΨ λ₆(log n) log⁷ n) update time. For the decremental setting, where only deletions are allowed, we first develop an efficient disk revealing structure: given two sets R and B of disks, we can delete disks from R, and upon each deletion, we receive a list of all disks in B that no longer intersect the union of R. Using this, we get decremental data structures with amortized query time O(log n/log log n) that support m deletions in O((nlog⁵ n + m log⁷ n) λ₆(log n) + nlog Ψ log⁴n) overall time for bounded radius ratio Ψ and O((nlog⁶ n + m log⁸n) λ₆(log n)) for arbitrary radii.
Cite as
Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Dynamic Connectivity in Disk Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 49:1-49:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kaplan_et_al:LIPIcs.SoCG.2022.49,
author = {Kaplan, Haim and Kauer, Alexander and Klost, Katharina and Knorr, Kristin and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul},
title = {{Dynamic Connectivity in Disk Graphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {49:1--49:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.49},
URN = {urn:nbn:de:0030-drops-160572},
doi = {10.4230/LIPIcs.SoCG.2022.49},
annote = {Keywords: Disk Graphs, Connectivity, Lower Envelopes}
}
Document
An (ℵ₀,k+2)-Theorem for k-Transversals
Abstract
A family ℱ of sets satisfies the (p,q)-property if among every p members of ℱ, some q can be pierced by a single point. The celebrated (p,q)-theorem of Alon and Kleitman asserts that for any p ≥ q ≥ d+1, any family ℱ of compact convex sets in ℝ^d that satisfies the (p,q)-property can be pierced by a finite number c(p,q,d) of points. A similar theorem with respect to piercing by (d-1)-dimensional flats, called (d-1)-transversals, was obtained by Alon and Kalai.
In this paper we prove the following result, which can be viewed as an (ℵ₀,k+2)-theorem with respect to k-transversals: Let ℱ be an infinite family of sets in ℝ^d such that each A ∈ ℱ contains a ball of radius r and is contained in a ball of radius R, and let 0 ≤ k < d. If among every ℵ₀ elements of ℱ, some k+2 can be pierced by a k-dimensional flat, then ℱ can be pierced by a finite number of k-dimensional flats.
This is the first (p,q)-theorem in which the assumption is weakened to an (∞,⋅) assumption. Our proofs combine geometric and topological tools.
Cite as
Chaya Keller and Micha A. Perles. An (ℵ₀,k+2)-Theorem for k-Transversals. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{keller_et_al:LIPIcs.SoCG.2022.50,
author = {Keller, Chaya and Perles, Micha A.},
title = {{An (\aleph₀,k+2)-Theorem for k-Transversals}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {50:1--50:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.50},
URN = {urn:nbn:de:0030-drops-160581},
doi = {10.4230/LIPIcs.SoCG.2022.50},
annote = {Keywords: convexity, (p,q)-theorem, k-transversal, infinite (p,q)-theorem}
}
Document
Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles
Abstract
We present an algorithm to compute the geodesic L₁ farthest-point Voronoi diagram of m point sites in the presence of n rectangular obstacles in the plane. It takes O(nm+n log n + mlog m) construction time using O(nm) space. This is the first optimal algorithm for constructing the farthest-point Voronoi diagram in the presence of obstacles. We can construct a data structure in the same construction time and space that answers a farthest-neighbor query in O(log(n+m)) time.
Cite as
Mincheol Kim, Chanyang Seo, Taehoon Ahn, and Hee-Kap Ahn. Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kim_et_al:LIPIcs.SoCG.2022.51,
author = {Kim, Mincheol and Seo, Chanyang and Ahn, Taehoon and Ahn, Hee-Kap},
title = {{Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {51:1--51:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.51},
URN = {urn:nbn:de:0030-drops-160596},
doi = {10.4230/LIPIcs.SoCG.2022.51},
annote = {Keywords: Geodesic distance, L₁ metric, farthest-point Voronoi diagram}
}
Document
Point Separation and Obstacle Removal by Finding and Hitting Odd Cycles
Abstract
Suppose we are given a pair of points s, t and a set 𝒮 of n geometric objects in the plane, called obstacles. We show that in polynomial time one can construct an auxiliary (multi-)graph G with vertex set 𝒮 and every edge labeled from {0, 1}, such that a set 𝒮_d ⊆ 𝒮 of obstacles separates s from t if and only if G[𝒮_d] contains a cycle whose sum of labels is odd. Using this structural characterization of separating sets of obstacles we obtain the following algorithmic results.
In the Obstacle-removal problem the task is to find a curve in the plane connecting s to t intersecting at most q obstacles. We give a 2.3146^q n^{O(1)} algorithm for Obstacle-removal, significantly improving upon the previously best known q^{O(q³)} n^{O(1)} algorithm of Eiben and Lokshtanov (SoCG'20). We also obtain an alternative proof of a constant factor approximation algorithm for Obstacle-removal, substantially simplifying the arguments of Kumar et al. (SODA'21).
In the Generalized Points-separation problem input consists of the set 𝒮 of obstacles, a point set A of k points and p pairs (s₁, t₁), … (s_p, t_p) of points from A. The task is to find a minimum subset 𝒮_r ⊆ 𝒮 such that for every i, every curve from s_i to t_i intersects at least one obstacle in 𝒮_r. We obtain 2^{O(p)} n^{O(k)}-time algorithm for Generalized Points-separation. This resolves an open problem of Cabello and Giannopoulos (SoCG'13), who asked about the existence of such an algorithm for the special case where (s₁, t₁), … (s_p, t_p) contains all the pairs of points in A. Finally, we improve the running time of our algorithm to f(p,k) ⋅ n^{O(√k)} when the obstacles are unit disks, where f(p,k) = 2^{O(p)} k^{O(k)}, and show that, assuming the Exponential Time Hypothesis (ETH), the running time dependence on k of our algorithms is essentially optimal.
Cite as
Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, Subhash Suri, and Jie Xue. Point Separation and Obstacle Removal by Finding and Hitting Odd Cycles. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kumar_et_al:LIPIcs.SoCG.2022.52,
author = {Kumar, Neeraj and Lokshtanov, Daniel and Saurabh, Saket and Suri, Subhash and Xue, Jie},
title = {{Point Separation and Obstacle Removal by Finding and Hitting Odd Cycles}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {52:1--52:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.52},
URN = {urn:nbn:de:0030-drops-160609},
doi = {10.4230/LIPIcs.SoCG.2022.52},
annote = {Keywords: points-separation, min color path, constraint removal, barrier resillience}
}
Document
A Universal Triangulation for Flat Tori
Abstract
A result due to Burago and Zalgaller states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space 𝔼³. A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. In a first part, we adapt the proof of Burago and Zalgaller, which is partially constructive, to produce PL isometric embeddings of flat tori. In practice, the resulting embeddings have a huge number of vertices, moreover distinct for every flat torus. In a second part, based on another construction of Zalgaller and on recent works by Arnoux et al., we exhibit a universal triangulation with 5974 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.
Cite as
Francis Lazarus and Florent Tallerie. A Universal Triangulation for Flat Tori. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lazarus_et_al:LIPIcs.SoCG.2022.53,
author = {Lazarus, Francis and Tallerie, Florent},
title = {{A Universal Triangulation for Flat Tori}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {53:1--53:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.53},
URN = {urn:nbn:de:0030-drops-160617},
doi = {10.4230/LIPIcs.SoCG.2022.53},
annote = {Keywords: Triangulation, flat torus, isometric embedding}
}
Document
Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound
Abstract
In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(n α_k(n)) edges, for any k≥2. The function α_k is the inverse of a certain Ackermann-style function at the ⌊k/2⌋th level of the primitive recursive hierarchy, where α₀(n)=⌈n/2⌉, α₁(n)=⌈√n⌉, α₂(n)=⌈log n⌉, α₃(n)=⌈log log n⌉, α₄(n)=log^* n, α₅(n)=⌊1/2 log^*n⌋, .... Roughly speaking, for k≥2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars. Also, α_{2α(n)+4}(n)≤4, where α(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function.
Whether or not this tradeoff is tight has remained open, even for the cases k=2 and k=3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ε>0, any (1+ε)-spanner for the uniform line metric with hop-diameter at most k must have at least Ω(n α_k(n)) edges.
Cite as
Hung Le, Lazar Milenković, and Shay Solomon. Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 54:1-54:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{le_et_al:LIPIcs.SoCG.2022.54,
author = {Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay},
title = {{Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {54:1--54:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.54},
URN = {urn:nbn:de:0030-drops-160629},
doi = {10.4230/LIPIcs.SoCG.2022.54},
annote = {Keywords: Euclidean spanners, hop-diameter, inverse-Ackermann, lower bounds, sparse spanners}
}
Document
Minimum Height Drawings of Ordered Trees in Polynomial Time: Homotopy Height of Tree Duals
Abstract
We consider drawings of graphs in the plane in which vertices are assigned distinct points in the plane and edges are drawn as simple curves connecting the vertices and such that the edges intersect only at their common endpoints. There is an intuitive quality measure for drawings of a graph that measures the height of a drawing ϕ : G↪ℝ² as follows. For a vertical line 𝓁 in ℝ², let the height of 𝓁 be the cardinality of the set 𝓁 ∩ ϕ(G). The height of a drawing of G is the maximum height over all vertical lines. In this paper, instead of abstract graphs, we fix a drawing and consider plane graphs. In other words, we are looking for a homeomorphism of the plane that minimizes the height of the resulting drawing. This problem is equivalent to the homotopy height problem in the plane, and the homotopic Fréchet distance problem. These problems were recently shown to lie in NP, but no polynomial-time algorithm or NP-hardness proof has been found since their formulation in 2009. We present the first polynomial-time algorithm for drawing trees with optimal height. This corresponds to a polynomial-time algorithm for the homotopy height where the triangulation has only one vertex (that is, a set of loops incident to a single vertex), so that its dual is a tree.
Cite as
Tim Ophelders and Salman Parsa. Minimum Height Drawings of Ordered Trees in Polynomial Time: Homotopy Height of Tree Duals. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 55:1-55:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ophelders_et_al:LIPIcs.SoCG.2022.55,
author = {Ophelders, Tim and Parsa, Salman},
title = {{Minimum Height Drawings of Ordered Trees in Polynomial Time: Homotopy Height of Tree Duals}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {55:1--55:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.55},
URN = {urn:nbn:de:0030-drops-160631},
doi = {10.4230/LIPIcs.SoCG.2022.55},
annote = {Keywords: Graph drawing, homotopy height}
}
Document
Disjointness Graphs of Short Polygonal Chains
Abstract
The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G).
Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have χ(G) ≤ (ω(G))³+ω(G).
Cite as
János Pach, Gábor Tardos, and Géza Tóth. Disjointness Graphs of Short Polygonal Chains. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{pach_et_al:LIPIcs.SoCG.2022.56,
author = {Pach, J\'{a}nos and Tardos, G\'{a}bor and T\'{o}th, G\'{e}za},
title = {{Disjointness Graphs of Short Polygonal Chains}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {56:1--56:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.56},
URN = {urn:nbn:de:0030-drops-160645},
doi = {10.4230/LIPIcs.SoCG.2022.56},
annote = {Keywords: chi-bounded, disjointness graph}
}
Document
Covering Points by Hyperplanes and Related Problems
Abstract
For a set P of n points in ℝ^d, for any d ≥ 2, a hyperplane h is called k-rich with respect to P if it contains at least k points of P. Answering and generalizing a question asked by Peyman Afshani, we show that if the number of k-rich hyperplanes in ℝ^d, d ≥ 3, is at least Ω(n^d/k^α + n/k), with a sufficiently large constant of proportionality and with d ≤ α < 2d-1, then there exists a (d-2)-flat that contains Ω(k^{(2d-1-α)/(d-1)}) points of P. We also present upper bound constructions that give instances in which the above lower bound is tight. An extension of our analysis yields similar lower bounds for k-rich spheres.
Cite as
Zuzana Patáková and Micha Sharir. Covering Points by Hyperplanes and Related Problems. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 57:1-57:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{patakova_et_al:LIPIcs.SoCG.2022.57,
author = {Pat\'{a}kov\'{a}, Zuzana and Sharir, Micha},
title = {{Covering Points by Hyperplanes and Related Problems}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {57:1--57:7},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.57},
URN = {urn:nbn:de:0030-drops-160652},
doi = {10.4230/LIPIcs.SoCG.2022.57},
annote = {Keywords: Rich hyperplanes, Incidences, Covering points by hyperplanes}
}
Document
The Degree-Rips Complexes of an Annulus with Outliers
Abstract
The degree-Rips bifiltration is the most computable of the parameter-free, density-sensitive bifiltrations in topological data analysis. It is known that this construction is stable to small perturbations of the input data, but its robustness to outliers is not well understood. In recent work, Blumberg-Lesnick prove a result in this direction using the Prokhorov distance and homotopy interleavings. Based on experimental evaluation, they argue that a more refined approach is desirable, and suggest the framework of homology inference. Motivated by these experiments, we consider a probability measure that is uniform with high density on an annulus, and uniform with low density on the disc inside the annulus. We compute the degree-Rips complexes of this probability space up to homotopy type, using the Adamaszek-Adams computation of the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the limit objects for the Blumberg-Lesnick experiments. We argue that the homology inference approach has strong explanatory power in this case, and suggest studying the limit objects directly as a strategy for further work.
Cite as
Alexander Rolle. The Degree-Rips Complexes of an Annulus with Outliers. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{rolle:LIPIcs.SoCG.2022.58,
author = {Rolle, Alexander},
title = {{The Degree-Rips Complexes of an Annulus with Outliers}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {58:1--58:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.58},
URN = {urn:nbn:de:0030-drops-160664},
doi = {10.4230/LIPIcs.SoCG.2022.58},
annote = {Keywords: multi-parameter persistent homology, stability, homology inference}
}
Document
Chains, Koch Chains, and Point Sets with Many Triangulations
Abstract
We introduce the abstract notion of a chain, which is a sequence of n points in the plane, ordered by x-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations.
We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have Ω(9.08ⁿ) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω(8.65ⁿ) for the maximum number of triangulations of planar point sets.
Cite as
Daniel Rutschmann and Manuel Wettstein. Chains, Koch Chains, and Point Sets with Many Triangulations. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{rutschmann_et_al:LIPIcs.SoCG.2022.59,
author = {Rutschmann, Daniel and Wettstein, Manuel},
title = {{Chains, Koch Chains, and Point Sets with Many Triangulations}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {59:1--59:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.59},
URN = {urn:nbn:de:0030-drops-160678},
doi = {10.4230/LIPIcs.SoCG.2022.59},
annote = {Keywords: Planar Point Set, Chain, Koch Chain, Triangulation, Maximum Number of Triangulations, Lower Bound}
}
Document
Nearly-Doubling Spaces of Persistence Diagrams
Abstract
The space of persistence diagrams under bottleneck distance is known to have infinite doubling dimension. Because many metric search algorithms and data structures have bounds that depend on the dimension of the search space, the high-dimensionality makes it difficult to analyze and compare asymptotic running times of metric search algorithms on this space.
We introduce the notion of nearly-doubling metrics, those that are Gromov-Hausdorff close to metric spaces of bounded doubling dimension and prove that bounded k-point persistence diagrams are nearly-doubling. This allows us to prove that in some ways, persistence diagrams can be expected to behave like a doubling metric space. We prove our results in great generality, studying a large class of quotient metrics (of which the persistence plane is just one example). We also prove bounds on the dimension of the k-point bottleneck space over such metrics.
The notion of being nearly-doubling in this Gromov-Hausdorff sense is likely of more general interest. Some algorithms that have a dependence on the dimension can be analyzed in terms of the dimension of the nearby metric rather than that of the metric itself. We give a specific example of this phenomenon by analyzing an algorithm to compute metric nets, a useful operation on persistence diagrams.
Cite as
Donald R. Sheehy and Siddharth S. Sheth. Nearly-Doubling Spaces of Persistence Diagrams. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{sheehy_et_al:LIPIcs.SoCG.2022.60,
author = {Sheehy, Donald R. and Sheth, Siddharth S.},
title = {{Nearly-Doubling Spaces of Persistence Diagrams}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {60:1--60:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.60},
URN = {urn:nbn:de:0030-drops-160686},
doi = {10.4230/LIPIcs.SoCG.2022.60},
annote = {Keywords: Topological Data Analysis, Persistence Diagrams, Gromov-Hausdorff Distance}
}
Document
From Geometry to Topology: Inverse Theorems for Distributed Persistence
Abstract
What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from injective. We therefore propose that, in many cases, the collection of persistence diagrams of many small subsets of X is a better invariant. This invariant, which we call "distributed persistence," is perfectly parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of metric spaces (with the quasi-isometry distance) to the space of distributed persistence invariants (with the Hausdorff-Bottleneck distance) is globally bi-Lipschitz. This is a much stronger property than simply being injective, as it implies that the inverse image of a small neighborhood is a small neighborhood, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the inverse Lipschitz constant depends on the size of the subsets taken, so that as the size of these subsets goes from small to large, the invariant interpolates between a purely geometric one and a topological one. Lastly, we note that our inverse results do not actually require considering all subsets of a fixed size (an enormous collection), but a relatively small collection satisfying simple covering properties. These theoretical results are complemented by synthetic experiments demonstrating the use of distributed persistence in practice.
Cite as
Elchanan Solomon, Alexander Wagner, and Paul Bendich. From Geometry to Topology: Inverse Theorems for Distributed Persistence. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{solomon_et_al:LIPIcs.SoCG.2022.61,
author = {Solomon, Elchanan and Wagner, Alexander and Bendich, Paul},
title = {{From Geometry to Topology: Inverse Theorems for Distributed Persistence}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {61:1--61:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.61},
URN = {urn:nbn:de:0030-drops-160690},
doi = {10.4230/LIPIcs.SoCG.2022.61},
annote = {Keywords: Applied Topology, Persistent Homology, Inverse Problems, Subsampling}
}
Document
A Positive Fraction Erdős-Szekeres Theorem and Its Applications
Abstract
A famous theorem of Erdős and Szekeres states that any sequence of n distinct real numbers contains a monotone subsequence of length at least √n. Here, we prove a positive fraction version of this theorem. For n > (k-1)², any sequence A of n distinct real numbers contains a collection of subsets A_1,…, A_k ⊂ A, appearing sequentially, all of size s = Ω(n/k²), such that every subsequence (a_1,…, a_k), with a_i ∈ A_i, is increasing, or every such subsequence is decreasing. The subsequence S = (A_1,…, A_k) described above is called block-monotone of depth k and block-size s. Our theorem is asymptotically best possible and follows from a more general Ramsey-type result for monotone paths, which we find of independent interest. We also show that for any positive integer k, any finite sequence of distinct real numbers can be partitioned into O(k²log k) block-monotone subsequences of depth at least k, upon deleting at most (k-1)² entries. We apply our results to mutually avoiding planar point sets and biarc diagrams in graph drawing.
Cite as
Andrew Suk and Ji Zeng. A Positive Fraction Erdős-Szekeres Theorem and Its Applications. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{suk_et_al:LIPIcs.SoCG.2022.62,
author = {Suk, Andrew and Zeng, Ji},
title = {{A Positive Fraction Erd\H{o}s-Szekeres Theorem and Its Applications}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {62:1--62:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.62},
URN = {urn:nbn:de:0030-drops-160703},
doi = {10.4230/LIPIcs.SoCG.2022.62},
annote = {Keywords: Erd\H{o}s-Szekeres, block-monotone, monotone biarc diagrams, mutually avoiding sets}
}
Document
Optimal Coreset for Gaussian Kernel Density Estimation
Abstract
Given a point set P ⊂ ℝ^d, the kernel density estimate of P is defined as
𝒢-_P(x) = 1/|P| ∑_{p ∈ P}e^{-∥x-p∥²}
for any x ∈ ℝ^d. We study how to construct a small subset Q of P such that the kernel density estimate of P is approximated by the kernel density estimate of Q. This subset Q is called a coreset. The main technique in this work is constructing a ± 1 coloring on the point set P by discrepancy theory and we leverage Banaszczyk’s Theorem. When d > 1 is a constant, our construction gives a coreset of size O(1/ε) as opposed to the best-known result of O(1/ε √{log 1/ε}). It is the first result to give a breakthrough on the barrier of √log factor even when d = 2.
Cite as
Wai Ming Tai. Optimal Coreset for Gaussian Kernel Density Estimation. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{tai:LIPIcs.SoCG.2022.63,
author = {Tai, Wai Ming},
title = {{Optimal Coreset for Gaussian Kernel Density Estimation}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {63:1--63:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.63},
URN = {urn:nbn:de:0030-drops-160719},
doi = {10.4230/LIPIcs.SoCG.2022.63},
annote = {Keywords: Discrepancy Theory, Kernel Density Estimation, Coreset}
}
Document
GPU Computation of the Euler Characteristic Curve for Imaging Data
Abstract
Persistent homology is perhaps the most popular and useful tool offered by topological data analysis - with point-cloud data being the most common setup. Its older cousin, the Euler characteristic curve (ECC) is less expressive - but far easier to compute. It is particularly suitable for analyzing imaging data, and is commonly used in fields ranging from astrophysics to biomedical image analysis. These fields are embracing GPU computations to handle increasingly large datasets.
We therefore propose an optimized GPU implementation of ECC computation for 2D and 3D grayscale images. The goal of this paper is twofold. First, we offer a practical tool, illustrating its performance with thorough experimentation - but also explain its inherent shortcomings. Second, this simple algorithm serves as a perfect backdrop for highlighting basic GPU programming techniques that make our implementation so efficient - and some common pitfalls we avoided. This is intended as a step towards a wider usage of GPU programming in computational geometry and topology software. We find this is particularly important as geometric and topological tools are used in conjunction with modern, GPU-accelerated machine learning frameworks.
Cite as
Fan Wang, Hubert Wagner, and Chao Chen. GPU Computation of the Euler Characteristic Curve for Imaging Data. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{wang_et_al:LIPIcs.SoCG.2022.64,
author = {Wang, Fan and Wagner, Hubert and Chen, Chao},
title = {{GPU Computation of the Euler Characteristic Curve for Imaging Data}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {64:1--64:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.64},
URN = {urn:nbn:de:0030-drops-160724},
doi = {10.4230/LIPIcs.SoCG.2022.64},
annote = {Keywords: topological data analysis, Euler characteristic, Euler characteristic curve, Betti curve, persistent homology, algorithms, parallel programming, algorithm engineering, GPU programming, imaging data}
}
Document
Media Exposition
Space Ants: Episode II - Coordinating Connected Catoms (Media Exposition)
Abstract
How can a set of identical mobile agents coordinate their motions to transform their arrangement from a given starting to a desired goal configuration? We consider this question in the context of actual physical devices called Catoms, which can perform reconfiguration, but need to maintain connectivity at all times to ensure communication and energy supply. We demonstrate and animate algorithmic results, in particular a proof of hardness, as well as an algorithm that guarantees constant stretch for certain classes of arrangements: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, then the total duration of our overall schedule is in 𝒪(d), which is optimal up to constant factors.
Cite as
Julien Bourgeois, Sándor P. Fekete, Ramin Kosfeld, Peter Kramer, Benoît Piranda, Christian Rieck, and Christian Scheffer. Space Ants: Episode II - Coordinating Connected Catoms (Media Exposition). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 65:1-65:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bourgeois_et_al:LIPIcs.SoCG.2022.65,
author = {Bourgeois, Julien and Fekete, S\'{a}ndor P. and Kosfeld, Ramin and Kramer, Peter and Piranda, Beno\^{i}t and Rieck, Christian and Scheffer, Christian},
title = {{Space Ants: Episode II - Coordinating Connected Catoms}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {65:1--65:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.65},
URN = {urn:nbn:de:0030-drops-160732},
doi = {10.4230/LIPIcs.SoCG.2022.65},
annote = {Keywords: Motion planning, parallel motion, bounded stretch, scaled shape, makespan, connectivity, swarm robotics}
}
Document
Media Exposition
A Cautionary Tale: Burning the Medial Axis Is Unstable (Media Exposition)
Abstract
The medial axis of a set consists of the points in the ambient space without a unique closest point on the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a topologically equivalent skeleton. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities various prunings of the medial axis have been proposed. Here, we examine one type of pruning, called burning. Because of the good experimental results, it was hoped that the burning method of simplifying the medial axis would be stable. In this work we show a simple example that dashes such hopes based on Bing’s house with two rooms, demonstrating an isotopy of a shape where the medial axis goes from collapsible to non-collapsible.
Cite as
Erin Chambers, Christopher Fillmore, Elizabeth Stephenson, and Mathijs Wintraecken. A Cautionary Tale: Burning the Medial Axis Is Unstable (Media Exposition). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 66:1-66:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chambers_et_al:LIPIcs.SoCG.2022.66,
author = {Chambers, Erin and Fillmore, Christopher and Stephenson, Elizabeth and Wintraecken, Mathijs},
title = {{A Cautionary Tale: Burning the Medial Axis Is Unstable}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {66:1--66:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.66},
URN = {urn:nbn:de:0030-drops-160744},
doi = {10.4230/LIPIcs.SoCG.2022.66},
annote = {Keywords: Medial axis, Collapse, Pruning, Burning, Stability}
}
Document
Media Exposition
Visualizing and Unfolding Nets of 4-Polytopes (Media Exposition)
Abstract
Over a decade ago, it was shown that every edge unfolding of the Platonic solids was without self-overlap, yielding a valid net. Recent work has extended this property to their higher-dimensional analogs: the 4-cube, 4-simplex, and 4-orthoplex. We present an interactive visualization that allows the user to unfold these polytopes by drawing on their dual 1-skeleton graph.
Cite as
Satyan L. Devadoss, Matthew S. Harvey, and Sam Zhang. Visualizing and Unfolding Nets of 4-Polytopes (Media Exposition). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 67:1-67:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{devadoss_et_al:LIPIcs.SoCG.2022.67,
author = {Devadoss, Satyan L. and Harvey, Matthew S. and Zhang, Sam},
title = {{Visualizing and Unfolding Nets of 4-Polytopes}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {67:1--67:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.67},
URN = {urn:nbn:de:0030-drops-160759},
doi = {10.4230/LIPIcs.SoCG.2022.67},
annote = {Keywords: unfoldings, nets, polytopes}
}
Document
Media Exposition
Visualizing WSPDs and Their Applications (Media Exposition)
Abstract
Introduced by Callahan and Kosaraju back in 1995, the concept of well-separated pair decomposition (WSPD) has occupied a special significance in computational geometry when it comes to solving distance problems in d-space. We present an in-browser tool that can be used to visualize WSPDs and several of their applications in 2-space. Apart from research, it can also be used by instructors for introducing WSPDs in a classroom setting. The tool will be permanently maintained by the third author at https://wisno33.github.io/VisualizingWSPDsAndTheirApplications/.
Cite as
Anirban Ghosh, FNU Shariful, and David Wisnosky. Visualizing WSPDs and Their Applications (Media Exposition). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 68:1-68:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ghosh_et_al:LIPIcs.SoCG.2022.68,
author = {Ghosh, Anirban and Shariful, FNU and Wisnosky, David},
title = {{Visualizing WSPDs and Their Applications}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {68:1--68:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.68},
URN = {urn:nbn:de:0030-drops-160760},
doi = {10.4230/LIPIcs.SoCG.2022.68},
annote = {Keywords: well-separated pair decomposition, nearest neighbor, geometric spanners, minimum spanning tree}
}
Document
Media Exposition
Subdivision Methods for Sum-Of-Distances Problems: Fermat-Weber Point, n-Ellipses and the Min-Sum Cluster Voronoi Diagram (Media Exposition)
Abstract
Given a set P of n points, the sum of distances function of a point x is d_{P}(x) : = ∑_{p ∈ P} ||x - p||. Using a subdivision approach with soft predicates we implement and visualize approximate solutions for three different problems involving the sum of distances function in ℝ². Namely, (1) finding the Fermat-Weber point, (2) constructing n-ellipses of a given set of points, and (3) constructing the nearest Voronoi diagram under the sum of distances function, given a set of point clusters as sites.
Cite as
Ioannis Mantas, Evanthia Papadopoulou, Martin Suderland, and Chee Yap. Subdivision Methods for Sum-Of-Distances Problems: Fermat-Weber Point, n-Ellipses and the Min-Sum Cluster Voronoi Diagram (Media Exposition). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 69:1-69:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{mantas_et_al:LIPIcs.SoCG.2022.69,
author = {Mantas, Ioannis and Papadopoulou, Evanthia and Suderland, Martin and Yap, Chee},
title = {{Subdivision Methods for Sum-Of-Distances Problems: Fermat-Weber Point, n-Ellipses and the Min-Sum Cluster Voronoi Diagram}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {69:1--69:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.69},
URN = {urn:nbn:de:0030-drops-160773},
doi = {10.4230/LIPIcs.SoCG.2022.69},
annote = {Keywords: Fermat point, geometric median, Weber point, Fermat distance, sum of distances, n-ellipse, multifocal ellipse, min-sum Voronoi diagram, cluster Voronoi diagram}
}
Document
Media Exposition
An Interactive Framework for Reconfiguration in the Sliding Square Model (Media Exposition)
Abstract
We describe SquareSlider, a software framework for visualizing reconfiguration algorithms of modular robots in the sliding square model. In this model, a robot consists of a configuration of squares in a rectangular grid, which can reconfigure through a fixed set of possible moves. SquareSlider is a web-based tool that implements an easy-to-use interface allowing the user to build a configuration, run a reconfiguration algorithm on it, and examine the results.
Cite as
Willem Sonke and Jules Wulms. An Interactive Framework for Reconfiguration in the Sliding Square Model (Media Exposition). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 70:1-70:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{sonke_et_al:LIPIcs.SoCG.2022.70,
author = {Sonke, Willem and Wulms, Jules},
title = {{An Interactive Framework for Reconfiguration in the Sliding Square Model}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {70:1--70:4},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.70},
URN = {urn:nbn:de:0030-drops-160781},
doi = {10.4230/LIPIcs.SoCG.2022.70},
annote = {Keywords: Modular robots, Implementation, Visualization}
}
Document
CG Challenge
Shadoks Approach to Minimum Partition into Plane Subgraphs (CG Challenge)
Abstract
We explain the heuristics used by the Shadoks team to win first place in the CG:SHOP 2022 challenge that considers the minimum partition into plane subgraphs. The goal is to partition a set of segments into as few subsets as possible such that segments in the same subset do not cross each other. The challenge has given 225 instances containing between 2500 and 75000 segments. For every instance, our solution was the best among all 32 participating teams.
Cite as
Loïc Crombez, Guilherme D. da Fonseca, Yan Gerard, and Aldo Gonzalez-Lorenzo. Shadoks Approach to Minimum Partition into Plane Subgraphs (CG Challenge). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 71:1-71:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{crombez_et_al:LIPIcs.SoCG.2022.71,
author = {Crombez, Lo\"{i}c and da Fonseca, Guilherme D. and Gerard, Yan and Gonzalez-Lorenzo, Aldo},
title = {{Shadoks Approach to Minimum Partition into Plane Subgraphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {71:1--71:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.71},
URN = {urn:nbn:de:0030-drops-160794},
doi = {10.4230/LIPIcs.SoCG.2022.71},
annote = {Keywords: Plane graphs, graph coloring, intersection graph, conflict optimizer, line segments, computational geometry}
}
Document
CG Challenge
Conflict-Based Local Search for Minimum Partition into Plane Subgraphs (CG Challenge)
Abstract
This paper examines the approach taken by team gitastrophe in the CG:SHOP 2022 challenge. The challenge was to partition the edges of a geometric graph, with vertices represented by points in the plane and edges as straight lines, into the minimum number of planar subgraphs. We used a simple variation of a conflict optimizer strategy used by team Shadoks in the previous year’s CG:SHOP to rank second in the challenge.
Cite as
Jack Spalding-Jamieson, Brandon Zhang, and Da Wei Zheng. Conflict-Based Local Search for Minimum Partition into Plane Subgraphs (CG Challenge). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 72:1-72:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{spaldingjamieson_et_al:LIPIcs.SoCG.2022.72,
author = {Spalding-Jamieson, Jack and Zhang, Brandon and Zheng, Da Wei},
title = {{Conflict-Based Local Search for Minimum Partition into Plane Subgraphs}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {72:1--72:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.72},
URN = {urn:nbn:de:0030-drops-160807},
doi = {10.4230/LIPIcs.SoCG.2022.72},
annote = {Keywords: local search, planar graph, graph colouring, geometric graph, conflict optimizer}
}
Document
CG Challenge
Local Search with Weighting Schemes for the CG:SHOP 2022 Competition (CG Challenge)
Abstract
This paper describes the heuristics used by the LASAOFOOFUBESTINNRRALLDECA team for the CG:SHOP 2022 challenge. We introduce a new greedy algorithm that exploits information about the challenge instances, and hybridize two classical local-search schemes with weighting schemes. We found 211/225 best-known solutions. Hence, with the algorithms presented in this article, our team was able to reach the 3rd place of the challenge, among 40 participating teams.
Cite as
Florian Fontan, Pascal Lafourcade, Luc Libralesso, and Benjamin Momège. Local Search with Weighting Schemes for the CG:SHOP 2022 Competition (CG Challenge). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 73:1-73:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{fontan_et_al:LIPIcs.SoCG.2022.73,
author = {Fontan, Florian and Lafourcade, Pascal and Libralesso, Luc and Mom\`{e}ge, Benjamin},
title = {{Local Search with Weighting Schemes for the CG:SHOP 2022 Competition}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {73:1--73:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.73},
URN = {urn:nbn:de:0030-drops-160811},
doi = {10.4230/LIPIcs.SoCG.2022.73},
annote = {Keywords: heuristics, vertex coloring, digital geometry}
}
Document
CG Challenge
SAT-Based Local Search for Plane Subgraph Partitions (CG Challenge)
Abstract
The Partition into Plane Subgraphs Problem (PPS) asks to partition the edges of a geometric graph with straight line segments into as few classes as possible, such that the line segments within a class do not cross. We discuss our approach GC-SLIM: a local search method that views PPS as a graph coloring problem and tackles it with a new and unique combination of propositional satisfiability (SAT) and tabu search, achieving the fourth place in the 2022 CG:SHOP Challenge.
Cite as
André Schidler. SAT-Based Local Search for Plane Subgraph Partitions (CG Challenge). In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 74:1-74:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{schidler:LIPIcs.SoCG.2022.74,
author = {Schidler, Andr\'{e}},
title = {{SAT-Based Local Search for Plane Subgraph Partitions}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {74:1--74:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.74},
URN = {urn:nbn:de:0030-drops-160829},
doi = {10.4230/LIPIcs.SoCG.2022.74},
annote = {Keywords: graph coloring, plane subgraphs, SAT, logic, SLIM, local improvement, large neighborhood search}
}