Volume
LIPIcs, Volume 164
36th International Symposium on Computational Geometry (SoCG 2020)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG)
Event
SoCG 2020, June 23-26, 2020, Zürich, SwitzerlandEditors
Publication Details
- published at: 2020-06-08
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-143-6
- DBLP: db/conf/compgeom/compgeom2020
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Complete Volume
LIPIcs, Volume 164, SoCG 2020, Complete Volume
Abstract
LIPIcs, Volume 164, SoCG 2020, Complete Volume
Cite as
36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 1-1222, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@Proceedings{cabello_et_al:LIPIcs.SoCG.2020,
title = {{LIPIcs, Volume 164, SoCG 2020, Complete Volume}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {1--1222},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020},
URN = {urn:nbn:de:0030-drops-121576},
doi = {10.4230/LIPIcs.SoCG.2020},
annote = {Keywords: LIPIcs, Volume 164, SoCG 2020, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cabello_et_al:LIPIcs.SoCG.2020.0,
author = {Cabello, Sergio and Chen, Danny Z.},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {0:i--0:xx},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.0},
URN = {urn:nbn:de:0030-drops-121587},
doi = {10.4230/LIPIcs.SoCG.2020.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons
Abstract
What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn-(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn-(m + ⌈ n/6 ⌉), for m ≥ n. We prove a new upper bound of mn-(m+n)+C for some constant C, which is optimal apart from the value of C.
Cite as
Eyal Ackerman, Balázs Keszegh, and Günter Rote. An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ackerman_et_al:LIPIcs.SoCG.2020.1,
author = {Ackerman, Eyal and Keszegh, Bal\'{a}zs and Rote, G\"{u}nter},
title = {{An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {1:1--1:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.1},
URN = {urn:nbn:de:0030-drops-121591},
doi = {10.4230/LIPIcs.SoCG.2020.1},
annote = {Keywords: Simple polygon, Ramsey theory, combinatorial geometry}
}
Document
Dynamic Geometric Set Cover and Hitting Set
Abstract
We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our paper are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in ℝ¹ and ℝ². The first framework uses bootstrapping and gives a (1+ε)-approximate data structure for dynamic interval set cover in ℝ¹ with O(n^α/ε) amortized update time for any constant α > 0; in ℝ², this method gives O(1)-approximate data structures for unit-square (and quadrant) set cover and hitting set with O(n^(1/2+α)) amortized update time. The second framework uses local modification, and leads to a (1+ε)-approximate data structure for dynamic interval hitting set in ℝ¹ with Õ(1/ε) amortized update time; in ℝ², it gives O(1)-approximate data structures for unit-square (and quadrant) set cover and hitting set in the partially dynamic settings with Õ(1) amortized update time.
Cite as
Pankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dynamic Geometric Set Cover and Hitting Set. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2020.2,
author = {Agarwal, Pankaj K. and Chang, Hsien-Chih and Suri, Subhash and Xiao, Allen and Xue, Jie},
title = {{Dynamic Geometric Set Cover and Hitting Set}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {2:1--2:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.2},
URN = {urn:nbn:de:0030-drops-121604},
doi = {10.4230/LIPIcs.SoCG.2020.2},
annote = {Keywords: Geometric set cover, Geometric hitting set, Dynamic data structures}
}
Document
The Parameterized Complexity of Guarding Almost Convex Polygons
Abstract
The Art Gallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16], the classic variant has an O(log k)-approximation algorithm [Bonnet and Miltzow, SoCG'17], and it may require irrational guards [Abrahamsen et al., SoCG'17]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be ∃ℝ-complete [Abrahamsen et al., STOC'18]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP.
Given the first discovery, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016]: Is Art Gallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of Art Gallery, this case can also be viewed as the classic Dominating Set problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-Vertex Art Gallery is solvable in time r^O(r²)n^O(1). Furthermore, our approach extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT as well. To this end, we utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions.
Cite as
Akanksha Agrawal, Kristine V. K. Knudsen, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. The Parameterized Complexity of Guarding Almost Convex Polygons. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{agrawal_et_al:LIPIcs.SoCG.2020.3,
author = {Agrawal, Akanksha and Knudsen, Kristine V. K. and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav},
title = {{The Parameterized Complexity of Guarding Almost Convex Polygons}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {3:1--3:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.3},
URN = {urn:nbn:de:0030-drops-121614},
doi = {10.4230/LIPIcs.SoCG.2020.3},
annote = {Keywords: Art Gallery, Reflex vertices, Monotone 2-CSP, Parameterized Complexity, Fixed Parameter Tractability}
}
Document
Euclidean TSP in Narrow Strips
Abstract
We investigate how the complexity of {Euclidean TSP} for point sets P inside the strip (-∞,+∞)×[0,δ] depends on the strip width δ. We obtain two main results.
- For the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in O(n log²n) time using an existing algorithm) is guaranteed to be a shortest tour overall when δ ⩽ 2√2, a bound which is best possible.
- We present an algorithm that is fixed-parameter tractable with respect to δ. More precisely, our algorithm has running time 2^{O(√δ)} n² for sparse point sets, where each 1×δ rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle [0,n]× [0,δ], it has an expected running time of 2^{O(√δ)} n² + O(n³).
Cite as
Henk Alkema, Mark de Berg, and Sándor Kisfaludi-Bak. Euclidean TSP in Narrow Strips. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{alkema_et_al:LIPIcs.SoCG.2020.4,
author = {Alkema, Henk and de Berg, Mark and Kisfaludi-Bak, S\'{a}ndor},
title = {{Euclidean TSP in Narrow Strips}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {4:1--4:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.4},
URN = {urn:nbn:de:0030-drops-121628},
doi = {10.4230/LIPIcs.SoCG.2020.4},
annote = {Keywords: Computational geometry, Euclidean TSP, bitonic TSP, fixed-parameter tractable algorithms}
}
Document
The ε-t-Net Problem
Abstract
We study a natural generalization of the classical ε-net problem (Haussler - Welzl 1987), which we call the ε-t-net problem: Given a hypergraph on n vertices and parameters t and ε ≥ t/n, find a minimum-sized family S of t-element subsets of vertices such that each hyperedge of size at least ε n contains a set in S. When t=1, this corresponds to the ε-net problem.
We prove that any sufficiently large hypergraph with VC-dimension d admits an ε-t-net of size O((1+log t)d/ε log 1/ε). For some families of geometrically-defined hypergraphs (such as the dual hypergraph of regions with linear union complexity), we prove the existence of O(1/ε)-sized ε-t-nets.
We also present an explicit construction of ε-t-nets (including ε-nets) for hypergraphs with bounded VC-dimension. In comparison to previous constructions for the special case of ε-nets (i.e., for t=1), it does not rely on advanced derandomization techniques. To this end we introduce a variant of the notion of VC-dimension which is of independent interest.
Cite as
Noga Alon, Bruno Jartoux, Chaya Keller, Shakhar Smorodinsky, and Yelena Yuditsky. The ε-t-Net Problem. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{alon_et_al:LIPIcs.SoCG.2020.5,
author = {Alon, Noga and Jartoux, Bruno and Keller, Chaya and Smorodinsky, Shakhar and Yuditsky, Yelena},
title = {{The \epsilon-t-Net Problem}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {5:1--5:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.5},
URN = {urn:nbn:de:0030-drops-121639},
doi = {10.4230/LIPIcs.SoCG.2020.5},
annote = {Keywords: epsilon-nets, geometric hypergraphs, VC-dimension, linear union complexity}
}
Document
Terrain Visibility Graphs: Persistence Is Not Enough
Abstract
In this paper, we consider the Visibility Graph Recognition and Reconstruction problems in the context of terrains. Here, we are given a graph G with labeled vertices v₀, v₁, …, v_{n-1} such that the labeling corresponds with a Hamiltonian path H. G also may contain other edges. We are interested in determining if there is a terrain T with vertices p₀, p₁, …, p_{n-1} such that G is the visibility graph of T and the boundary of T corresponds with H. G is said to be persistent if and only if it satisfies the so-called X-property and Bar-property. It is known that every "pseudo-terrain" has a persistent visibility graph and that every persistent graph is the visibility graph for some pseudo-terrain. The connection is not as clear for (geometric) terrains. It is known that the visibility graph of any terrain T is persistent, but it has been unclear whether every persistent graph G has a terrain T such that G is the visibility graph of T. There actually have been several papers that claim this to be the case (although no formal proof has ever been published), and recent works made steps towards building a terrain reconstruction algorithm for any persistent graph. In this paper, we show that there exists a persistent graph G that is not the visibility graph for any terrain T. This means persistence is not enough by itself to characterize the visibility graphs of terrains, and implies that pseudo-terrains are not stretchable.
Cite as
Safwa Ameer, Matt Gibson-Lopez, Erik Krohn, Sean Soderman, and Qing Wang. Terrain Visibility Graphs: Persistence Is Not Enough. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{ameer_et_al:LIPIcs.SoCG.2020.6,
author = {Ameer, Safwa and Gibson-Lopez, Matt and Krohn, Erik and Soderman, Sean and Wang, Qing},
title = {{Terrain Visibility Graphs: Persistence Is Not Enough}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {6:1--6:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.6},
URN = {urn:nbn:de:0030-drops-121640},
doi = {10.4230/LIPIcs.SoCG.2020.6},
annote = {Keywords: Terrains, Visibility Graph Characterization, Visibility Graph Recognition}
}
Document
On β-Plurality Points in Spatial Voting Games
Abstract
Let V be a set of n points in ℝ^d, called voters. A point p ∈ ℝ^d is a plurality point for V when the following holds: for every q ∈ ℝ^d the number of voters closer to p than to q is at least the number of voters closer to q than to p. Thus, in a vote where each v ∈ V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q. For most voter sets a plurality point does not exist. We therefore introduce the concept of β-plurality points, which are defined similarly to regular plurality points except that the distance of each voter to p (but not to q) is scaled by a factor β, for some constant 0<β⩽1. We investigate the existence and computation of β-plurality points, and obtain the following results.
- Define β^*_d := sup{β : any finite multiset V in ℝ^d admits a β-plurality point}. We prove that β^*₂ = √3/2, and that 1/√d ⩽ β^*_d ⩽ √3/2 for all d⩾3.
- Define β(V) := sup {β : V admits a β-plurality point}. We present an algorithm that, given a voter set V in {ℝ}^d, computes an (1-ε)⋅ β(V) plurality point in time O(n²/ε^(3d-2) ⋅ log(n/ε^(d-1)) ⋅ log²(1/ε)).
Cite as
Boris Aronov, Mark de Berg, Joachim Gudmundsson, and Michael Horton. On β-Plurality Points in Spatial Voting Games. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{aronov_et_al:LIPIcs.SoCG.2020.7,
author = {Aronov, Boris and de Berg, Mark and Gudmundsson, Joachim and Horton, Michael},
title = {{On \beta-Plurality Points in Spatial Voting Games}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {7:1--7:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.7},
URN = {urn:nbn:de:0030-drops-121651},
doi = {10.4230/LIPIcs.SoCG.2020.7},
annote = {Keywords: Computational geometry, Spatial voting theory, Plurality point, Computational social choice}
}
Document
Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets
Abstract
We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in A×B×C, a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses only roughly O(n^(28/15)) constant-degree polynomial sign tests, for the special case where two of the sets lie on one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to any other problem where we seek a triple that satisfies a single polynomial equation, e.g., determining whether A× B× C contains a triple spanning a unit-area triangle.
This result extends recent work by Barba et al. [Luis Barba et al., 2019] and by Chan [Timothy M. Chan, 2020], where all three sets A, B, and C are assumed to be one-dimensional. While there are common features in the high-level approaches, here and in [Luis Barba et al., 2019], the actual analysis in this work becomes more involved and requires new methods and techniques, involving polynomial partitions and other related tools.
As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple (a,b,c) ∈ A×B×C that simultaneously satisfies two real polynomial equations. For example, this is the setup when testing for the existence of pairs of similar triangles spanned by the input points, in various contexts discussed later in the paper. We show that problems of this kind can be solved with roughly O(n^(24/13)) constant-degree polynomial sign tests. These problems can be extended to higher dimensions in various ways, and we present subquadratic solutions to some of these extensions, in the algebraic decision-tree model.
Cite as
Boris Aronov, Esther Ezra, and Micha Sharir. Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{aronov_et_al:LIPIcs.SoCG.2020.8,
author = {Aronov, Boris and Ezra, Esther and Sharir, Micha},
title = {{Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {8:1--8:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.8},
URN = {urn:nbn:de:0030-drops-121666},
doi = {10.4230/LIPIcs.SoCG.2020.8},
annote = {Keywords: Algebraic decision tree, Polynomial partition, Collinearity testing, 3SUM-hard problems, Polynomials vanishing on Cartesian products}
}
Document
Extending Drawings of Graphs to Arrangements of Pseudolines
Abstract
In the recent study of crossing numbers, drawings of graphs that can be extended to an arrangement of pseudolines (pseudolinear drawings) have played an important role as they are a natural combinatorial extension of rectilinear (or straight-line) drawings. A characterization of the pseudolinear drawings of K_n was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.
Cite as
Alan Arroyo, Julien Bensmail, and R. Bruce Richter. Extending Drawings of Graphs to Arrangements of Pseudolines. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{arroyo_et_al:LIPIcs.SoCG.2020.9,
author = {Arroyo, Alan and Bensmail, Julien and Richter, R. Bruce},
title = {{Extending Drawings of Graphs to Arrangements of Pseudolines}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {9:1--9:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.9},
URN = {urn:nbn:de:0030-drops-121672},
doi = {10.4230/LIPIcs.SoCG.2020.9},
annote = {Keywords: graphs, graph drawings, geometric graph drawings, arrangements of pseudolines, crossing numbers, stretchability}
}
Document
Dimensionality Reduction for k-Distance Applied to Persistent Homology
Abstract
Given a set P of n points and a constant k, we are interested in computing the persistent homology of the Čech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy [Proc. SoCG, 2014]. We show that any linear transformation that preserves pairwise distances up to a (1±ε) multiplicative factor, must preserve the persistent homology of the Čech filtration up to a factor of (1-ε)^{-1}. Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the Čech filtration for the approximate k-distance of Buchet et al. are preserved up to a (1±ε) factor.
We also prove extensions of our main theorem, for point sets (i) lying in a region of bounded Gaussian width or (ii) on a low-dimensional manifold, obtaining the target dimension bounds of Lotz [Proc. Roy. Soc. , 2019] and Clarkson [Proc. SoCG, 2008 ] respectively.
Cite as
Shreya Arya, Jean-Daniel Boissonnat, Kunal Dutta, and Martin Lotz. Dimensionality Reduction for k-Distance Applied to Persistent Homology. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{arya_et_al:LIPIcs.SoCG.2020.10,
author = {Arya, Shreya and Boissonnat, Jean-Daniel and Dutta, Kunal and Lotz, Martin},
title = {{Dimensionality Reduction for k-Distance Applied to Persistent Homology}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {10:1--10:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.10},
URN = {urn:nbn:de:0030-drops-121682},
doi = {10.4230/LIPIcs.SoCG.2020.10},
annote = {Keywords: Dimensionality reduction, Johnson-Lindenstrauss lemma, Topological Data Analysis, Persistent Homology, k-distance, distance to measure}
}
Document
Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study
Abstract
Persistent homology is a common tool of topological data analysis, whose main descriptor, the persistence diagram, aims at computing and encoding the geometry and topology of given datasets. In this article, we present a novel application of persistent homology to characterize the spatial arrangement of immune and epithelial (tumor) cells within the breast cancer immune microenvironment. More specifically, quantitative and robust characterizations are built by computing persistence diagrams out of a staining technique (quantitative multiplex immunofluorescence) which allows us to obtain spatial coordinates and stain intensities on individual cells. The resulting persistence diagrams are evaluated as characteristic biomarkers of cancer subtype and prognostic biomarker of overall survival. For a cohort of approximately 700 breast cancer patients with median 8.5-year clinical follow-up, we show that these persistence diagrams outperform and complement the usual descriptors which capture spatial relationships with nearest neighbor analysis. This provides new insights and possibilities on the general problem of building (topology-based) biomarkers that are characteristic and predictive of cancer subtype, overall survival and response to therapy.
Cite as
Andrew Aukerman, Mathieu Carrière, Chao Chen, Kevin Gardner, Raúl Rabadán, and Rami Vanguri. Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{aukerman_et_al:LIPIcs.SoCG.2020.11,
author = {Aukerman, Andrew and Carri\`{e}re, Mathieu and Chen, Chao and Gardner, Kevin and Rabad\'{a}n, Ra\'{u}l and Vanguri, Rami},
title = {{Persistent Homology Based Characterization of the Breast Cancer Immune Microenvironment: A Feasibility Study}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {11:1--11:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.11},
URN = {urn:nbn:de:0030-drops-121695},
doi = {10.4230/LIPIcs.SoCG.2020.11},
annote = {Keywords: Topological data analysis, persistence diagrams}
}
Document
Homotopic Curve Shortening and the Affine Curve-Shortening Flow
Abstract
We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set P⊂ ℝ² of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent.
We find experimentally that, if the initial curve is held fixed and P is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between HCS and ACSF generalizes the link between "grid peeling" and the ACSF observed by Eppstein et al. (2017), which applied only to convex curves, and which was studied only for regular grids.
We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.
Cite as
Sergey Avvakumov and Gabriel Nivasch. Homotopic Curve Shortening and the Affine Curve-Shortening Flow. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{avvakumov_et_al:LIPIcs.SoCG.2020.12,
author = {Avvakumov, Sergey and Nivasch, Gabriel},
title = {{Homotopic Curve Shortening and the Affine Curve-Shortening Flow}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {12:1--12:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.12},
URN = {urn:nbn:de:0030-drops-121708},
doi = {10.4230/LIPIcs.SoCG.2020.12},
annote = {Keywords: affine curve-shortening flow, shortest homotopic path, integer grid, convex-layer decomposition}
}
Document
Empty Squares in Arbitrary Orientation Among Points
Abstract
This paper studies empty squares in arbitrary orientation among a set P of n points in the plane. We prove that the number of empty squares with four contact pairs is between Ω(n) and O(n²), and that these bounds are tight, provided P is in a certain general position. A contact pair of a square is a pair of a point p ∈ P and a side 𝓁 of the square with p ∈ 𝓁. The upper bound O(n²) also applies to the number of empty squares with four contact points, while we construct a point set among which there is no square of four contact points. We then present an algorithm that maintains a combinatorial structure of the L_∞ Voronoi diagram of P, while the axes of the plane continuously rotate by 90 degrees, and simultaneously reports all empty squares with four contact pairs among P in an output-sensitive way within O(slog n) time and O(n) space, where s denotes the number of reported squares. Several new algorithmic results are also obtained: a largest empty square among P and a square annulus of minimum width or minimum area that encloses P over all orientations can be computed in worst-case O(n² log n) time.
Cite as
Sang Won Bae and Sang Duk Yoon. Empty Squares in Arbitrary Orientation Among Points. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bae_et_al:LIPIcs.SoCG.2020.13,
author = {Bae, Sang Won and Yoon, Sang Duk},
title = {{Empty Squares in Arbitrary Orientation Among Points}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {13:1--13:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.13},
URN = {urn:nbn:de:0030-drops-121716},
doi = {10.4230/LIPIcs.SoCG.2020.13},
annote = {Keywords: empty square, arbitrary orientation, Erd\H{o}s - Szekeres problem, L\underline∞ Voronoi diagram, largest empty square problem, square annulus}
}
Document
Holes and Islands in Random Point Sets
Abstract
For d ∈ ℕ, let S be a finite set of points in ℝ^d in general position. A set H of k points from S is a k-hole in S if all points from H lie on the boundary of the convex hull conv(H) of H and the interior of conv(H) does not contain any point from S. A set I of k points from S is a k-island in S if conv(I) ∩ S = I. Note that each k-hole in S is a k-island in S.
For fixed positive integers d, k and a convex body K in ℝ^d with d-dimensional Lebesgue measure 1, let S be a set of n points chosen uniformly and independently at random from K. We show that the expected number of k-islands in S is in O(n^d). In the case k=d+1, we prove that the expected number of empty simplices (that is, (d+1)-holes) in S is at most 2^(d-1) ⋅ d! ⋅ binom(n,d). Our results improve and generalize previous bounds by Bárány and Füredi [I. Bárány and Z. Füredi, 1987], Valtr [P. Valtr, 1995], Fabila-Monroy and Huemer [Fabila-Monroy and Huemer, 2012], and Fabila-Monroy, Huemer, and Mitsche [Fabila-Monroy et al., 2015].
Cite as
Martin Balko, Manfred Scheucher, and Pavel Valtr. Holes and Islands in Random Point Sets. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{balko_et_al:LIPIcs.SoCG.2020.14,
author = {Balko, Martin and Scheucher, Manfred and Valtr, Pavel},
title = {{Holes and Islands in Random Point Sets}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {14:1--14:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.14},
URN = {urn:nbn:de:0030-drops-121722},
doi = {10.4230/LIPIcs.SoCG.2020.14},
annote = {Keywords: stochastic geometry, random point set, Erd\H{o}s-Szekeres type problem, k-hole, k-island, empty polytope, convex position, Horton set}
}
Document
The Reeb Graph Edit Distance Is Universal
Abstract
We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal.
Cite as
Ulrich Bauer, Claudia Landi, and Facundo Mémoli. The Reeb Graph Edit Distance Is Universal. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bauer_et_al:LIPIcs.SoCG.2020.15,
author = {Bauer, Ulrich and Landi, Claudia and M\'{e}moli, Facundo},
title = {{The Reeb Graph Edit Distance Is Universal}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {15:1--15:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.15},
URN = {urn:nbn:de:0030-drops-121730},
doi = {10.4230/LIPIcs.SoCG.2020.15},
annote = {Keywords: Reeb graphs, topological descriptors, edit distance, interleaving distance}
}
Document
Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages
Abstract
An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages. Our main contribution is a technique that generalizes this result to a much wider family of nonplanar graphs, which is characterized by a biconnected skeleton of crossing-free edges whose faces have bounded degree. Notably, this family includes all 1-planar and all optimal 2-planar graphs as subgraphs. We prove that this family of graphs has bounded book thickness, and as a corollary, we obtain the first constant upper bound for the book thickness of optimal 2-planar graphs.
Cite as
Michael A. Bekos, Giordano Da Lozzo, Svenja M. Griesbach, Martin Gronemann, Fabrizio Montecchiani, and Chrysanthi Raftopoulou. Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{bekos_et_al:LIPIcs.SoCG.2020.16,
author = {Bekos, Michael A. and Da Lozzo, Giordano and Griesbach, Svenja M. and Gronemann, Martin and Montecchiani, Fabrizio and Raftopoulou, Chrysanthi},
title = {{Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {16:1--16:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.16},
URN = {urn:nbn:de:0030-drops-121749},
doi = {10.4230/LIPIcs.SoCG.2020.16},
annote = {Keywords: Book embeddings, Book thickness, Nonplanar graphs, Planar skeleton}
}
Document
Parallel Computation of Alpha Complexes for Biomolecules
Abstract
The alpha complex, a subset of the Delaunay triangulation, has been extensively used as the underlying representation for biomolecular structures. We propose a GPU-based parallel algorithm for the computation of the alpha complex, which exploits the knowledge of typical spatial distribution and sizes of atoms in a biomolecule. Unlike existing methods, this algorithm does not require prior construction of the Delaunay triangulation. The algorithm computes the alpha complex in two stages. The first stage proceeds in a bottom-up fashion and computes a superset of the edges, triangles, and tetrahedra belonging to the alpha complex. The false positives from this estimation stage are removed in a subsequent pruning stage to obtain the correct alpha complex. Computational experiments on several biomolecules demonstrate the superior performance of the algorithm, up to a factor of 50 when compared to existing methods that are optimized for biomolecules.
Cite as
Talha Bin Masood, Tathagata Ray, and Vijay Natarajan. Parallel Computation of Alpha Complexes for Biomolecules. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{masood_et_al:LIPIcs.SoCG.2020.17,
author = {Masood, Talha Bin and Ray, Tathagata and Natarajan, Vijay},
title = {{Parallel Computation of Alpha Complexes for Biomolecules}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {17:1--17:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.17},
URN = {urn:nbn:de:0030-drops-121758},
doi = {10.4230/LIPIcs.SoCG.2020.17},
annote = {Keywords: Delaunay triangulation, parallel algorithms, biomolecules, GPU}
}
Document
Relative Persistent Homology
Abstract
The alpha complex efficiently computes persistent homology of a point cloud X in Euclidean space when the dimension d is low. Given a subset A of X, relative persistent homology can be computed as the persistent homology of the relative Čech complex Č(X, A). But this is not computationally feasible for larger point clouds X. The aim of this note is to present a method for efficient computation of relative persistent homology in low dimensional Euclidean space. We introduce the relative Delaunay-Čech complex DelČ(X, A) whose homology is the relative persistent homology. It is constructed from the Delaunay complex of an embedding of X in (d+1)-dimensional Euclidean space.
Cite as
Nello Blaser and Morten Brun. Relative Persistent Homology. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 18:1-18:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{blaser_et_al:LIPIcs.SoCG.2020.18,
author = {Blaser, Nello and Brun, Morten},
title = {{Relative Persistent Homology}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {18:1--18:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.18},
URN = {urn:nbn:de:0030-drops-121762},
doi = {10.4230/LIPIcs.SoCG.2020.18},
annote = {Keywords: topological data analysis, relative homology, Delaunay-\v{C}ech complex, alpha complex}
}
Document
Edge Collapse and Persistence of Flag Complexes
Abstract
In this article, we extend the notions of dominated vertex and strong collapse of a simplicial complex as introduced by J. Barmak and E. Miniam. We say that a simplex (of any dimension) is dominated if its link is a simplicial cone. Domination of edges appears to be a very powerful concept, especially when applied to flag complexes. We show that edge collapse (removal of dominated edges) in a flag complex can be performed using only the 1-skeleton of the complex. Furthermore, the residual complex is a flag complex as well. Next we show that, similar to the case of strong collapses, we can use edge collapses to reduce a flag filtration ℱ to a smaller flag filtration ℱ^c with the same persistence. Here again, we only use the 1-skeletons of the complexes. The resulting method to compute ℱ^c is simple and extremely efficient and, when used as a preprocessing for persistence computation, leads to gains of several orders of magnitude w.r.t the state-of-the-art methods (including our previous approach using strong collapse). The method is exact, irrespective of dimension, and improves performance of persistence computation even in low dimensions. This is demonstrated by numerous experiments on publicly available data.
Cite as
Jean-Daniel Boissonnat and Siddharth Pritam. Edge Collapse and Persistence of Flag Complexes. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{boissonnat_et_al:LIPIcs.SoCG.2020.19,
author = {Boissonnat, Jean-Daniel and Pritam, Siddharth},
title = {{Edge Collapse and Persistence of Flag Complexes}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {19:1--19:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.19},
URN = {urn:nbn:de:0030-drops-121777},
doi = {10.4230/LIPIcs.SoCG.2020.19},
annote = {Keywords: Computational Topology, Topological Data Analysis, Edge Collapse, Simple Collapse, Persistent homology}
}
Document
The Topological Correctness of PL-Approximations of Isomanifolds
Abstract
Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.
Cite as
Jean-Daniel Boissonnat and Mathijs Wintraecken. The Topological Correctness of PL-Approximations of Isomanifolds. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{boissonnat_et_al:LIPIcs.SoCG.2020.20,
author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs},
title = {{The Topological Correctness of PL-Approximations of Isomanifolds}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {20:1--20:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.20},
URN = {urn:nbn:de:0030-drops-121787},
doi = {10.4230/LIPIcs.SoCG.2020.20},
annote = {Keywords: PL-approximations, isomanifolds, solution manifolds, topological correctness}
}
Document
Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes
Abstract
We study two optimization problems on simplicial complexes with homology over ℤ₂, the minimum bounded chain problem: given a d-dimensional complex 𝒦 embedded in ℝ^(d+1) and a null-homologous (d-1)-cycle C in 𝒦, find the minimum d-chain with boundary C, and the minimum homologous chain problem: given a (d+1)-manifold ℳ and a d-chain D in ℳ, find the minimum d-chain homologous to D. We show strong hardness results for both problems even for small values of d; d = 2 for the former problem, and d=1 for the latter problem. We show that both problems are APX-hard, and hard to approximate within any constant factor assuming the unique games conjecture. On the positive side, we show that both problems are fixed-parameter tractable with respect to the size of the optimal solution. Moreover, we provide an O(√{log β_d})-approximation algorithm for the minimum bounded chain problem where β_d is the dth Betti number of 𝒦. Finally, we provide an O(√{log n_{d+1}})-approximation algorithm for the minimum homologous chain problem where n_{d+1} is the number of (d+1)-simplices in ℳ.
Cite as
Glencora Borradaile, William Maxwell, and Amir Nayyeri. Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{borradaile_et_al:LIPIcs.SoCG.2020.21,
author = {Borradaile, Glencora and Maxwell, William and Nayyeri, Amir},
title = {{Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {21:1--21:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.21},
URN = {urn:nbn:de:0030-drops-121799},
doi = {10.4230/LIPIcs.SoCG.2020.21},
annote = {Keywords: computational topology, algorithmic complexity, simplicial complexes}
}
Document
On Rectangle-Decomposable 2-Parameter Persistence Modules
Abstract
This paper addresses two questions: (1) can we identify a sensible class of 2-parameter persistence modules on which the rank invariant is complete? (2) can we determine efficiently whether a given 2-parameter persistence module belongs to this class? We provide positive answers to both questions, and our class of interest is that of rectangle-decomposable modules. Our contributions include: (a) a proof that the rank invariant is complete on rectangle-decomposable modules, together with an inclusion-exclusion formula for counting the multiplicities of the summands; (b) algorithms to check whether a module induced in homology by a bifiltration is rectangle-decomposable, and to decompose it in the affirmative, with a better complexity than state-of-the-art decomposition methods for general 2-parameter persistence modules. Our algorithms are backed up by a new structure theorem, whereby a 2-parameter persistence module is rectangle-decomposable if, and only if, its restrictions to squares are. This local condition is key to the efficiency of our algorithms, and it generalizes previous conditions from the class of block-decomposable modules to the larger one of rectangle-decomposable modules. It also admits an algebraic formulation that turns out to be a weaker version of the one for block-decomposability. Our analysis focuses on the case of modules indexed over finite grids, the more general cases are left as future work.
Cite as
Magnus Bakke Botnan, Vadim Lebovici, and Steve Oudot. On Rectangle-Decomposable 2-Parameter Persistence Modules. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{botnan_et_al:LIPIcs.SoCG.2020.22,
author = {Botnan, Magnus Bakke and Lebovici, Vadim and Oudot, Steve},
title = {{On Rectangle-Decomposable 2-Parameter Persistence Modules}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.22},
URN = {urn:nbn:de:0030-drops-121802},
doi = {10.4230/LIPIcs.SoCG.2020.22},
annote = {Keywords: topological data analysis, multiparameter persistence, rank invariant}
}
Document
Robust Anisotropic Power-Functions-Based Filtrations for Clustering
Abstract
We consider robust power-distance functions that approximate the distance function to a compact set, from a noisy sample. We pay particular interest to robust power-distance functions that are anisotropic, in the sense that their sublevel sets are unions of ellipsoids, and not necessarily unions of balls. Using persistence homology on such power-distance functions provides robust clustering schemes. We investigate such clustering schemes and compare the different procedures on synthetic and real datasets. In particular, we enhance the good performance of the anisotropic method for some cases for which classical methods fail.
Cite as
Claire Brécheteau. Robust Anisotropic Power-Functions-Based Filtrations for Clustering. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{brecheteau:LIPIcs.SoCG.2020.23,
author = {Br\'{e}cheteau, Claire},
title = {{Robust Anisotropic Power-Functions-Based Filtrations for Clustering}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {23:1--23:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.23},
URN = {urn:nbn:de:0030-drops-121818},
doi = {10.4230/LIPIcs.SoCG.2020.23},
annote = {Keywords: Power functions, Filtrations, Hierarchical Clustering, Ellipsoids}
}
Document
Geometric Secluded Paths and Planar Satisfiability
Abstract
We consider paths with low exposure to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between integral exposure (when we care about how long the path sees every point of the domain) and 0/1 exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT.
Cite as
Kevin Buchin, Valentin Polishchuk, Leonid Sedov, and Roman Voronov. Geometric Secluded Paths and Planar Satisfiability. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{buchin_et_al:LIPIcs.SoCG.2020.24,
author = {Buchin, Kevin and Polishchuk, Valentin and Sedov, Leonid and Voronov, Roman},
title = {{Geometric Secluded Paths and Planar Satisfiability}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {24:1--24:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.24},
URN = {urn:nbn:de:0030-drops-121827},
doi = {10.4230/LIPIcs.SoCG.2020.24},
annote = {Keywords: Visibility, Route planning, Security/privacy, Planar satisfiability}
}
Document
The Next 350 Million Knots
Abstract
The tabulation of all prime knots up to a given number of crossings was one of the founding problems of knot theory in the 1800s, and continues to be of interest today. Here we extend the tables from 16 to 19 crossings, with a total of 352 152 252 distinct non-trivial prime knots.
The tabulation has two major stages: (1) a combinatorial enumeration stage, which involves generating a provably sufficient set of candidate knot diagrams; and (2) a computational topology stage, which involves identifying and removing duplicate knots, and certifying that all knots that remain are topologically distinct. In this paper we describe the many different algorithmic components in this process, which draw on graph theory, hyperbolic geometry, knot polynomials, normal surface theory, and computational algebra. We also discuss the algorithm engineering challenges in solving difficult topological problems systematically and reliably on hundreds of millions of inputs, despite the fact that no reliably fast algorithms for these problems are known.
Cite as
Benjamin A. Burton. The Next 350 Million Knots. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{burton:LIPIcs.SoCG.2020.25,
author = {Burton, Benjamin A.},
title = {{The Next 350 Million Knots}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {25:1--25:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.25},
URN = {urn:nbn:de:0030-drops-121831},
doi = {10.4230/LIPIcs.SoCG.2020.25},
annote = {Keywords: Computational topology, knots, 3-manifolds, implementation}
}
Document
Elder-Rule-Staircodes for Augmented Metric Spaces
Abstract
An augmented metric space (X, d_X, f_X) is a metric space (X, d_X) equipped with a function f_X: X → ℝ. It arises commonly in practice, e.g, a point cloud X in ℝ^d where each point x∈ X has a density function value f_X(x) associated to it. Such an augmented metric space naturally gives rise to a 2-parameter filtration. However, the resulting 2-parameter persistence module could still be of wild representation type, and may not have simple indecomposables.
In this paper, motivated by the elder-rule for the zeroth homology of a 1-parameter filtration, we propose a barcode-like summary, called the elder-rule-staircode, as a way to encode the zeroth homology of the 2-parameter filtration induced by a finite augmented metric space. Specifically, given a finite (X, d_X, f_X), its elder-rule-staircode consists of n = |X| number of staircase-like blocks in the plane. We show that the fibered barcode, the fibered merge tree, and the graded Betti numbers associated to the zeroth homology of the 2-parameter filtration induced by (X, d_X, f_X) can all be efficiently computed once the elder-rule-staircode is given. Furthermore, for certain special cases, this staircode corresponds exactly to the set of indecomposables of the zeroth homology of the 2-parameter filtration. Finally, we develop and implement an efficient algorithm to compute the elder-rule-staircode in O(n²log n) time, which can be improved to O(n²α(n)) if X is from a fixed dimensional Euclidean space ℝ^d, where α(n) is the inverse Ackermann function.
Cite as
Chen Cai, Woojin Kim, Facundo Mémoli, and Yusu Wang. Elder-Rule-Staircodes for Augmented Metric Spaces. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cai_et_al:LIPIcs.SoCG.2020.26,
author = {Cai, Chen and Kim, Woojin and M\'{e}moli, Facundo and Wang, Yusu},
title = {{Elder-Rule-Staircodes for Augmented Metric Spaces}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {26:1--26:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.26},
URN = {urn:nbn:de:0030-drops-121848},
doi = {10.4230/LIPIcs.SoCG.2020.26},
annote = {Keywords: Persistent homology, Multiparameter persistence, Barcodes, Elder rule, Hierarchical clustering, Graded Betti numbers}
}
Document
Faster Approximation Algorithms for Geometric Set Cover
Abstract
We improve the running times of O(1)-approximation algorithms for the set cover problem in geometric settings, specifically, covering points by disks in the plane, or covering points by halfspaces in three dimensions. In the unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized O(n log⁴n)-time, O(1)-approximation algorithm, by using variants of the multiplicative weight update (MWU) method combined with geometric data structures. We simplify the data structure requirement in one of their methods and obtain a deterministic O(n log³n log log n)-time algorithm. With further new ideas, we obtain a still faster randomized O(n log n(log log n)^O(1))-time algorithm.
For the weighted problem, we also give a randomized O(n log⁴n log log n)-time, O(1)-approximation algorithm, by simple modifications to the MWU method and the quasi-uniform sampling technique.
Cite as
Timothy M. Chan and Qizheng He. Faster Approximation Algorithms for Geometric Set Cover. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chan_et_al:LIPIcs.SoCG.2020.27,
author = {Chan, Timothy M. and He, Qizheng},
title = {{Faster Approximation Algorithms for Geometric Set Cover}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {27:1--27:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.27},
URN = {urn:nbn:de:0030-drops-121856},
doi = {10.4230/LIPIcs.SoCG.2020.27},
annote = {Keywords: Set cover, approximation algorithms, multiplicate weight update method, random sampling, shallow cuttings}
}
Document
Further Results on Colored Range Searching
Abstract
We present a number of new results about range searching for colored (or "categorical") data:
1) For a set of n colored points in three dimensions, we describe randomized data structures with O(n polylog n) space that can report the distinct colors in any query orthogonal range (axis-aligned box) in O(k polyloglog n) expected time, where k is the number of distinct colors in the range, assuming that coordinates are in {1,…,n}. Previous data structures require O((log n)/(log log n) + k) query time. Our result also implies improvements in higher constant dimensions.
2) Our data structures can be adapted to halfspace ranges in three dimensions (or circular ranges in two dimensions), achieving O(k log n) expected query time. Previous data structures require O(k log²n) query time.
3) For a set of n colored points in two dimensions, we describe a data structure with O(n polylog n) space that can answer colored "type-2" range counting queries: report the number of occurrences of every distinct color in a query orthogonal range. The query time is O((log n)/(log log n) + k log log n), where k is the number of distinct colors in the range. Naively performing k uncolored range counting queries would require O(k (log n)/(log log n)) time.
Our data structures are designed using a variety of techniques, including colored variants of randomized incremental construction (which may be of independent interest), colored variants of shallow cuttings, and bit-packing tricks.
Cite as
Timothy M. Chan, Qizheng He, and Yakov Nekrich. Further Results on Colored Range Searching. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{chan_et_al:LIPIcs.SoCG.2020.28,
author = {Chan, Timothy M. and He, Qizheng and Nekrich, Yakov},
title = {{Further Results on Colored Range Searching}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {28:1--28:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.28},
URN = {urn:nbn:de:0030-drops-121868},
doi = {10.4230/LIPIcs.SoCG.2020.28},
annote = {Keywords: Range searching, geometric data structures, randomized incremental construction, random sampling, word RAM}
}
Document
A Generalization of Self-Improving Algorithms
Abstract
Ailon et al. [SICOMP'11] proposed self-improving algorithms for sorting and Delaunay triangulation (DT) when the input instances x₁,⋯,x_n follow some unknown product distribution. That is, x_i comes from a fixed unknown distribution 𝒟_i, and the x_i’s are drawn independently. After spending O(n^{1+ε}) time in a learning phase, the subsequent expected running time is O((n+ H)/ε), where H ∈ {H_S,H_DT}, and H_S and H_DT are the entropies of the distributions of the sorting and DT output, respectively. In this paper, we allow dependence among the x_i’s under the group product distribution. There is a hidden partition of [1,n] into groups; the x_i’s in the k-th group are fixed unknown functions of the same hidden variable u_k; and the u_k’s are drawn from an unknown product distribution. We describe self-improving algorithms for sorting and DT under this model when the functions that map u_k to x_i’s are well-behaved. After an O(poly(n))-time training phase, we achieve O(n + H_S) and O(nα(n) + H_DT) expected running times for sorting and DT, respectively, where α(⋅) is the inverse Ackermann function.
Cite as
Siu-Wing Cheng, Man-Kwun Chiu, Kai Jin, and Man Ting Wong. A Generalization of Self-Improving Algorithms. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 29:1-29:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cheng_et_al:LIPIcs.SoCG.2020.29,
author = {Cheng, Siu-Wing and Chiu, Man-Kwun and Jin, Kai and Wong, Man Ting},
title = {{A Generalization of Self-Improving Algorithms}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {29:1--29:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.29},
URN = {urn:nbn:de:0030-drops-121873},
doi = {10.4230/LIPIcs.SoCG.2020.29},
annote = {Keywords: expected running time, entropy, sorting, Delaunay triangulation}
}
Document
Dynamic Distribution-Sensitive Point Location
Abstract
We propose a dynamic data structure for the distribution-sensitive point location problem. Suppose that there is a fixed query distribution in ℝ², and we are given an oracle that can return in O(1) time the probability of a query point falling into a polygonal region of constant complexity. We can maintain a convex subdivision S with n vertices such that each query is answered in O(OPT) expected time, where OPT is the minimum expected time of the best linear decision tree for point location in S. The space and construction time are O(n log² n). An update of S as a mixed sequence of k edge insertions and deletions takes O(k log⁵ n) amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of n sites can be performed in O(n log⁵ n) expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.
Cite as
Siu-Wing Cheng and Man-Kit Lau. Dynamic Distribution-Sensitive Point Location. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{cheng_et_al:LIPIcs.SoCG.2020.30,
author = {Cheng, Siu-Wing and Lau, Man-Kit},
title = {{Dynamic Distribution-Sensitive Point Location}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {30:1--30:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.30},
URN = {urn:nbn:de:0030-drops-121882},
doi = {10.4230/LIPIcs.SoCG.2020.30},
annote = {Keywords: dynamic planar point location, convex subdivision, linear decision tree}
}
Document
No-Dimensional Tverberg Theorems and Algorithms
Abstract
Tverberg’s theorem states that for any k ≥ 2 and any set P ⊂ ℝ^d of at least (d + 1)(k - 1) + 1 points, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem lies in the complexity class PPAD ∩ PLS, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa [SODA 2019] gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm.
We present a deterministic algorithm that finds for any n-point set P ⊂ ℝ^d and any k ∈ {2, … , n} in O(nd ⌈log k⌉) time a k-partition of P such that there is a ball of radius O((k/√n)diam(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, and that there are no approximation algorithms that are truly polynomial in any dimension, our result provides a remarkably efficient and simple new notion of approximation.
Our main contribution is to generalize Sarkaria’s method [Israel Journal Math., 1992] to reduce the Tverberg problem to the Colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.
Cite as
Aruni Choudhary and Wolfgang Mulzer. No-Dimensional Tverberg Theorems and Algorithms. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
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@InProceedings{choudhary_et_al:LIPIcs.SoCG.2020.31,
author = {Choudhary, Aruni and Mulzer, Wolfgang},
title = {{No-Dimensional Tverberg Theorems and Algorithms}},
booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
pages = {31:1--31:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-143-6},
ISSN = {1868-8969},
year = {2020},
volume = {164},
editor = {Cabello, Sergio and Chen, Danny Z.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.31},
URN = {urn:nbn:de:0030-drops-121893},
doi = {10.4230/LIPIcs.SoCG.2020.31},
annote = {Keywords: Tverberg’s theorem, Colorful Carath\'{e}odory Theorem, Tensor lifting}
}
Document
Lexicographic Optimal Homologous Chains and Applications to Point Cloud Triangulations
Abstract
This paper considers a particular case of the Optimal Homologous Chain Problem (OHCP) for integer modulo 2 coefficients, where optimality is meant as a minimal lexicographic order on chains induced by a total order on simplices. The matrix reduction algorithm used for persistent homology is used to derive polynomial algorithms solving this problem instance, whereas OHCP is NP-hard in the general case. The complexity is further improved to a quasilinear algorithm by leveraging a dual graph minimum cut formulation w