eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
1
1412
10.4230/LIPIcs.SoCG.2024
article
LIPIcs, Volume 293, SoCG 2024, Complete Volume
Mulzer, Wolfgang
1
https://orcid.org/0000-0002-1948-5840
Phillips, Jeff M.
2
https://orcid.org/0000-0003-1169-2965
Freie Universität Berlin, Germany
University of Utah, USA
LIPIcs, Volume 293, SoCG 2024, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024/LIPIcs.SoCG.2024.pdf
LIPIcs, Volume 293, SoCG 2024, Complete Volume
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
0:i
0:xxii
10.4230/LIPIcs.SoCG.2024.0
article
Front Matter, Table of Contents, Preface, Conference Organization
Mulzer, Wolfgang
1
https://orcid.org/0000-0002-1948-5840
Phillips, Jeff M.
2
https://orcid.org/0000-0003-1169-2965
Freie Universität Berlin, Germany
University of Utah, USA
Front Matter, Table of Contents, Preface, Conference Organization
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.0/LIPIcs.SoCG.2024.0.pdf
Front Matter
Table of Contents
Preface
Conference Organization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
1:1
1:14
10.4230/LIPIcs.SoCG.2024.1
article
A Universal In-Place Reconfiguration Algorithm for Sliding Cube-Shaped Robots in a Quadratic Number of Moves
Abel, Zachary
1
A. Akitaya, Hugo
2
https://orcid.org/0000-0002-6827-2200
Kominers, Scott Duke
3
4
Korman, Matias
5
Stock, Frederick
2
Massachusetts Institute of Technology, Cambridge, MA, USA
University of Massachusetts Lowell, MA, USA
Harvard University, Cambridge, MA, USA
a16z crypto, New York, NY, USA
Siemens Electronic Design Automation, Wilsonville, OR, USA
In the modular robot reconfiguration problem, we are given n cube-shaped modules (or robots) as well as two configurations, i.e., placements of the n modules so that their union is face-connected. The goal is to find a sequence of moves that reconfigures the modules from one configuration to the other using "sliding moves," in which a module slides over the face or edge of a neighboring module, maintaining connectivity of the configuration at all times.
For many years it has been known that certain module configurations in this model require at least Ω(n²) moves to reconfigure between them. In this paper, we introduce the first universal reconfiguration algorithm - i.e., we show that any n-module configuration can reconfigure itself into any specified n-module configuration using just sliding moves. Our algorithm achieves reconfiguration in O(n²) moves, making it asymptotically tight. We also present a variation that reconfigures in-place, it ensures that throughout the reconfiguration process, all modules, except for one, will be contained in the union of the bounding boxes of the start and end configuration.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.1/LIPIcs.SoCG.2024.1.pdf
modular reconfigurable robots
sliding cube model
reconfiguration
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
2:1
2:15
10.4230/LIPIcs.SoCG.2024.2
article
Clustering with Few Disks to Minimize the Sum of Radii
Abrahamsen, Mikkel
1
de Berg, Sarita
2
Meijer, Lucas
2
Nusser, André
1
Theocharous, Leonidas
3
University of Copenhagen, Denmark
Utrecht University, The Netherlands
Eindhoven University of Technology, The Netherlands
Given a set of n points in the Euclidean plane, the k-MinSumRadius problem asks to cover this point set using k disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV '12]; however, the running time of this algorithm is 𝒪(n^881), and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the k-MinSumRadius problem is that of small k. For the 2-MinSumRadius problem, a near-quadratic time algorithm with expected running time 𝒪(n² log² n log² log n) was given over 30 years ago [Eppstein '92].
We present the first improvement of this result, namely, a near-linear time algorithm to compute the 2-MinSumRadius that runs in expected 𝒪(n log² n log² log n) time. We generalize this result to any constant dimension d, for which we give an 𝒪(n^{2-1/(⌈d/2⌉ + 1) + ε}) time algorithm. Additionally, we give a near-quadratic time algorithm for 3-MinSumRadius in the plane that runs in expected 𝒪(n² log² n log² log n) time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.2/LIPIcs.SoCG.2024.2.pdf
geometric clustering
minimize sum of radii
covering points with disks
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
3:1
3:14
10.4230/LIPIcs.SoCG.2024.3
article
On the Number of Digons in Arrangements of Pairwise Intersecting Circles
Ackerman, Eyal
1
Damásdi, Gábor
2
3
https://orcid.org/0000-0002-6390-5419
Keszegh, Balázs
2
3
https://orcid.org/0000-0002-3839-5103
Pinchasi, Rom
4
5
Raffay, Rebeka
6
Department of Mathematics, Physics and Computer Science, University of Haifa at Oranim, Tivon 36006, Israel
HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
ELTE Eötvös Loránd University, Budapest, Hungary
Technion - Israel Institute of Technology, Haifa, Israel
Visiting professor at EPFL, Lausanne, Switzerland
École Polytechnique Fédérale de Lausanne, Switzerland
A long-standing open conjecture of Branko Grünbaum from 1972 states that any arrangement of n pairwise intersecting pseudocircles in the plane can have at most 2n-2 digons. Agarwal et al. proved this conjecture for arrangements in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Grünbaum’s conjecture is true for arrangements of pseudocircles in which there are three pseudocircles every pair of which creates a digon. In this paper we prove this over 50-year-old conjecture of Grünbaum for any arrangement of pairwise intersecting circles in the plane.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.3/LIPIcs.SoCG.2024.3.pdf
Arrangement of pseudocircles
Counting touchings
Counting digons
Grünbaum’s conjecture
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
4:1
4:15
10.4230/LIPIcs.SoCG.2024.4
article
Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane
Agarwal, Pankaj K.
1
https://orcid.org/0000-0002-9439-181X
Ezra, Esther
2
https://orcid.org/0000-0001-8133-1335
Sharir, Micha
3
https://orcid.org/0000-0002-2541-3763
Department of Computer Science, Duke University, Durham, NC, USA
Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel
School of Computer Science, Tel Aviv University, Tel Aviv, Israel
Let P be a set of m points in ℝ², let Σ be a set of n semi-algebraic sets of constant complexity in ℝ², let (S,+) be a semigroup, and let w: P → S be a weight function on the points of P. We describe a randomized algorithm for computing w(P∩σ) for every σ ∈ Σ in overall expected time O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n), where s > 0 is a constant that bounds the maximum complexity of the regions of Σ, and where the O^*(⋅) notation hides subpolynomial factors. For s ≥ 3, surprisingly, this bound is smaller than the best-known bound for answering m such queries in an on-line manner. The latter takes O^*(m^{s/(2s-1)}n^{(2s-2)/(2s-1)} + m + n) time.
Let Φ: Σ × P → {0,1} be the Boolean predicate (of constant complexity) such that Φ(σ,p) = 1 if p ∈ σ and 0 otherwise, and let Σ_Φ P = {(σ,p) ∈ Σ× P ∣ Φ(σ,p) = 1}. Our algorithm actually computes a partition ℬ_Φ of Σ_Φ P into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n). It is straightforward to compute w(P∩σ) for all σ ∈ Σ from ℬ_Φ. Similarly, if η: Σ → S is a weight function on the regions of Σ, ∑_{σ ∈ Σ: p ∈ σ} η(σ), for every point p ∈ P, can be computed from ℬ_Φ in a straightforward manner. We also mention a few other applications of computing ℬ_Φ.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.4/LIPIcs.SoCG.2024.4.pdf
Range-searching
semi-algebraic sets
pseudo-lines
duality
geometric cuttings
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
5:1
5:17
10.4230/LIPIcs.SoCG.2024.5
article
Communication Complexity and Discrepancy of Halfplanes
Ahmed, Manasseh
1
Cheung, Tsun-Ming
2
Hatami, Hamed
2
Sareen, Kusha
2
Marianopolis College, Montreal, Canada
School of Computer Science, McGill University, Montreal, Canada
We study the discrepancy of the following communication problem. Alice receives a halfplane, and Bob receives a point in the plane, and their goal is to determine whether Bob’s point belongs to Alice’s halfplane. This communication task corresponds to determining whether x₁y₁+y₂ ≥ x₂, where the first player knows (x₁,x₂) and the second player knows (y₁,y₂).
Denoting n = m³, we show that when the inputs are chosen from [m] × [m²], the communication discrepancy of the above problem is O(n^{-1/6} log^{3/2} n).
On the other hand, through the connections to the notion of hereditary discrepancy by Matoušek, Nikolov, and Tawler (IMRN 2020) and a classical result of Matoušek (Discrete Comput. Geom. 1995), we show that the communication discrepancy of every set of n points and n halfplanes is at least Ω(n^{-1/4} log^{-1} n).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.5/LIPIcs.SoCG.2024.5.pdf
Randomized communication complexity
Discrepancy theory
factorization norm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
6:1
6:19
10.4230/LIPIcs.SoCG.2024.6
article
Probabilistic Analysis of Multiparameter Persistence Decompositions into Intervals
Alonso, Ángel Javier
1
https://orcid.org/0000-0002-5822-546X
Kerber, Michael
1
https://orcid.org/0000-0002-8030-9299
Skraba, Primoz
2
Institute of Geometry, Graz University of Technology, Austria
Queen Mary University of London, United Kingdom
Multiparameter persistence modules can be uniquely decomposed into indecomposable summands. Among these indecomposables, intervals stand out for their simplicity, making them preferable for their ease of interpretation in practical applications and their computational efficiency. Empirical observations indicate that modules that decompose into only intervals are rare. To support this observation, we show that for numerous common multiparameter constructions, such as density- or degree-Rips bifiltrations, and across a general category of point samples, the probability of the homology-induced persistence module decomposing into intervals goes to zero as the sample size goes to infinity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.6/LIPIcs.SoCG.2024.6.pdf
Topological Data Analysis
Multi-Parameter Persistence
Decomposition of persistence modules
Poisson point processes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
7:1
7:15
10.4230/LIPIcs.SoCG.2024.7
article
ETH-Tight Algorithm for Cycle Packing on Unit Disk Graphs
An, Shinwoo
1
Oh, Eunjin
1
POSTECH, Pohang, South Korea
In this paper, we consider the Cycle Packing problem on a unit disk graph defined as follows. Given a unit disk graph G with n vertices and an integer k, the goal is to find a set of k vertex-disjoint cycles of G if it exists. Our algorithm runs in time 2^O(√k) n^O(1). This improves the 2^O(√klog k) n^O(1)-time algorithm by Fomin et al. [SODA 2012, ICALP 2017]. Moreover, our algorithm is optimal assuming the exponential-time hypothesis.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.7/LIPIcs.SoCG.2024.7.pdf
Unit disk graphs
cycle packing
tree decomposition
parameterized algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
8:1
8:15
10.4230/LIPIcs.SoCG.2024.8
article
Eight-Partitioning Points in 3D, and Efficiently Too
Aronov, Boris
1
https://orcid.org/0000-0003-3110-4702
Basit, Abdul
2
https://orcid.org/0000-0003-0754-3203
Ramesh, Indu
1
https://orcid.org/0009-0008-9967-0819
Tasinato, Gianluca
3
https://orcid.org/0009-0008-7231-5753
Wagner, Uli
3
https://orcid.org/0000-0002-1494-0568
Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
School of Mathematics, Monash University, Clayton, Australia
Institute of Science and Technology Austria, Klosterneuburg, Austria
An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in ℝ³ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in ℝ³ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument.
We prove the following variant of this result: Any mass distribution (or point set) in ℝ³ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction.
Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in ℝ³ (with prescribed normal direction of one of the planes) in time O^*(n^{5/2}).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.8/LIPIcs.SoCG.2024.8.pdf
Mass partitions
partitions of points in three dimensions
Borsuk-Ulam Theorem
Ham-Sandwich Theorem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
9:1
9:15
10.4230/LIPIcs.SoCG.2024.9
article
A Clique-Based Separator for Intersection Graphs of Geodesic Disks in ℝ²
Aronov, Boris
1
https://orcid.org/0000-0003-3110-4702
de Berg, Mark
2
https://orcid.org/0000-0001-5770-3784
Theocharous, Leonidas
2
https://orcid.org/0000-0002-1707-6787
Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of ℝ² and let 𝒟 = {D_1,…,D_n} be a set of geodesic disks with respect to the metric d. We prove that 𝒢^×(𝒟), the intersection graph of the disks in 𝒟, has a clique-based separator consisting of O(n^{3/4+ε}) cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators.
Our clique-based separator yields an algorithm for q-Coloring that runs in time 2^O(n^{3/4+ε}), assuming the boundaries of the disks D_i can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses O(n^{7/4+ε}) storage and can report the hop distance between any two nodes in 𝒢^×(𝒟) in O(n^{3/4+ε}) time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.9/LIPIcs.SoCG.2024.9.pdf
Computational geometry
intersection graphs
separator theorems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
10:1
10:14
10.4230/LIPIcs.SoCG.2024.10
article
Discrete Fréchet Distance Oracles
Aronov, Boris
1
https://orcid.org/0000-0003-3110-4702
Farhana, Tsuri
2
Katz, Matthew J.
2
https://orcid.org/0000-0002-0672-729X
Ramesh, Indu
1
https://orcid.org/0009-0008-9967-0819
Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
It is unlikely that the discrete Fréchet distance between two curves of length n can be computed in strictly subquadratic time. We thus consider the setting where one of the curves, P, is known in advance. In particular, we wish to construct data structures (distance oracles) of near-linear size that support efficient distance queries with respect to P in sublinear time. Since there is evidence that this is impossible for query curves of length Θ(n^α), for any α > 0, we focus on query curves of (small) constant length, for which we are able to devise distance oracles with the desired bounds.
We extend our tools to handle subcurves of the given curve, and even arbitrary vertex-to-vertex subcurves of a given geometric tree. That is, we construct an oracle that can quickly compute the distance between a short polygonal path (the query) and a path in the preprocessed tree between two query-specified vertices. Moreover, we define a new family of geometric graphs, t-local graphs (which strictly contains the family of geometric spanners with constant stretch), for which a similar oracle exists: we can preprocess a graph G in the family, so that, given a query segment and a pair u,v of vertices in G, one can quickly compute the smallest discrete Fréchet distance between the segment and any (u,v)-path in G. The answer is exact, if t = 1, and approximate if t > 1.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.10/LIPIcs.SoCG.2024.10.pdf
discrete Fréchet distance
distance oracle
heavy-path decomposition
t-local graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
11:1
11:19
10.4230/LIPIcs.SoCG.2024.11
article
Tight Bounds for the Learning of Homotopy à la Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds
Attali, Dominique
1
Dal Poz Kouřimská, Hana
2
https://orcid.org/0000-0001-7841-0091
Fillmore, Christopher
2
https://orcid.org/0000-0001-7631-2885
Ghosh, Ishika
2
3
https://orcid.org/0000-0002-7901-5912
Lieutier, André
4
Stephenson, Elizabeth
2
https://orcid.org/0000-0002-6862-208X
Wintraecken, Mathijs
5
https://orcid.org/0000-0002-7472-2220
Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France
IST Austria, Klosterneuburg, Austria
Michigan State University, East Lansing, MI, USA
Aix-en-Provence, France
Inria Sophia Antipolis, Université Côte d'Azur, Sophia Antipolis, France
In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways:
- As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature.
- In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach.
- The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to 𝒮. We exhibit their tightness by an explicit construction.
We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.11/LIPIcs.SoCG.2024.11.pdf
Homotopy
Inference
Sets of positive reach
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
12:1
12:15
10.4230/LIPIcs.SoCG.2024.12
article
An O(n log n)-Time Approximation Scheme for Geometric Many-To-Many Matching
Bandyapadhyay, Sayan
1
https://orcid.org/0000-0001-8875-0102
Xue, Jie
2
https://orcid.org/0000-0001-7015-1988
Department of Computer Science, Portland State University, OR, USA
Department of Computer Science, New York University Shanghai, China
Geometric matching is an important topic in computational geometry and has been extensively studied over decades. In this paper, we study a geometric-matching problem, known as geometric many-to-many matching. In this problem, the input is a set S of n colored points in ℝ^d, which implicitly defines a graph G = (S,E(S)) where E(S) = {(p,q): p,q ∈ S have different colors}, and the goal is to compute a minimum-cost subset E^* ⊆ E(S) of edges that cover all points in S. Here the cost of E^* is the sum of the costs of all edges in E^*, where the cost of a single edge e is the Euclidean distance (or more generally, the L_p-distance) between the two endpoints of e. Our main result is a (1+ε)-approximation algorithm with an optimal running time O_ε(n log n) for geometric many-to-many matching in any fixed dimension, which works under any L_p-norm. This is the first near-linear approximation scheme for the problem in any d ≥ 2. Prior to this work, only the bipartite case of geometric many-to-many matching was considered in ℝ¹ and ℝ², and the best known approximation scheme in ℝ² takes O_ε(n^{1.5} ⋅ poly(log n)) time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.12/LIPIcs.SoCG.2024.12.pdf
many-to-many matching
geometric optimization
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
13:1
13:13
10.4230/LIPIcs.SoCG.2024.13
article
Topological k-Metrics
Barkan, Willow
1
Bennett, Huck
2
Nayyeri, Amir
1
Oregon State University, Corvallis, OR, USA
University of Colorado, Boulder, CO, USA
Metric spaces (X, d) are ubiquitous objects in mathematics and computer science that allow for capturing pairwise distance relationships d(x, y) between points x, y ∈ X. Because of this, it is natural to ask what useful generalizations there are of metric spaces for capturing "k-wise distance relationships" d(x_1, …, x_k) among points x_1, …, x_k ∈ X for k > 2. To that end, Gähler (Math. Nachr., 1963) (and perhaps others even earlier) defined k-metric spaces, which generalize metric spaces, and most notably generalize the triangle inequality d(x₁, x₂) ≤ d(x₁, y) + d(y, x₂) to the "simplex inequality" d(x_1, …, x_k) ≤ ∑_{i=1}^k d(x_1, …, x_{i-1}, y, x_{i+1}, …, x_k). (The definition holds for any fixed k ≥ 2, and a 2-metric space is just a (standard) metric space.)
In this work, we introduce strong k-metric spaces, k-metric spaces that satisfy a topological condition stronger than the simplex inequality, which makes them "behave nicely." We also introduce coboundary k-metrics, which generalize 𝓁_p metrics (and in fact all finite metric spaces induced by norms) and minimum bounding chain k-metrics, which generalize shortest path metrics (and capture all strong k-metrics). Using these definitions, we prove analogs of a number of fundamental results about embedding finite metric spaces including Fréchet embedding (isometric embedding into 𝓁_∞) and isometric embedding of all tree metrics into 𝓁₁. We also study relationships between families of (strong) k-metrics, and show that natural quantities, like simplex volume, are strong k-metrics.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.13/LIPIcs.SoCG.2024.13.pdf
k-metrics
metric embeddings
computational topology
simplicial complexes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
14:1
14:19
10.4230/LIPIcs.SoCG.2024.14
article
Totally Geodesic Surfaces in Hyperbolic 3-Manifolds: Algorithms and Examples
Basilio, Brannon
1
Lee, Chaeryn
1
Malionek, Joseph
1
Department of Mathematics, University of Illinois Urbana-Champaign, IL, USA
Finding a totally geodesic surface, an embedded surface where the geodesics in the surface are also geodesics in the surrounding manifold, has been a problem of interest in the study of 3-manifolds. This has especially been of interest in hyperbolic 3-manifolds and knot complements, complements of piecewise-linearly embedded circles in the 3-sphere. This is due to Menasco-Reid’s conjecture stating that hyperbolic knot complements do not contain such surfaces. Here, we present an algorithm that determines whether a given surface is totally geodesic and an algorithm that checks whether a given 3-manifold contains a totally geodesic surface. We applied our algorithm on over 150,000 3-manifolds and discovered nine 3-manifolds with totally geodesic surfaces. Additionally, we verified Menasco-Reid’s conjecture for knots up to 12 crossings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.14/LIPIcs.SoCG.2024.14.pdf
totally geodesic
Fuchsian group
hyperbolic
knot complement
computational topology
low-dimensional topology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
15:1
15:16
10.4230/LIPIcs.SoCG.2024.15
article
Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory
Bauer, Ulrich
1
https://orcid.org/0000-0002-9683-0724
Roll, Fabian
2
https://orcid.org/0000-0002-3604-4545
Department of Mathematics and Munich Data Science Institute, Technische Universität München, Germany
Department of Mathematics, Technische Universität München, Germany
We study the connection between discrete Morse theory and persistent homology in the context of shape reconstruction methods. Specifically, we consider the construction of Wrap complexes, introduced by Edelsbrunner as a subcomplex of the Delaunay complex, and the construction of lexicographic optimal homologous cycles, also considered by Cohen–Steiner, Lieutier, and Vuillamy in a similar setting. We show that for any cycle in a Delaunay complex for a given radius parameter, the lexicographically optimal homologous cycle is supported on the Wrap complex for the same parameter, thereby establishing a close connection between the two methods. We obtain this result by establishing a fundamental connection between reduction of cycles in the computation of persistent homology and gradient flows in the algebraic generalization of discrete Morse theory.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.15/LIPIcs.SoCG.2024.15.pdf
persistent homology
discrete Morse theory
apparent pairs
Wrap complex
lexicographic optimal chains
shape reconstruction
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
16:1
16:16
10.4230/LIPIcs.SoCG.2024.16
article
A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon
de Berg, Mark
1
https://orcid.org/0000-0001-5770-3784
Theocharous, Leonidas
1
https://orcid.org/0000-0002-1707-6787
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Let 𝒫 be a simple polygon with m vertices and let P be a set of n points inside 𝒫. We prove that there exists, for any ε > 0, a set C ⊂ P of size O(1/ε²) such that the following holds: for any query point q inside the polygon 𝒫, the geodesic distance from q to its furthest neighbor in C is at least 1-ε times the geodesic distance to its further neighbor in P. Thus the set C can be used for answering ε-approximate furthest-neighbor queries with a data structure whose storage requirement is independent of the size of P. The coreset can be constructed in O(1/(ε) (nlog(1/ε) + (n+m)log(n+m))) time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.16/LIPIcs.SoCG.2024.16.pdf
Furthest-neighbor queries
polygons
geodesic distance
coreset
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
17:1
17:16
10.4230/LIPIcs.SoCG.2024.17
article
Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain
de Berg, Sarita
1
Miltzow, Tillmann
1
Staals, Frank
1
Department of Information and Computing Sciences, Utrecht University, The Netherlands
We devise a data structure that can answer shortest path queries for two query points in a polygonal domain P on n vertices. For any ε > 0, the space complexity of the data structure is O(n^{10+ε}) and queries can be answered in O(log n) time. Alternatively, we can achieve a space complexity of O(n^{9+ε}) by relaxing the query time to O(log² n). This is the first improvement upon a conference paper by Chiang and Mitchell from 1999. They presented a data structure with O(n^{11}) space complexity and O(log n) query time. Our main result can be extended to include a space-time trade-off. Specifically, we devise data structures with O(n^{9+ε}/𝓁^{4+O(ε)}) space complexity and O(𝓁 log² n) query time, for any integer 1 ≤ 𝓁 ≤ n.
Furthermore, we present improved data structures for the special case where we restrict one (or both) of the query points to lie on the boundary of P. When one of the query points is restricted to lie on the boundary, and the other query point is unrestricted, the space complexity becomes O(n^{6+ε}) and the query time O(log²n). When both query points are on the boundary, the space complexity is decreased further to O(n^{4+ε}) and the query time to O(log n), thereby improving an earlier result of Bae and Okamoto.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.17/LIPIcs.SoCG.2024.17.pdf
data structure
polygonal domain
geodesic distance
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
18:1
18:16
10.4230/LIPIcs.SoCG.2024.18
article
Plane Hamiltonian Cycles in Convex Drawings
Bergold, Helena
1
https://orcid.org/0000-0002-9622-8936
Felsner, Stefan
2
https://orcid.org/0000-0002-6150-1998
M. Reddy, Meghana
3
https://orcid.org/0000-0001-9185-1246
Orthaber, Joachim
4
https://orcid.org/0000-0002-9982-0070
Scheucher, Manfred
2
https://orcid.org/0000-0002-1657-9796
Institut für Informatik, Freie Universität Berlin, Germany
Institut für Mathematik, Technische Universität Berlin, Germany
Department of Computer Science, ETH Zürich, Switzerland
Institute of Software Technology, Graz University of Technology, Austria
A conjecture by Rafla from 1988 asserts that every simple drawing of the complete graph K_n admits a plane Hamiltonian cycle. It turned out that already the existence of much simpler non-crossing substructures in such drawings is hard to prove. Recent progress was made by Aichholzer et al. and by Suk and Zeng who proved the existence of a plane path of length Ω(log n / log log n) and of a plane matching of size Ω(n^{1/2}) in every simple drawing of K_n.
Instead of studying simpler substructures, we prove Rafla’s conjecture for the subclass of convex drawings, the most general class in the convexity hierarchy introduced by Arroyo et al. Moreover, we show that every convex drawing of K_n contains a plane Hamiltonian path between each pair of vertices (Hamiltonian connectivity) and a plane k-cycle for each 3 ≤ k ≤ n (pancyclicity), and present further results on maximal plane subdrawings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.18/LIPIcs.SoCG.2024.18.pdf
simple drawing
convexity hierarchy
plane pancyclicity
plane Hamiltonian connectivity
maximal plane subdrawing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
19:1
19:16
10.4230/LIPIcs.SoCG.2024.19
article
Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time
Bhore, Sujoy
1
https://orcid.org/0000-0003-0104-1659
Nöllenburg, Martin
2
https://orcid.org/0000-0003-0454-3937
Tóth, Csaba D.
3
4
https://orcid.org/0000-0002-8769-3190
Wulms, Jules
5
https://orcid.org/0000-0002-9314-8260
Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India
Institute of Logic and Computation, Algorithms and Complexity Group, TU Wien, Austria
Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
Department of Computer Science, Tufts University, Medford, MA, USA
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
A fundamental question is whether one can maintain a maximum independent set (MIS) in polylogarithmic update time for a dynamic collection of geometric objects in Euclidean space. For a set of intervals, it is known that no dynamic algorithm can maintain an exact MIS in sublinear update time. Therefore, the typical objective is to explore the trade-off between update time and solution size. Substantial efforts have been made in recent years to understand this question for various families of geometric objects, such as intervals, hypercubes, hyperrectangles, and fat objects.
We present the first fully dynamic approximation algorithm for disks of arbitrary radii in the plane that maintains a constant-factor approximate MIS in polylogarithmic expected amortized update time. Moreover, for a fully dynamic set of n unit disks in the plane, we show that a 12-approximate MIS can be maintained with worst-case update time O(log n), and optimal output-sensitive reporting. This result generalizes to fat objects of comparable sizes in any fixed dimension d, where the approximation ratio depends on the dimension and the fatness parameter. Further, we note that, even for a dynamic set of disks of unit radius in the plane, it is impossible to maintain O(1+ε)-approximate MIS in truly sublinear update time, under standard complexity assumptions.
Our results build on two recent technical tools: (i) The MIX algorithm by Cardinal et al. (ESA 2021) that can smoothly transition from one independent set to another; hence it suffices to maintain a family of independent sets where the largest one is an O(1)-approximate MIS. (ii) A dynamic nearest/farthest neighbor data structure for disks by Kaplan et al. (DCG 2020) and Liu (SICOMP 2022), which generalizes the dynamic convex hull data structure by Chan (JACM 2010), and quickly yields a "replacement" disk (if any) when a disk in one of our independent sets is deleted.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.19/LIPIcs.SoCG.2024.19.pdf
Dynamic algorithm
Independent set
Geometric intersection graph
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
20:1
20:16
10.4230/LIPIcs.SoCG.2024.20
article
Constrained and Ordered Level Planarity Parameterized by the Number of Levels
Blažej, Václav
1
https://orcid.org/0000-0001-9165-6280
Klemz, Boris
2
https://orcid.org/0000-0002-4532-3765
Klesen, Felix
2
https://orcid.org/0000-0003-1136-5673
Sieper, Marie Diana
2
https://orcid.org/0009-0003-7491-2811
Wolff, Alexander
2
https://orcid.org/0000-0001-5872-718X
Zink, Johannes
2
https://orcid.org/0000-0002-7398-718X
University of Warwick, Coventry, UK
Institut für Informatik, Universität Würzburg, Germany
The problem Level Planarity asks for a crossing-free drawing of a graph in the plane such that vertices are placed at prescribed y-coordinates (called levels) and such that every edge is realized as a y-monotone curve. In the variant Constrained Level Planarity (CLP), each level y is equipped with a partial order ≺_y on its vertices and in the desired drawing the left-to-right order of vertices on level y has to be a linear extension of ≺_y. Ordered Level Planarity (OLP) corresponds to the special case of CLP where the given partial orders ≺_y are total orders. Previous results by Brückner and Rutter [SODA 2017] and Klemz and Rote [ACM Trans. Alg. 2019] state that both CLP and OLP are NP-hard even in severely restricted cases. In particular, they remain NP-hard even when restricted to instances whose width (the maximum number of vertices that may share a common level) is at most two. In this paper, we focus on the other dimension: we study the parameterized complexity of CLP and OLP with respect to the height (the number of levels).
We show that OLP parameterized by the height is complete with respect to the complexity class XNLP, which was first studied by Elberfeld, Stockhusen, and Tantau [Algorithmica 2015] (under a different name) and recently made more prominent by Bodlaender, Groenland, Nederlof, and Swennenhuis [FOCS 2021]. It contains all parameterized problems that can be solved nondeterministically in time f(k)⋅ n^O(1) and space f(k)⋅ log n (where f is a computable function, n is the input size, and k is the parameter). If a problem is XNLP-complete, it lies in XP, but is W[t]-hard for every t.
In contrast to the fact that OLP parameterized by the height lies in XP, it turns out that CLP is NP-hard even when restricted to instances of height 4. We complement this result by showing that CLP can be solved in polynomial time for instances of height at most 3.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.20/LIPIcs.SoCG.2024.20.pdf
Parameterized Complexity
Graph Drawing
XNLP
XP
W[t]-hard
Level Planarity
Planar Poset Diagram
Computational Geometry
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
21:1
21:16
10.4230/LIPIcs.SoCG.2024.21
article
On Edge Collapse of Random Simplicial Complexes
Boissonnat, Jean-Daniel
1
Dutta, Kunal
2
Dutta, Soumik
2
Pritam, Siddharth
3
Université Côte d'Azur, INRIA, Sophia Antipolis, France
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Chennai Mathematical Institute, India
We consider the edge collapse (introduced in [Boissonnat, Pritam. SoCG 2020]) process on the Erdős-Rényi random clique complex X(n,c/√n) on n vertices with edge probability c/√n such that c > √η₂ where η₂ = inf{η | x = e^{-η(1-x)²} has a solution in (0,1)}. For a given c > √η₂, we show that after t iterations of maximal edge collapsing phases, the remaining subcomplex, or t-core, has at most (1+o(1))binom(n,2)p(1-(c²/3)(1-(1-γ_t)³)) and at least (1+o(1)) binom(n,2) p(1-γ_{t+1}-c²γ_t(1-γ_t)²) edges asymptotically almost surely (a.a.s.), where {γ_t}_{t ≥ 0} is recursively determined by γ_{t+1} = e^{-c²(1-γ_t)²} and γ_0 = 0. We also determine the upper and lower bound on the final core with explicit formulas. If c < √{η₂} then we show that the final core contains o(n√n) edges. On the other hand, if, instead of c being a constant with respect to n, c > √{2log n} then the edge collapse process is no more effective in reducing the size of the complex.
Our proof is based on the notion of local weak convergence [Aldous, Steele. In Probability on discrete structures. Springer, 2004] together with two new components. Firstly, we identify the critical combinatorial structures that control the outcome of the edge collapse process. By controlling the expected number of these structures during the edge collapse process we establish a.a.s. bounds on the size of the core. We also give a new concentration inequality for typically Lipschitz functions on random graphs which improves on the bound of [Warnke. Combinatorics, Probability and Computing, 2016] and is, therefore, of independent interest. The proof of our lower bound is via the recursive technique of [Linial and Peled. A Journey Through Discrete Mathematics. 2017] to simulate cycles in infinite trees. These are the first theoretical results proved for edge collapses on random (or non-random) simplicial complexes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.21/LIPIcs.SoCG.2024.21.pdf
Computational Topology
Topological Data Analysis
Strong Collapse
Simple Collapse
Persistent homology
Random simplicial complexes
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
22:1
22:17
10.4230/LIPIcs.SoCG.2024.22
article
Reconfiguration of Plane Trees in Convex Geometric Graphs
Bousquet, Nicolas
1
https://orcid.org/0000-0003-0170-0503
de Meyer, Lucas
1
https://orcid.org/0000-0001-5865-4697
Pierron, Théo
1
https://orcid.org/0000-0002-5586-5613
Wesolek, Alexandra
1
https://orcid.org/0000-0003-4841-5937
Université de Lyon, LIRIS, CNRS, Université Claude Bernard Lyon 1, France
A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most 2n-4 between any pair of non-crossing spanning trees (where n denotes the number of points). Hernando et al. proved that the length of a minimal flip sequence can be of length at least (3/2) n. Two recent results of Aichholzer et al. and Bousquet et al. improved the Avis and Fukuda upper bound by proving that there always exists a flip sequence of length respectively at most 2n-log n and 2n-√n when the points are in convex position.
We pursue the investigation of the convex case by improving the upper bound by a linear factor for the first time in 30 years. We prove that there always exists a flip sequence between any pair of non-crossing spanning trees T₁,T₂ of length at most c n where c ≈ 1.95. Our result is actually stronger since we prove that, for any two trees T₁,T₂, there exists a flip sequence from T₁ to T₂ of length at most c |T₁ ⧵ T₂|.
We also improve the best lower bound in terms of the symmetric difference by proving that there exists a pair of trees T₁,T₂ such that a minimal flip sequence has length (5/3) |T₁ ⧵ T₂|, improving the lower bound of Hernando et al. by considering the symmetric difference instead of the number of vertices.
We generalize this lower bound construction to non-crossing flips (where we close the gap between upper and lower bounds) and rotations.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.22/LIPIcs.SoCG.2024.22.pdf
Reconfiguration
Non-crossing trees
flip distance
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
23:1
23:18
10.4230/LIPIcs.SoCG.2024.23
article
A Canonical Tree Decomposition for Chirotopes
Bouvel, Mathilde
1
Feray, Valentin
2
Goaoc, Xavier
1
Koechlin, Florent
3
Université de Lorraine, CNRS, INRIA, LORIA, F-54000 Nancy, France
Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
Université Sorbonne Paris Nord, LIPN, CNRS UMR 7030, F-93340 Villetaneuse, France
We introduce and study a notion of decomposition of planar point sets (or rather of their chirotopes) as trees decorated by smaller chirotopes. This decomposition is based on the concept of mutually avoiding sets, and adapts in some sense the modular decomposition of graphs in the world of chirotopes. The associated tree always exists and is unique up to some appropriate constraints. We also show how to compute the number of triangulations of a chirotope efficiently, starting from its tree and the (weighted) numbers of triangulations of its parts.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.23/LIPIcs.SoCG.2024.23.pdf
Order type
modular decomposition
counting triangulations
mutually avoiding point sets
generating functions
rewriting systems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
24:1
24:15
10.4230/LIPIcs.SoCG.2024.24
article
Dynamic Convex Hulls for Simple Paths
Brewer, Bruce
1
https://orcid.org/0009-0008-2995-148X
Brodal, Gerth Stølting
2
https://orcid.org/0000-0001-9054-915X
Wang, Haitao
1
https://orcid.org/0000-0001-8134-7409
Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Department of Computer Science, Aarhus University, Denmark
We consider two restricted cases of the planar dynamic convex hull problem with point insertions and deletions. We assume all updates are performed on a deque (double-ended queue) of points. The first case considers the monotonic path case, where all points are sorted in a given direction, say horizontally left-to-right, and only the leftmost and rightmost points can be inserted and deleted. The second case, which is more general, assumes that the points in the deque constitute a simple path. For both cases, we present solutions supporting deque insertions and deletions in worst-case constant time and standard queries on the convex hull of the points in O(log n) time, where n is the number of points in the current point set. The convex hull of the current point set can be reported in O(h+log n) time, where h is the number of edges of the convex hull. For the 1-sided monotone path case, where updates are only allowed on one side, the reporting time can be reduced to O(h), and queries on the convex hull are supported in O(log h) time. All our time bounds are worst case. In addition, we prove lower bounds that match these time bounds, and thus our results are optimal.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.24/LIPIcs.SoCG.2024.24.pdf
Dynamic convex hull
convex hull queries
simple paths
path updates
deque
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
25:1
25:17
10.4230/LIPIcs.SoCG.2024.25
article
Fine-Grained Complexity of Earth Mover’s Distance Under Translation
Bringmann, Karl
1
Staals, Frank
2
Węgrzycki, Karol
3
van Wordragen, Geert
4
Saarland University and Max-Planck-Institute for Informatics, Saarbrücken, Germany
Department of Information and Computing Sciences, Utrecht University, The Netherlands
Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany
Department of Computer Science, Aalto University, Espoo, Finland
The Earth Mover’s Distance is a popular similarity measure in several branches of computer science. It measures the minimum total edge length of a perfect matching between two point sets. The Earth Mover’s Distance under Translation (EMDuT) is a translation-invariant version thereof. It minimizes the Earth Mover’s Distance over all translations of one point set.
For EMDuT in ℝ¹, we present an 𝒪̃(n²)-time algorithm. We also show that this algorithm is nearly optimal by presenting a matching conditional lower bound based on the Orthogonal Vectors Hypothesis. For EMDuT in ℝ^d, we present an 𝒪̃(n^{2d+2})-time algorithm for the L₁ and L_∞ metric. We show that this dependence on d is asymptotically tight, as an n^o(d)-time algorithm for L_1 or L_∞ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.25/LIPIcs.SoCG.2024.25.pdf
Earth Mover’s Distance
Earth Mover’s Distance under Translation
Fine-Grained Complexity
Maximum Weight Bipartite Matching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
26:1
26:14
10.4230/LIPIcs.SoCG.2024.26
article
Approximating the Geometric Knapsack Problem in Near-Linear Time and Dynamically
Buchem, Moritz
1
https://orcid.org/0000-0002-1590-346X
Deuker, Paul
1
Wiese, Andreas
1
https://orcid.org/0000-0003-3705-016X
Technische Universität München, Germany
One important goal in algorithm design is determining the best running time for solving a problem (approximately). For some problems, we know the optimal running time, assuming certain conditional lower bounds. In this paper, we study the d-dimensional geometric knapsack problem in which we are far from this level of understanding. We are given a set of weighted d-dimensional geometric items like squares, rectangles, or hypercubes and a knapsack which is a square or a (hyper-)cube. Our goal is to select a subset of the given items that fit non-overlappingly inside the knapsack, maximizing the total profit of the packed items. We make a significant step towards determining the best running time for solving these problems approximately by presenting approximation algorithms whose running times are near-linear, i.e., O(n⋅poly(log n)), for any constant d and any parameter ε > 0 (the exponent of log n depends on d and 1/ε).
In the case of (hyper)-cubes, we present a (1+ε)-approximation algorithm. This improves drastically upon the currently best known algorithm which is a (1+ε)-approximation algorithm with a running time of n^{O_{ε,d}(1)} where the exponent of n depends exponentially on 1/ε and d. In particular, our algorithm is an efficient polynomial time approximation scheme (EPTAS). Moreover, we present a (2+ε)-approximation algorithm for rectangles in the setting without rotations and a (17/9+ε)≈ 1.89-approximation algorithm if we allow rotations by 90 degrees. The best known polynomial time algorithms for this setting have approximation ratios of 17/9+ε and 1.5+ε, respectively, and running times in which the exponent of n depends exponentially on 1/ε. In addition, we give dynamic algorithms with polylogarithmic query and update times, having the same approximation guarantees as our other algorithms above.
Key to our results is a new family of structured packings which we call easily guessable packings. They are flexible enough to guarantee the existence of profitable solutions while providing enough structure so that we can compute these solutions very quickly.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.26/LIPIcs.SoCG.2024.26.pdf
Geometric packing
approximation algorithms
dynamic algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
27:1
27:15
10.4230/LIPIcs.SoCG.2024.27
article
Map-Matching Queries Under Fréchet Distance on Low-Density Spanners
Buchin, Kevin
1
https://orcid.org/0000-0002-3022-7877
Buchin, Maike
2
https://orcid.org/0000-0002-3446-4343
Gudmundsson, Joachim
3
https://orcid.org/0000-0002-6778-7990
Popov, Aleksandr
4
https://orcid.org/0000-0002-0158-1746
Wong, Sampson
5
https://orcid.org/0000-0003-3803-3804
Department of Computer Science, TU Dortmund, Germany
Faculty of Computer Science, Ruhr-Universität Bochum, Germany
School of Computer Science, University of Sydney, Australia
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Department of Computer Science, University of Copenhagen, Denmark
Map matching is a common task when analysing GPS tracks, such as vehicle trajectories. The goal is to match a recorded noisy polygonal curve to a path on the map, usually represented as a geometric graph. The Fréchet distance is a commonly used metric for curves, making it a natural fit. The map-matching problem is well-studied, yet until recently no-one tackled the data structure question: preprocess a given graph so that one can query the minimum Fréchet distance between all graph paths and a polygonal curve. Recently, Gudmundsson, Seybold, and Wong [Gudmundsson et al., 2023] studied this problem for arbitrary query polygonal curves and c-packed graphs. In this paper, we instead require the graphs to be λ-low-density t-spanners, which is significantly more representative of real-world networks. We also show how to report a path that minimises the distance efficiently rather than only returning the minimal distance, which was stated as an open problem in their paper.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.27/LIPIcs.SoCG.2024.27.pdf
Map Matching
Fréchet Distance
Data Structures
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
28:1
28:16
10.4230/LIPIcs.SoCG.2024.28
article
Computing Shortest Closed Curves on Non-Orientable Surfaces
Bulavka, Denys
1
Colin de Verdière, Éric
2
Fuladi, Niloufar
3
Einstein Institute of Mathematics, Hebrew University, Jerusalem, Israel
LIGM, CNRS, Univ. Gustave Eiffel, F-77454 Marne-la-Vallée, France
LORIA, CNRS, INRIA, Université de Lorraine, F-54000 Nancy, France
We initiate the study of computing shortest non-separating simple closed curves with some given topological properties on non-orientable surfaces. While, for orientable surfaces, any two non-separating simple closed curves are related by a self-homeomorphism of the surface, and computing shortest such curves has been vastly studied, for non-orientable ones the classification of non-separating simple closed curves up to ambient homeomorphism is subtler, depending on whether the curve is one-sided or two-sided, and whether it is orienting or not (whether it cuts the surface into an orientable one).
We prove that computing a shortest orienting (weakly) simple closed curve on a non-orientable combinatorial surface is NP-hard but fixed-parameter tractable in the genus of the surface. In contrast, we can compute a shortest non-separating non-orienting (weakly) simple closed curve with given sidedness in g^{O(1)} ⋅ n log n time, where g is the genus and n the size of the surface.
For these algorithms, we develop tools that can be of independent interest, to compute a variation on canonical systems of loops for non-orientable surfaces based on the computation of an orienting curve, and some covering spaces that are essentially quotients of homology covers.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.28/LIPIcs.SoCG.2024.28.pdf
Surface
Graph
Algorithm
Non-orientable surface
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
29:1
29:23
10.4230/LIPIcs.SoCG.2024.29
article
Practical Software for Triangulating and Simplifying 4-Manifolds
Burke, Rhuaidi Antonio
1
https://orcid.org/0000-0001-8886-1881
The University of Queensland, Brisbane, Australia
Dimension 4 is the first dimension in which exotic smooth manifold pairs appear - manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and triangulated 4-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this paper we introduce new software tools to make this possible, including a software implementation of an algorithm which enables us to build triangulations of 4-manifolds from Kirby diagrams, as well as a new heuristic for simplifying 4-manifold triangulations. Using these tools, we present new triangulations of several bounded exotic pairs, corks and plugs (objects responsible for "exoticity"), as well as the smallest known triangulation of the fundamental K3 surface. The small size of these triangulations benefit us by revealing fine structural features in 4-manifold triangulations.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.29/LIPIcs.SoCG.2024.29.pdf
computational low-dimensional topology
triangulations
4-manifolds
exotic 4-manifolds
mathematical software
experiments in low-dimensional topology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
30:1
30:16
10.4230/LIPIcs.SoCG.2024.30
article
Effective Computation of the Heegaard Genus of 3-Manifolds
Burton, Benjamin A.
1
https://orcid.org/0000-0002-3478-3301
Thompson, Finn
1
https://orcid.org/0000-0003-3241-8332
The University of Queensland, Brisbane, Australia
The Heegaard genus is a fundamental invariant of 3-manifolds. However, computing the Heegaard genus of a triangulated 3-manifold is NP-hard, and while algorithms exist, little work has been done in making such an algorithm efficient and practical for implementation. Current algorithms use almost normal surfaces, which are an extension of the algorithm-friendly normal surface theory but which add considerable complexity for both running time and implementation.
Here we take a different approach: instead of working with almost normal surfaces, we give a general method of modifying the input triangulation that allows us to avoid almost normal surfaces entirely. The cost is just four new tetrahedra, and the benefit is that important surfaces that were once almost normal can be moved to the simpler setting of normal surfaces in the new triangulation. We apply this technique to the computation of Heegaard genus, where we develop algorithms and heuristics that prove successful in practice when applied to a data set of 3,000 closed hyperbolic 3-manifolds; we precisely determine the genus for at least 2,705 of these.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.30/LIPIcs.SoCG.2024.30.pdf
3-manifolds
triangulations
normal surfaces
computational topology
Heegaard genus
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
31:1
31:15
10.4230/LIPIcs.SoCG.2024.31
article
Geometric Matching and Bottleneck Problems
Cabello, Sergio
1
2
https://orcid.org/0000-0002-3183-4126
Cheng, Siu-Wing
3
https://orcid.org/0000-0002-3557-9935
Cheong, Otfried
4
https://orcid.org/0000-0003-4467-7075
Knauer, Christian
5
University of Ljubljana, Ljubljana, Slovenia
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
HKUST, Hong Kong, China
SCALGO, Aarhus, Denmark
University of Bayreuth, Germany
Let P be a set of at most n points and let R be a set of at most n geometric ranges, such as disks and rectangles, where each p ∈ P has an associated supply s_{p} > 0, and each r ∈ R has an associated demand d_r > 0. A (many-to-many) matching is a set 𝒜 of ordered triples (p,r,a_{pr}) ∈ P × R × ℝ_{> 0} such that p ∈ r and the a_{pr}’s satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing ∑_{(p,r,a_{pr}) ∈ 𝒜} a_{pr}.
Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of n red points P and a set of n blue points Q that minimizes the length of the longest edge. For the L_∞-metric, we can do this in time O(n^{1+ε}) in any fixed dimension, for the L₂-metric in the plane in time O(n^{4/3 + ε}), for any ε > 0.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.31/LIPIcs.SoCG.2024.31.pdf
Many-to-many matching
bipartite
planar
geometric
approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
32:1
32:19
10.4230/LIPIcs.SoCG.2024.32
article
SCARST: Schnyder Compact and Regularity Sensitive Triangulation Data Structure
Castelli Aleardi, Luca
1
https://orcid.org/0000-0002-1142-2562
Devillers, Olivier
2
https://orcid.org/0000-0003-4275-5068
LIX, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
We consider the design of fast and compact representations of the connectivity information of triangle meshes. Although traditional data structures (Half-Edge, Corner Table) are fast and user-friendly, they tend to be memory-expensive. On the other hand, compression schemes, while meeting information-theoretic lower bounds, do not support navigation within the mesh structure. Compact representations provide an advantageous balance for representing large meshes, enabling a judicious compromise between memory consumption and fast implementation of navigational operations. We propose new representations that are sensitive to the regularity of the graph while still having worst case guarantees. For all our data structures we have both an interesting storage cost, typically 2 or 3 r.p.v. (references per vertex) in the case of very regular triangulations, and provable upper bounds in the worst case scenario. One of our solutions has a worst case cost of 3.33 r.p.v., which is currently the best-known bound improving the previous 4 r.p.v. [Castelli et al. 2018]. Our representations have slightly slower running times (factors 1.5 to 4) than classical data structures. In our experiments we compare on various meshes runtime and memory performance of our representations with those of the most efficient existing solutions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.32/LIPIcs.SoCG.2024.32.pdf
Meshes
compression
triangulations
compact representations
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
33:1
33:15
10.4230/LIPIcs.SoCG.2024.33
article
Semialgebraic Range Stabbing, Ray Shooting, and Intersection Counting in the Plane
Chan, Timothy M.
1
https://orcid.org/0000-0002-8093-0675
Cheng, Pingan
2
https://orcid.org/0000-0002-8131-847X
Zheng, Da Wei
1
https://orcid.org/0000-0002-0844-9457
University of Illinois Urbana-Champaign, Urbana, IL, USA
Aarhus University, Denmark
Polynomial partitioning techniques have recently led to improved geometric data structures for a variety of fundamental problems related to semialgebraic range searching and intersection searching in 3D and higher dimensions (e.g., see [Agarwal, Aronov, Ezra, and Zahl, SoCG 2019; Ezra and Sharir, SoCG 2021; Agarwal, Aronov, Ezra, Katz, and Sharir, SoCG 2022]). They have also led to improved algorithms for offline versions of semialgebraic range searching in 2D, via lens-cutting [Sharir and Zahl (2017)]. In this paper, we show that these techniques can yield new data structures for a number of other 2D problems even for online queries:
1) Semialgebraic range stabbing. We present a data structure for n semialgebraic ranges in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of ranges containing a query point in O(n^{1/4+ε}) time, for an arbitrarily small constant ε > 0. (The query time bound is likely close to tight for this space bound.)
2) Ray shooting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can find the first arc hit by a query (straight-line) ray in O(n^{1/4+ε}) time. (The query bound is again likely close to tight for this space bound, and they improve a result by Ezra and Sharir with near n^{3/2} space and near √n query time.)
3) Intersection counting amid algebraic arcs. We present a data structure for n algebraic arcs in 2D of constant description complexity with O(n^{3/2+ε}) preprocessing time and space, so that we can count the number of intersection points with a query algebraic arc of constant description complexity in O(n^{1/2+ε}) time. In particular, this implies an O(n^{3/2+ε})-time algorithm for counting intersections between two sets of n algebraic arcs in 2D. (This generalizes a classical O(n^{3/2+ε})-time algorithm for circular arcs by Agarwal and Sharir from SoCG 1991.)
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.33/LIPIcs.SoCG.2024.33.pdf
Computational geometry
range searching
intersection searching
semialgebraic sets
data structures
polynomial partitioning
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
34:1
34:15
10.4230/LIPIcs.SoCG.2024.34
article
Convex Polygon Containment: Improving Quadratic to Near Linear Time
Chan, Timothy M.
1
https://orcid.org/0000-0002-8093-0675
Hair, Isaac M.
2
https://orcid.org/0000-0001-6992-4488
Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Department of Computer Science, University of California, Santa Barbara, CA, USA
We revisit a standard polygon containment problem: given a convex k-gon P and a convex n-gon Q in the plane, find a placement of P inside Q under translation and rotation (if it exists), or more generally, find the largest copy of P inside Q under translation, rotation, and scaling.
Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required Ω(n²) time, even in the simplest k = 3 case. We present a significantly faster new algorithm for k = 3 achieving O(n polylog n) running time. Moreover, we extend the result for general k, achieving O(k^O(1/ε) n^{1+ε}) running time for any ε > 0.
Along the way, we also prove a new O(k^O(1) n polylog n) bound on the number of similar copies of P inside Q that have 4 vertices of P in contact with the boundary of Q (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.34/LIPIcs.SoCG.2024.34.pdf
Polygon containment
convex polygons
translations
rotations
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
35:1
35:15
10.4230/LIPIcs.SoCG.2024.35
article
Enclosing Points with Geometric Objects
Chan, Timothy M.
1
https://orcid.org/0000-0002-8093-0675
He, Qizheng
1
https://orcid.org/0000-0002-2518-1114
Xue, Jie
2
https://orcid.org/0000-0001-7015-1988
Department of Computer Science, University of Illinois Urbana-Champaign, IL, USA
Department of Computer Science, New York University Shanghai, China
Let X be a set of points in ℝ² and 𝒪 be a set of geometric objects in ℝ², where |X| + |𝒪| = n. We study the problem of computing a minimum subset 𝒪^* ⊆ 𝒪 that encloses all points in X. Here a point x ∈ X is enclosed by 𝒪^* if it lies in a bounded connected component of ℝ²∖(⋃_{O ∈ 𝒪^*} O). We propose two algorithmic frameworks to design polynomial-time approximation algorithms for the problem. The first framework is based on sparsification and min-cut, which results in O(1)-approximation algorithms for unit disks, unit squares, etc. The second framework is based on LP rounding, which results in an O(α(n)log n)-approximation algorithm for segments, where α(n) is the inverse Ackermann function, and an O(log n)-approximation algorithm for disks.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.35/LIPIcs.SoCG.2024.35.pdf
obstacle placement
geometric optimization
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
36:1
36:13
10.4230/LIPIcs.SoCG.2024.36
article
Dynamic Geometric Connectivity in the Plane with Constant Query Time
Chan, Timothy M.
1
https://orcid.org/0000-0002-8093-0675
Huang, Zhengcheng
1
Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
We present the first fully dynamic connectivity data structures for geometric intersection graphs achieving constant query time and sublinear amortized update time for many classes of geometric objects in 2D . Our data structures can answer connectivity queries between two objects, as well as "global" connectivity queries (e.g., deciding whether the entire graph is connected). Previously, the data structure by Afshani and Chan (ESA'06) achieved such bounds only in the special case of axis-aligned line segments or rectangles but did not work for arbitrary line segments or disks, whereas the data structures by Chan, Pătraşcu, and Roditty (FOCS'08) worked for more general classes of geometric objects but required n^{Ω(1)} query time and could not handle global connectivity queries.
Specifically, we obtain new data structures with O(1) query time and amortized update time near n^{4/5}, n^{7/8}, and n^{20/21} for axis-aligned line segments, disks, and arbitrary line segments respectively. Besides greatly reducing the query time, our data structures also improve the previous update times for axis-aligned line segments by Afshani and Chan (from near n^{10/11} to n^{4/5}) and for disks by Chan, Pătraşcu, and Roditty (from near n^{20/21} to n^{7/8}).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.36/LIPIcs.SoCG.2024.36.pdf
Connectivity
dynamic data structures
geometric intersection graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
37:1
37:15
10.4230/LIPIcs.SoCG.2024.37
article
Optimal Euclidean Tree Covers
Chang, Hsien-Chih
1
Conroy, Jonathan
1
Le, Hung
2
Milenković, Lazar
3
Solomon, Shay
3
Than, Cuong
2
Department of Computer Science, Dartmouth College, Hanover, NH, USA
Manning CICS, UMass Amherst, MA, USA
Tel Aviv University, Israel
A (1+e)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1+e)-stretch path in one of the trees. The celebrated Dumbbell Theorem [Arya et al. STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+e)-stretch tree cover with O_d(e^{-d} ⋅ log(1/e)) trees, where the O_d notation suppresses terms that depend solely on the dimension d. The running time of their construction is O_d(n log n ⋅ log(1/e)/e^d + n ⋅ e^{-2d}). Since the same point may occur in multiple levels of the tree, the maximum degree of a point in the tree cover may be as large as Ω(log Φ), where Φ is the aspect ratio of the input point set.
In this work we present a (1+e)-stretch tree cover with O_d(e^{-d+1} ⋅ log(1/e)) trees, which is optimal (up to the log(1/e) factor). Moreover, the maximum degree of points in any tree is an absolute constant for any d. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a (1+e)-stretch Steiner tree cover (that may use Steiner points) with O_d(e^{(-d+1)/2} ⋅ log(1/e)) trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive O_d(n log n) term; this improves over the running time underlying the Dumbbell Theorem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.37/LIPIcs.SoCG.2024.37.pdf
Tree cover
spanner
Steiner point
routing
bounded-degree
quadtree
net-tree
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
38:1
38:14
10.4230/LIPIcs.SoCG.2024.38
article
Computing Diameter+2 in Truly-Subquadratic Time for Unit-Disk Graphs
Chang, Hsien-Chih
1
https://orcid.org/0000-0001-6714-7988
Gao, Jie
2
https://orcid.org/0000-0001-5083-6082
Le, Hung
3
https://orcid.org/0000-0001-8223-9944
Department of Computer Science, Dartmouth College, Hanover, NH, USA
Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Manning CICS, University of Massachusetts Amherst, MA, USA
Finding the diameter of a graph in general cannot be done in truly subquadratic assuming the Strong Exponential Time Hypothesis (SETH), even when the underlying graph is unweighted and sparse. When restricting to concrete classes of graphs and assuming SETH, planar graphs and minor-free graphs admit truly subquadratic algorithms, while geometric intersection graphs of unit balls, congruent equilateral triangles, and unit segments do not. Unit-disk graphs is one of the major open cases where the complexity of diameter computation remains unknown. More generally, it is conjectured that a truly subquadratic time algorithm exists for pseudo-disk graphs where each pair of objects has at most two intersections on the boundary.
In this paper, we show a truly-subquadratic algorithm of running time O^~(n^{2-1/18}), for finding the diameter in a unit-disk graph, whose output differs from the optimal solution by at most 2. This is the first algorithm that provides an additive guarantee in distortion, independent of the size or the diameter of the graph. Our algorithm requires two important technical elements. First, we show that for the intersection graph of pseudo-disks, the graph VC-dimension - either of k-hop balls or the distance encoding vectors - is 4. This contrasts to the VC dimension of the pseudo-disks themselves as geometric ranges (which is known to be 3). Second, we introduce a clique-based r-clustering for geometric intersection graphs, which is an analog of the r-division construction for planar graphs. We also showcase the new techniques by establishing new results for distance oracles for unit-disk graphs with subquadratic storage and O(1) query time. The results naturally extend to unit L₁ or L_∞-disks and fat pseudo-disks of similar size. Last, if the pseudo-disks additionally have bounded ply, we have a truly subquadratic algorithm to find the exact diameter.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.38/LIPIcs.SoCG.2024.38.pdf
Unit-disk graph
pseudo-disks
r-division
VC-dimension
distance oracle
clique-based separator
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
39:1
39:11
10.4230/LIPIcs.SoCG.2024.39
article
Nearly Orthogonal Sets over Finite Fields
Chawin, Dror
1
Haviv, Ishay
1
School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel
For a field 𝔽 and integers d and k, a set of vectors of 𝔽^d is called k-nearly orthogonal if its members are non-self-orthogonal and every k+1 of them include an orthogonal pair. We prove that for every prime p there exists a positive constant δ = δ (p), such that for every field 𝔽 of characteristic p and for all integers k ≥ 2 and d ≥ k^{1/(p-1)}, there exists a k-nearly orthogonal set of at least d^{δ ⋅ k^{1/(p-1)} / log k} vectors of 𝔽^d. In particular, for the binary field we obtain a set of d^Ω(k/log k) vectors, and this is tight up to the log k term in the exponent. For comparison, the best known lower bound over the reals is d^Ω(log k / log log k)} (Alon and Szegedy, Graphs and Combin., 1999). The proof combines probabilistic and spectral arguments.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.39/LIPIcs.SoCG.2024.39.pdf
Nearly orthogonal sets
Finite fields
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
40:1
40:15
10.4230/LIPIcs.SoCG.2024.40
article
Optimal Algorithm for the Planar Two-Center Problem
Cho, Kyungjin
1
https://orcid.org/0000-0003-2223-4273
Oh, Eunjin
1
https://orcid.org/0000-0003-0798-2580
Wang, Haitao
2
https://orcid.org/0000-0001-8134-7409
Xue, Jie
3
https://orcid.org/0000-0001-7015-1988
Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Department of Computer Science, New York University Shanghai, China
We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set S of n points in the plane and the goal is to find two smallest congruent disks whose union contains all points of S. A longstanding open problem has been to obtain an O(nlog n)-time algorithm for planar two-center, matching the Ω(nlog n) lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in O(nlog² n) time. In this paper, we present an O(nlog n)-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.40/LIPIcs.SoCG.2024.40.pdf
two-center
r-coverage
disk coverage
circular hulls
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
41:1
41:15
10.4230/LIPIcs.SoCG.2024.41
article
GPU Algorithm for Enumerating PL Spheres of Picard Number 4: Application to Toric Topology
Choi, Suyoung
1
Jang, Hyeontae
1
Vallée, Mathieu
2
https://orcid.org/0000-0001-6336-9225
Ajou University, Suwon, South Korea
LIPN, CNRS UMR 7030, Université Sorbonne Paris Nord, Villetaneuse, France
The fundamental theorem for toric geometry states a toric manifold is encoded by a complete non-singular fan, whose combinatorial structure is the one of a PL sphere together with the set of generators of its rays. The wedge operation on a PL sphere increases its dimension without changing its Picard number. The seeds are the PL spheres that are not wedges. A PL sphere is toric colorable if it comes from a complete rational fan. A result of Choi and Park tells us that the set of toric seeds with a fixed Picard number p is finite. In fact, a toric PL sphere needs its facets to be bases of some binary matroids of corank p with neither coloops, nor cocircuits of size 2. We present and use a GPU-friendly and computationally efficient algorithm to enumerate this set of seeds, up to simplicial isomorphism. Explicitly, it allows us to obtain this set of seeds for Picard number 4 which is of main importance in toric topology for the characterization of toric manifolds with small Picard number. This follows the work of Kleinschmidt (1988) and Batyrev (1991) who fully classified toric manifolds with Picard number ≤ 3.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.41/LIPIcs.SoCG.2024.41.pdf
PL sphere
simplicial sphere
toric manifold
Picard number
weak pseudo-manifold
characteristic map
binary matroid
parallel computing
GPU programming
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
42:1
42:17
10.4230/LIPIcs.SoCG.2024.42
article
Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping
Conradi, Jacobus
1
https://orcid.org/0000-0002-8259-1187
Kolbe, Benedikt
2
https://orcid.org/0009-0005-0440-4912
Psarros, Ioannis
3
https://orcid.org/0000-0002-5079-5003
Rohde, Dennis
1
https://orcid.org/0000-0001-8984-1962
University of Bonn, Germany
Hausdorff Center for Mathematics, University of Bonn, Germany
Archimedes, Athena Research Center, Greece
We present algorithms for the computation of ε-coresets for k-median clustering of point sequences in ℝ^d under the p-dynamic time warping (DTW) distance. Coresets under DTW have not been investigated before, and the analysis is not directly accessible to existing methods as DTW is not a metric. The three main ingredients that allow our construction of coresets are the adaptation of the ε-coreset framework of sensitivity sampling, bounds on the VC dimension of approximations to the range spaces of balls under DTW, and new approximation algorithms for the k-median problem under DTW. We achieve our results by investigating approximations of DTW that provide a trade-off between the provided accuracy and amenability to known techniques. In particular, we observe that given n curves under DTW, one can directly construct a metric that approximates DTW on this set, permitting the use of the wealth of results on metric spaces for clustering purposes. The resulting approximations are the first with polynomial running time and achieve a very similar approximation factor as state-of-the-art techniques. We apply our results to produce a practical algorithm approximating (k,𝓁)-median clustering under DTW.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.42/LIPIcs.SoCG.2024.42.pdf
Dynamic time warping
coreset
median clustering
approximation algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
43:1
43:18
10.4230/LIPIcs.SoCG.2024.43
article
An Improved Lower Bound on the Number of Pseudoline Arrangements
Cortés Kühnast, Fernando
1
https://orcid.org/0009-0008-1847-362X
Dallant, Justin
2
https://orcid.org/0000-0001-5539-9037
Felsner, Stefan
1
https://orcid.org/0000-0002-6150-1998
Scheucher, Manfred
1
https://orcid.org/0000-0002-1657-9796
Institute of Mathematics, Technische Universität Berlin, Germany
Algorithms Research Group, Université libre de Bruxelles, Belgium
Arrangements of pseudolines are classic objects in discrete and computational geometry. They have been studied with increasing intensity since their introduction almost 100 years ago. The study of the number B_n of non-isomorphic simple arrangements of n pseudolines goes back to Goodman and Pollack, Knuth, and others. It is known that B_n is in the order of 2^Θ(n²) and finding asymptotic bounds on b_n = log₂(B_n)/n² remains a challenging task. In 2011, Felsner and Valtr showed that 0.1887 ≤ b_n ≤ 0.6571 for sufficiently large n. The upper bound remains untouched but in 2020 Dumitrescu and Mandal improved the lower bound constant to 0.2083. Their approach utilizes the known values of B_n for up to n = 12.
We tackle the lower bound by utilizing dynamic programming and the Lindström–Gessel–Viennot lemma. Our new bound is b_n ≥ 0.2721 for sufficiently large n. The result is based on a delicate interplay of theoretical ideas and computer assistance.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.43/LIPIcs.SoCG.2024.43.pdf
counting
pseudoline arrangement
recursive construction
bipermutation
divide and conquer
dynamic programming
computer-assisted proof
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
44:1
44:17
10.4230/LIPIcs.SoCG.2024.44
article
Stability and Approximations for Decorated Reeb Spaces
Curry, Justin
1
https://orcid.org/0000-0003-2504-8388
Mio, Washington
2
Needham, Tom
2
https://orcid.org/0000-0001-6165-3433
Okutan, Osman Berat
3
Russold, Florian
4
University at Albany, State University of New York, NY, USA
Florida State University, Tallahassee, FL, USA
Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Graz University of Technology, Austria
Given a map f:X → M from a topological space X to a metric space M, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when M = ℝ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of f. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.44/LIPIcs.SoCG.2024.44.pdf
Reeb spaces
Gromov-Hausdorff distance
Persistent homology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
45:1
45:15
10.4230/LIPIcs.SoCG.2024.45
article
Sweeping Arrangements of Non-Piercing Regions in the Plane
Dalal, Suryendu
1
Gangopadhyay, Rahul
2
Raman, Rajiv
1
Ray, Saurabh
3
IIIT-Delhi, India
Moscow Institute of Physics and Technology, Russia
NYU Abu Dhai, UAE
Let Γ be a finite set of Jordan curves in the plane. For any curve γ ∈ Γ, we denote the bounded region enclosed by γ as γ̃. We say that Γ is a non-piercing family if for any two curves α , β ∈ Γ, α̃ ⧵ β̃ is a connected region. A non-piercing family of curves generalizes a family of 2-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger ("Sweeping Arrangements of Curves", SoCG '89) proved that if we are given a family Γ of 2-intersecting curves and a sweep curve γ ∈ Γ, then the arrangement can be swept by γ while always maintaining the 2-intersecting property of the curves. We generalize the result of Snoeyink and Hershberger to the setting of non-piercing curves. We show that given an arrangement of non-piercing curves Γ, and a sweep curve γ ∈ Γ, the arrangement can be swept by γ so that the arrangement remains non-piercing throughout the process. We also give a shorter and simpler proof of the result of Snoeyink and Hershberger, and give an eclectic set of applications.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.45/LIPIcs.SoCG.2024.45.pdf
Sweeping
Pseudodisks
Discrete Geometry
Topology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
46:1
46:14
10.4230/LIPIcs.SoCG.2024.46
article
Saturation Results Around the Erdős-Szekeres Problem
Damásdi, Gábor
1
2
https://orcid.org/0000-0002-6390-5419
Dong, Zichao
1
3
Scheucher, Manfred
4
https://orcid.org/0000-0002-1657-9796
Zeng, Ji
5
1
HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
ELTE Eötvös Loránd University, Budapest, Hungary
Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS), Daejeon, South Korea
Institut für Mathematik, Technische Universität Berlin, Germany
University of California San Diego, La Jolla, CA, USA
In this paper, we consider saturation problems related to the celebrated Erdős-Szekeres convex polygon problem. For each n ≥ 7, we construct a planar point set of size (7/8) ⋅ 2^{n-2} which is saturated for convex n-gons. That is, the set contains no n points in convex position while the addition of any new point creates such a configuration. This demonstrates that the saturation number is smaller than the Ramsey number for the Erdős-Szekeres problem. The proof also shows that the original Erdős-Szekeres construction is indeed saturated.
Our construction is based on a similar improvement for the saturation version of the cups-versus-caps theorem. Moreover, we consider the generalization of the cups-versus-caps theorem to monotone paths in ordered hypergraphs. In contrast to the geometric setting, we show that this abstract saturation number is always equal to the corresponding Ramsey number.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.46/LIPIcs.SoCG.2024.46.pdf
Convex polygon
Cups-versus-caps
Monotone path
Saturation problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
47:1
47:17
10.4230/LIPIcs.SoCG.2024.47
article
Robustly Guarding Polygons
Das, Rathish
1
https://orcid.org/0000-0002-2416-6422
Filtser, Omrit
2
https://orcid.org/0000-0002-3978-1428
Katz, Matthew J.
3
https://orcid.org/0000-0002-0672-729X
Mitchell, Joseph S.B.
4
https://orcid.org/0000-0002-0152-2279
University of Houston, TX, USA
The Open University of Israel, Israel
Ben-Gurion University of the Negev, Beer-Sheva, Israel
Stony Brook University, NY, USA
We propose precise notions of what it means to guard a domain "robustly", under a variety of models. While approximation algorithms for minimizing the number of (precise) point guards in a polygon is a notoriously challenging area of investigation, we show that imposing various degrees of robustness on the notion of visibility coverage leads to a more tractable (and realistic) problem for which we can provide approximation algorithms with constant factor guarantees.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.47/LIPIcs.SoCG.2024.47.pdf
geometric optimization
approximation algorithms
guarding
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
48:1
48:18
10.4230/LIPIcs.SoCG.2024.48
article
Hopf Arborescent Links, Minor Theory, and Decidability of the Genus Defect
Dehornoy, Pierre
1
https://orcid.org/0000-0003-3450-8315
Lunel, Corentin
2
https://orcid.org/0009-0009-2060-3582
de Mesmay, Arnaud
2
https://orcid.org/0000-0002-7301-3799
Aix-Marseille Université, CNRS, I2M, Marseille, France
LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France
While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a class of knots and links called Hopf arborescent links, which are obtained as the boundaries of some iterated plumbings of Hopf bands. We show that for such links, computing the genus defects, which measure how much the four-dimensional genera differ from the classical genus, is decidable. Our proof is non-constructive, and is obtained by proving that Seifert surfaces of Hopf arborescent links under a relation of minors defined by containment of their Seifert surfaces form a well-quasi-order.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.48/LIPIcs.SoCG.2024.48.pdf
Knot Theory
Genus
Slice Genus
Hopf Arborescent Links
Well-Quasi-Order
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
49:1
49:15
10.4230/LIPIcs.SoCG.2024.49
article
Computing Zigzag Vineyard Efficiently Including Expansions and Contractions
Dey, Tamal K.
1
Hou, Tao
2
Department of Computer Science, Purdue University, West Lafayette, IN, USA
School of Computing, DePaul University, Chicago, IL, USA
Vines and vineyard connecting a stack of persistence diagrams have been introduced in the non-zigzag setting by Cohen-Steiner et al. [Cohen-Steiner et al., 2006]. We consider computing these vines over changing filtrations for zigzag persistence while incorporating two more operations: expansions and contractions in addition to the transpositions considered in the non-zigzag setting. Although expansions and contractions can be implemented in quadratic time in the non-zigzag case by utilizing the linear-time transpositions, it is not obvious how they can be carried out under the zigzag framework with the same complexity. While transpositions alone can be easily conducted in linear time using the recent FastZigzag algorithm [Tamal K. Dey and Tao Hou, 2022], expansions and contractions pose difficulty in breaking the barrier of cubic complexity [Dey and Hou, 2022]. Our main result is that, the half-way constructed up-down filtration in the FastZigzag algorithm indeed can be used to achieve linear time complexity for transpositions and quadratic time complexity for expansions and contractions, matching the time complexity of all corresponding operations in the non-zigzag case.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.49/LIPIcs.SoCG.2024.49.pdf
zigzag persistence
vines and vineyard
update operations
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
50:1
50:15
10.4230/LIPIcs.SoCG.2024.50
article
Cup Product Persistence and Its Efficient Computation
Dey, Tamal K.
1
https://orcid.org/0000-0001-5160-9738
Rathod, Abhishek
2
https://orcid.org/0000-0003-2533-3699
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Department of Computer Science, Ben Gurion University, Beersheba, Israel
It is well-known that the cohomology ring has a richer structure than homology groups. However, until recently, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Some of the recently introduced invariants, namely, persistent cup-length, persistent cup modules and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we devise an O(d n⁴) algorithm for computing the persistent k-cup modules for all k ∈ {2, … , d}, where d denotes the dimension of the filtered complex, and n denotes its size. Moreover, we note that since the persistent cup length can be obtained as a byproduct of our computations, this leads to a faster algorithm for computing it for d ≥ 3. Finally, we introduce a new stable invariant called partition modules of cup product that is more discriminative than persistent cup modules and devise an O(c(d)n⁴) algorithm for computing it, where c(d) is subexponential in d.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.50/LIPIcs.SoCG.2024.50.pdf
Persistent cohomology
cup product
image persistence
persistent cup module
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
51:1
51:18
10.4230/LIPIcs.SoCG.2024.51
article
Efficient Algorithms for Complexes of Persistence Modules with Applications
Dey, Tamal K.
1
https://orcid.org/0000-0001-5160-9738
Russold, Florian
2
Samaga, Shreyas N.
1
https://orcid.org/0000-0002-4128-3946
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Institute of Geometry, Graz University of Technology, Austria
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.51/LIPIcs.SoCG.2024.51.pdf
Persistent (co)homology
Persistence modules
Sheaves
Presentations
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
52:1
52:13
10.4230/LIPIcs.SoCG.2024.52
article
Colorful Intersections and Tverberg Partitions
Dobbins, Michael Gene
1
https://orcid.org/0000-0003-1428-406X
Holmsen, Andreas F.
2
3
Lee, Dohyeon
2
3
https://orcid.org/0009-0009-3238-5236
Department of Mathematics and Statistics, Binghamton University, NY, USA
Department of Mathematical Sciences, KAIST, Daejeon, South Korea
Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
The colorful Helly theorem and Tverberg’s theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers d ≥ m ≥ 1 and k a prime power. Suppose F₁, F₂, … , F_m are families of convex sets in ℝ^d, each of size n > (d/m+1)(k-1), such that for any choice C_i ∈ F_i we have ⋂_{i = 1}^m C_i ≠ ∅. Then, one of the families F_i admits a Tverberg k-partition. That is, one of the F_i can be partitioned into k nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning r-dimensional transversals to families of convex sets in ℝ^d that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.52/LIPIcs.SoCG.2024.52.pdf
Tverberg’s theorem
geometric transversals
topological combinatorics
configuration space/test map
discrete Morse theory
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
53:1
53:14
10.4230/LIPIcs.SoCG.2024.53
article
Maximum Betti Numbers of Čech Complexes
Edelsbrunner, Herbert
1
https://orcid.org/0000-0002-9823-6833
Pach, János
2
1
https://orcid.org/0000-0002-2389-2035
Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Rényi Institute of Mathematics, Budapest, Hungary
The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.53/LIPIcs.SoCG.2024.53.pdf
Discrete geometry
computational topology
Čech complexes
Delaunay mosaics
Alpha complexes
Betti numbers
extremal questions
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
54:1
54:8
10.4230/LIPIcs.SoCG.2024.54
article
An Improved Bound on Sums of Square Roots via the Subspace Theorem
Eisenbrand, Friedrich
1
Haeberle, Matthieu
1
Singer, Neta
1
EPFL, Lausanne, Switzerland
The sum of square roots is as follows: Given x_1,… ,x_n ∈ ℤ and a₁,… ,a_n ∈ ℕ decide whether E = ∑_{i=1}^n x_i √{a_i} ≥ 0. It is a prominent open problem (Problem 33 of the Open Problems Project), whether this can be decided in polynomial time. The state-of-the-art methods rely on separation bounds, which are lower bounds on the minimum nonzero absolute value of E. The current best bound shows that |E| ≥ (n ⋅ max_i (|x_i| ⋅√{a_i})) ^{-2ⁿ}, which is doubly exponentially small.
We provide a new bound of the form |E| ≥ γ ⋅ (n ⋅ max_i |x_i|)^{-2n} where γ is a constant depending on a₁,… ,a_n. This is singly exponential in n for fixed a_1,… ,a_n. The constant γ is not explicit and stems from the subspace theorem, a deep result in the geometry of numbers.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.54/LIPIcs.SoCG.2024.54.pdf
Exact computing
Separation Bounds
Computational Geometry
Geometry of Numbers
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
55:1
55:13
10.4230/LIPIcs.SoCG.2024.55
article
Dimensionality of Hamming Metrics and Rademacher Type
Eskenazis, Alexandros
1
2
https://orcid.org/0000-0002-1601-8307
CNRS, Institut de Mathématiques de Jussieu, Sorbonne Université, France
Trinity College, University of Cambridge, UK
Let X be a finite-dimensional normed space. We prove that if the Hamming cube {-1,1}ⁿ embeds into X with bi-Lipschitz distortion at most D ≥ 1, then dim(X) ≳ sup_{p ∈ [1,2]} n^p/(D^p 𝖳_p(X)^p), where 𝖳_p(X) is the Rademacher type p constant of X. This estimate yields a mutual refinement of distortion lower bounds which follow from works of Oleszkiewicz (1996) and Ivanisvili, van Handel and Volberg (2020). The proof relies on a combination of semigroup techniques on the biased hypercube with the Borsuk-Ulam theorem from algebraic topology.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.55/LIPIcs.SoCG.2024.55.pdf
Hamming cube
Rademacher type
metric embeddings
Borsuk-Ulam theorem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
56:1
56:15
10.4230/LIPIcs.SoCG.2024.56
article
Light, Reliable Spanners
Filtser, Arnold
1
Gitlitz, Yuval
2
Neiman, Ofer
2
Bar-Ilan University, Ramat-Gan, Israel
Ben-Gurion University of the Negev, Be'er Sheva, Israel
A ν-reliable spanner of a metric space (X,d), is a (dominating) graph H, such that for any possible failure set B ⊆ X, there is a set B^+ just slightly larger |B^+| ≤ (1+ν)⋅|B|, and all distances between pairs in X⧵B^+ are (approximately) preserved in H⧵B. Recently, there have been several works on sparse reliable spanners in various settings, but so far, the weight of such spanners has not been analyzed at all. In this work, we initiate the study of light reliable spanners, whose weight is proportional to that of the Minimum Spanning Tree (MST) of X.
We first observe that unlike sparsity, the lightness of any deterministic reliable spanner is huge, even for the metric of the simple path graph. Therefore, randomness must be used: an oblivious reliable spanner is a distribution over spanners, and the bound on |B^+| holds in expectation.
We devise an oblivious ν-reliable (2+2/(k-1))-spanner for any k-HST, whose lightness is ≈ ν^{-2}. We demonstrate a matching Ω(ν^{-2}) lower bound on the lightness (for any finite stretch). We also note that any stretch below 2 must incur linear lightness.
For general metrics, doubling metrics, and metrics arising from minor-free graphs, we construct light tree covers, in which every tree is a k-HST of low weight. Combining these covers with our results for k-HSTs, we obtain oblivious reliable light spanners for these metric spaces, with nearly optimal parameters. In particular, for doubling metrics we get an oblivious ν-reliable (1+ε)-spanner with lightness ε^{-O(ddim)} ⋅ Õ(ν^{-2}⋅log n), which is best possible (up to lower order terms).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.56/LIPIcs.SoCG.2024.56.pdf
light spanner
reliable spanner
HST cover
doubling metric
minor free graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
57:1
57:15
10.4230/LIPIcs.SoCG.2024.57
article
Multicut Problems in Embedded Graphs: The Dependency of Complexity on the Demand Pattern
Focke, Jacob
1
https://orcid.org/0000-0002-6895-755X
Hörsch, Florian
1
Li, Shaohua
1
Marx, Dániel
1
https://orcid.org/0000-0002-5686-8314
CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
The Multicut problem asks for a minimum cut separating certain pairs of vertices: formally, given a graph G and a demand graph H on a set T ⊆ V(G) of terminals, the task is to find a minimum-weight set C of edges of G such that whenever two vertices of T are adjacent in H, they are in different components of G⧵ C. Colin de Verdière [Algorithmica, 2017] showed that Multicut with t terminals on a graph G of genus g can be solved in time f(t,g) n^O(√{g²+gt+t}). Cohen-Addad et al. [JACM, 2021] proved a matching lower bound showing that the exponent of n is essentially best possible (for every fixed value of t and g), even in the special case of Multiway Cut, where the demand graph H is a complete graph.
However, this lower bound tells us nothing about other special cases of Multicut such as Group 3-Terminal Cut (where three groups of terminals need to be separated from each other). We show that if the demand pattern is, in some sense, close to being a complete bipartite graph, then Multicut can be solved faster than f(t,g) n^{O(√{g²+gt+t})}, and furthermore this is the only property that allows such an improvement. Formally, for a class ℋ of graphs, Multicut(ℋ) is the special case where the demand graph H is in ℋ. For every fixed class ℋ (satisfying some mild closure property), fixed g, and fixed t, our main result gives tight upper and lower bounds on the exponent of n in algorithms solving Multicut(ℋ).
In addition, we investigate a similar setting where, instead of parameterizing by the genus g of G, we parameterize by the minimum number k of edges of G that need to be deleted to obtain a planar graph. Interestingly, in this setting it makes a significant difference whether the graph G is weighted or unweighted: further nontrivial algorithmic techniques give substantial improvements in the unweighted case.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.57/LIPIcs.SoCG.2024.57.pdf
MultiCut
Multiway Cut
Parameterized Complexity
Tight Bounds
Embedded Graph
Planar Graph
Genus
Surface
Exponential Time Hypothesis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
58:1
58:15
10.4230/LIPIcs.SoCG.2024.58
article
Fréchet Edit Distance
Fox, Emily
1
Nayyeri, Amir
2
Perry, Jonathan James
1
https://orcid.org/0009-0003-0042-249X
Raichel, Benjamin
1
https://orcid.org/0000-0001-6584-4843
Department of Computer Science, University of Texas at Dallas, TX, USA
School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA
We define and investigate the Fréchet edit distance problem. Given two polygonal curves π and σ and a threshhold value δ > 0, we seek the minimum number of edits to σ such that the Fréchet distance between the edited σ and π is at most δ. For the edit operations we consider three cases, namely, deletion of vertices, insertion of vertices, or both. For this basic problem we consider a number of variants. Specifically, we provide polynomial time algorithms for both discrete and continuous Fréchet edit distance variants, as well as hardness results for weak Fréchet edit distance variants.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.58/LIPIcs.SoCG.2024.58.pdf
Fréchet distance
Edit distance
Hardness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
59:1
59:14
10.4230/LIPIcs.SoCG.2024.59
article
A Structure Theorem for Pseudo-Segments and Its Applications
Fox, Jacob
1
Pach, János
2
Suk, Andrew
3
Department of Mathematics, Stanford University, CA, USA
HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Department of Mathematics, University of California San Diego, La Jolla, CA, USA
We prove a far-reaching strengthening of Szemerédi’s regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such graphs can be partitioned into a bounded number of parts of roughly the same size such that almost all of the bipartite graphs between pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most n(log n)^O(log k) edges.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.59/LIPIcs.SoCG.2024.59.pdf
Regularity lemma
pseudo-segments
intersection graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
60:1
60:14
10.4230/LIPIcs.SoCG.2024.60
article
Near Optimal Locality Sensitive Orderings in Euclidean Space
Gao, Zhimeng
1
https://orcid.org/0009-0003-3668-9305
Har-Peled, Sariel
2
https://orcid.org/0000-0003-2638-9635
Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Kowloon, Hong Kong
Department of Computer Science, University of Illinois, Urbana, IL, USA
For a parameter ε ∈ (0,1), a set of ε-locality-sensitive orderings (LSOs) has the property that for any two points, p,q ∈ [0,1)^d, there exist an order in the set such that all the points between p and q (in the order) are ε-close to either p or q. Since the original construction of LSOs can not be (significantly) improved, we present a construction of modified LSOs, that yields a smaller set, while preserving their usefulness. Specifically, the resulting set of LSOs has size M = O(ℰ^{d-1} log ℰ), where ℰ = 1/ε. This improves over previous work by a factor of ℰ, and is optimal up to a factor of log ℰ.
As a consequence we get a flotilla of improved dynamic geometric algorithms, such as maintaining bichromatic closest pair, and spanners, among others. In particular, for geometric dynamic spanners the new result matches (up to the aforementioned log ℰ factor) the lower bound, Specifically, this is a near-optimal simple dynamic data-structure for maintaining spanners under insertions and deletions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.60/LIPIcs.SoCG.2024.60.pdf
Orderings
approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
61:1
61:16
10.4230/LIPIcs.SoCG.2024.61
article
Approximating the Maximum Independent Set of Convex Polygons with a Bounded Number of Directions
Grandoni, Fabrizio
1
https://orcid.org/0000-0002-9676-4931
Husić, Edin
1
https://orcid.org/0000-0002-6708-5112
Mari, Mathieu
2
https://orcid.org/0000-0001-8074-0241
Tinguely, Antoine
1
https://orcid.org/0009-0000-7321-5457
IDSIA, USI-SUPSI, Lugano, Switzerland
LIRMM, University of Montpellier, CNRS, Montpellier, France
In the maximum independent set of convex polygons problem, we are given a set of n convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the problem where the edges of the polygons can take at most d fixed directions. We present an 8d/3-approximation algorithm for this problem running in time O((nd)^O(d4^d)). The previous-best polynomial-time approximation (for constant d) was a classical n^ε approximation by Fox and Pach [SODA'11] that has recently been improved to a OPT^ε-approximation algorithm by Cslovjecsek, Pilipczuk and Węgrzycki [SODA '24], which also extends to an arbitrary set of convex polygons.
Our result builds on, and generalizes the recent constant factor approximation algorithms for the maximum independent set of axis-parallel rectangles problem (which is a special case of our problem with d = 2) by Mitchell [FOCS'21] and Gálvez, Khan, Mari, Mömke, Reddy, and Wiese [SODA'22].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.61/LIPIcs.SoCG.2024.61.pdf
Approximation algorithms
packing
independent set
polygons
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
62:1
62:14
10.4230/LIPIcs.SoCG.2024.62
article
Approximating Multiplicatively Weighted Voronoi Diagrams: Efficient Construction with Linear Size
Gudmundsson, Joachim
1
https://orcid.org/0000-0002-6778-7990
Seybold, Martin P.
2
https://orcid.org/0000-0001-6901-3035
Wong, Sampson
3
https://orcid.org/0000-0003-3803-3804
University of Sydney, Australia
Faculty of Computer Science, Theory and Applications of Algorithms, University of Vienna, Austria
University of Copenhagen, Denmark
Given a set of n sites from ℝ^d, each having some positive weight factor, the Multiplicatively Weighted Voronoi Diagram is a subdivision of space that associates each cell to the site whose weighted Euclidean distance is minimal for all points in the cell.
We give novel approximation algorithms that output a cube-based subdivision such that the weighted distance of a point with respect to the associated site is at most (1+ε) times the minimum weighted distance, for any fixed parameter ε ∈ (0,1). The diagram size is O_d(n log(1/ε)/ε^{d-1}) and the construction time is within an O_D(log(n)/ε^{(d+5)/2})-factor of the size bound. We also prove a matching lower bound for the size, showing that the proposed method is the first to achieve optimal size, up to Θ(1)^d-factors. In particular, the obscure log(1/ε) factor is unavoidable. As a by-product, we obtain a factor d^{O(d)} improvement in size for the unweighted case and O(d log(n) + d² log(1/ε)) point-location time in the subdivision, improving the known query bound by one d-factor.
The key ingredients of our approximation algorithms are the study of convex regions that we call cores, an adaptive refinement algorithm to obtain optimal size, and a novel notion of bisector coresets, which may be of independent interest. In particular, we show that coresets with O_d(1/ε^{(d+3)/2}) worst-case size can be computed in near-linear time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.62/LIPIcs.SoCG.2024.62.pdf
Multiplicatively Weighted Voronoi Diagram
Compressed QuadTree
Adaptive Refinement
Bisector Coresets
Semi-Separated Pair Decomposition
Lower Bound
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
63:1
63:15
10.4230/LIPIcs.SoCG.2024.63
article
Faster Fréchet Distance Approximation Through Truncated Smoothing
van der Horst, Thijs
1
2
Ophelders, Tim
1
2
Department of Information and Computing Sciences, Utrecht University, The Netherlands
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
The Fréchet distance is a popular distance measure for curves. Computing the Fréchet distance between two polygonal curves of n vertices takes roughly quadratic time, and conditional lower bounds suggest that even approximating to within a factor 3 cannot be done in strongly-subquadratic time, even in one dimension. The current best approximation algorithms present trade-offs between approximation quality and running time. Recently, van der Horst et al. (SODA, 2023) presented an O((n²/α) log³ n) time α-approximate algorithm for curves in arbitrary dimensions, for any α ∈ [1, n]. Our main contribution is an approximation algorithm for curves in one dimension, with a significantly faster running time of O(n log³ n + (n²/α³) log²n log log n). Additionally, we give an algorithm for curves in arbitrary dimensions that improves upon the state-of-the-art running time by a logarithmic factor, to O((n²/α) log² n). Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to O(n²/α) without making sacrifices in the asymptotic approximation factor.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.63/LIPIcs.SoCG.2024.63.pdf
Frécht distance
approximation algorithms
simplification
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
64:1
64:16
10.4230/LIPIcs.SoCG.2024.64
article
Moderate Dimension Reduction for k-Center Clustering
Jiang, Shaofeng H.-C.
1
https://orcid.org/0000-0001-7972-827X
Krauthgamer, Robert
2
https://orcid.org/0009-0003-8154-3735
Sapir, Shay
2
https://orcid.org/0000-0001-7531-685X
Peking University, Beijing, China
Weizmann Institute of Science, Rehovot, Israel
The Johnson-Lindenstrauss (JL) Lemma introduced the concept of dimension reduction via a random linear map, which has become a fundamental technique in many computational settings. For a set of n points in ℝ^d and any fixed ε > 0, it reduces the dimension d to O(log n) while preserving, with high probability, all the pairwise Euclidean distances within factor 1+ε. Perhaps surprisingly, the target dimension can be lower if one only wishes to preserve the optimal value of a certain problem on the pointset, e.g., Euclidean max-cut or k-means. However, for some notorious problems, like diameter (aka furthest pair), dimension reduction via the JL map to below O(log n) does not preserve the optimal value within factor 1+ε.
We propose to focus on another regime, of moderate dimension reduction, where a problem’s value is preserved within factor α > 1 using target dimension (log n)/poly(α). We establish the viability of this approach and show that the famous k-center problem is α-approximated when reducing to dimension O({log n}/α² + log k). Along the way, we address the diameter problem via the special case k = 1. Our result extends to several important variants of k-center (with outliers, capacities, or fairness constraints), and the bound improves further with the input’s doubling dimension.
While our poly(α)-factor improvement in the dimension may seem small, it actually has significant implications for streaming algorithms, and easily yields an algorithm for k-center in dynamic geometric streams, that achieves O(α)-approximation using space poly(kdn^{1/α²}). This is the first algorithm to beat O(n) space in high dimension d, as all previous algorithms require space at least exp(d). Furthermore, it extends to the k-center variants mentioned above.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.64/LIPIcs.SoCG.2024.64.pdf
Johnson-Lindenstrauss transform
dimension reduction
clustering
streaming algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
65:1
65:15
10.4230/LIPIcs.SoCG.2024.65
article
On the Parameterized Complexity of Motion Planning for Rectangular Robots
Kanj, Iyad
1
https://orcid.org/0000-0003-1698-8829
Parsa, Salman
1
https://orcid.org/0000-0002-8179-9322
School of Computing, DePaul University, Chicago, IL, USA
We study computationally-hard fundamental motion planning problems where the goal is to translate k axis-aligned rectangular robots from their initial positions to their final positions without collision, and with the minimum number of translation moves. Our aim is to understand the interplay between the number of robots and the geometric complexity of the input instance measured by the input size, which is the number of bits needed to encode the coordinates of the rectangles' vertices. We focus on axis-aligned translations, and more generally, translations restricted to a given set of directions, and we study the two settings where the robots move in the free plane, and where they are confined to a bounding box. We also consider two modes of motion: serial and parallel. We obtain fixed-parameter tractable (FPT) algorithms parameterized by k for all the settings under consideration.
In the case where the robots move serially (i.e., one in each time step) and axis-aligned, we prove a structural result stating that every problem instance admits an optimal solution in which the moves are along a grid, whose size is a function of k, that can be defined based on the input instance. This structural result implies that the problem is fixed-parameter tractable parameterized by k.
We also consider the case in which the robots move in parallel (i.e., multiple robots can move during the same time step), and which falls under the category of Coordinated Motion Planning problems. Our techniques for the axis-aligned motion here differ from those for the case of serial motion. We employ a search tree approach and perform a careful examination of the relative geometric positions of the robots that allow us to reduce the problem to FPT-many Linear Programming instances, thus obtaining an FPT algorithm.
Finally, we show that, when the robots move in the free plane, the FPT results for the serial motion case carry over to the case where the translations are restricted to any given set of directions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.65/LIPIcs.SoCG.2024.65.pdf
motion planning of rectangular robots
coordinated motion planing of rectangular robots
parameterized complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
66:1
66:15
10.4230/LIPIcs.SoCG.2024.66
article
Zarankiewicz’s Problem via ε-t-Nets
Keller, Chaya
1
https://orcid.org/0000-0001-6400-3946
Smorodinsky, Shakhar
2
https://orcid.org/0000-0003-3038-6955
Department of Computer Science, Ariel University, Israel
Department of Computer Science, Ben-Gurion University of the NEGEV, Beer Sheva, Israel
The classical Zarankiewicz’s problem asks for the maximum number of edges in a bipartite graph on n vertices which does not contain the complete bipartite graph K_{t,t}. Kővári, Sós and Turán proved an upper bound of O(n^{2-1/t}). Fox et al. obtained an improved bound of O(n^{2-1/d}) for graphs of VC-dimension d (where d < t). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. Chan and Har-Peled further improved Basit et al.’s bounds and presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of O(n log log n) for the incidence graph of points and pseudo-discs in the plane.
In this paper we present a new approach to Zarankiewicz’s problem, via ε-t-nets - a recently introduced generalization of the classical notion of ε-nets. Using the new approach, we obtain a sharp bound of O(n) for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain a short proof of the O(n^{2-1/d}) bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp O(n {log n}/{log log n}) bound for the intersection graph of two families of axis-parallel rectangles.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.66/LIPIcs.SoCG.2024.66.pdf
Zarankiewicz’s Problem
ε-t-nets
pseudo-discs
VC-dimension
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
67:1
67:17
10.4230/LIPIcs.SoCG.2024.67
article
Separator Theorem and Algorithms for Planar Hyperbolic Graphs
Kisfaludi-Bak, Sándor
1
Masaříková, Jana
2
van Leeuwen, Erik Jan
3
Walczak, Bartosz
4
Węgrzycki, Karol
5
6
Department of Computer Science, Aalto University, Espoo, Finland
Institute of Informatics, University of Warsaw, Poland
Department of Information and Computing Sciences, Utrecht University, The Netherlands
Department of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Saarland University, Saarbrücken, Germany
Max Planck Institute for Informatics, Saarbrücken, Germany
The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity.
Our main technical contribution is a novel balanced separator theorem for planar δ-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed δ ⩾ 0, we can find a small balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph.
An important advantage of our separator is that the union of our separator (vertex set Z) with any subset of the connected components of G - Z induces again a planar δ-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that the size of the separator is poly(δ) ⋅ log n.
As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar δ-hyperbolic graphs. We prove that both Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant δ, running in n polylog(n) ⋅ 2^𝒪(δ²) ⋅ ε^{-𝒪(δ)} time.
We also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no n^{o(δ)}-time algorithm on planar δ-hyperbolic graphs, unless ETH fails.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.67/LIPIcs.SoCG.2024.67.pdf
Hyperbolic metric
Planar Graphs
r-Division
Approximation Algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
68:1
68:15
10.4230/LIPIcs.SoCG.2024.68
article
A Quadtree, a Steiner Spanner, and Approximate Nearest Neighbours in Hyperbolic Space
Kisfaludi-Bak, Sándor
1
https://orcid.org/0000-0002-6856-2902
van Wordragen, Geert
1
https://orcid.org/0000-0002-2650-638X
Department of Computer Science, Aalto University, Finland
We propose a data structure in d-dimensional hyperbolic space that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces.
Using these quadtrees and the L-order we build geometric spanners. Near-linear size (1+ε)-spanners do not exist in hyperbolic spaces, but we create a Steiner spanner that achieves a spanning ratio of 1+ε with O_{d,ε}(n) edges, using a simple construction that can be maintained dynamically. As a corollary we also get a (2+ε)-spanner (in the classical sense) of the same size, where the spanning ratio 2+ε is almost optimal among spanners of subquadratic size.
Finally, we show that our Steiner spanner directly provides an elegant solution to the approximate nearest neighbour problem: given a point set P in d-dimensional hyperbolic space we build the data structure in O_{d,ε}(nlog n) time, using O_{d,ε}(n) space. Then for any query point q we can find a point p ∈ P that is at most 1+ε times farther from q than its nearest neighbour in P in O_{d,ε}(log n) time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time O_{d,ε}(log n). This is the first dynamic nearest neighbour data structure in hyperbolic space with proven efficiency guarantees.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.68/LIPIcs.SoCG.2024.68.pdf
hyperbolic geometry
Steiner spanner
dynamic approximate nearest neighbours
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
69:1
69:18
10.4230/LIPIcs.SoCG.2024.69
article
The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms
Dal Poz Kouřimská, Hana
1
https://orcid.org/0000-0001-7841-0091
Lieutier, André
2
Wintraecken, Mathijs
3
https://orcid.org/0000-0002-7472-2220
IST Austria, Klosterneuburg, Austria
Aix-en-Provence, France
Inria Sophia Antipolis, Université Côte d'Azur, Sophia Antipolis, France
We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.69/LIPIcs.SoCG.2024.69.pdf
Medial axis
Hausdorff distance
Lipschitz continuity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
70:1
70:17
10.4230/LIPIcs.SoCG.2024.70
article
Strange Random Topology of the Circle
Lim, Uzu
1
https://orcid.org/0000-0002-4563-4846
Mathematical Institute, University of Oxford, UK
A paradigm in topological data analysis asserts that persistent homology should be computed to recover the homology of a data manifold. But could there be more to persistent homology? In this paper I bound probabilities that a random m Čech complex built on a circle attains high-dimensional topology. This builds on the known result that any nerve complex of circular arcs has the homotopy type of a bouquet of spheres. We observe a phase transition going from one 1-sphere, bouquet of 2-spheres, one 3-sphere, bouquet of 4-spheres, and so on. Furthermore, the even-dimensional Betti numbers become arbitrarily large over shrinking intervals. Our main tool is an exact computation of the expected Euler characteristic, combined with constraints on homotopy types. The systematic behaviour we observe cannot be regarded as a "topological noise", and calls for deeper investigations from the TDA community.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.70/LIPIcs.SoCG.2024.70.pdf
Topological data analysis
persistent homology
stochastic topology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
71:1
71:15
10.4230/LIPIcs.SoCG.2024.71
article
Beyond Chromatic Threshold via (p,q)-Theorem, and Blow-Up Phenomenon
Liu, Hong
1
Shangguan, Chong
2
3
https://orcid.org/0000-0002-3206-3968
Skokan, Jozef
4
Xu, Zixiang
1
https://orcid.org/0000-0003-0155-6783
Extremal Combinatorics and Probability Group, Institute for Basic Science, Daejeon, South Korea
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, China
Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Qingdao, China
Department of Mathematics, London School of Economics, UK
We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated (p,q)-theorem in discrete geometry. In particular, for a graph G with bounded clique number and a natural density condition, we prove a (p,q)-theorem for an abstract convexity space associated with G. Our result strengthens those of Thomassen and Nikiforov on the chromatic threshold of cliques. Our (p,q)-theorem can also be viewed as a χ-boundedness result for (what we call) ultra maximal K_r-free graphs.
We further show that the graphs under study are blow-ups of constant size graphs, improving a result of Oberkampf and Schacht on homomorphism threshold of cliques. Our result unravels the cause underpinning such a blow-up phenomenon, differentiating the chromatic and homomorphism threshold problems for cliques. Our result implies that for the homomorphism threshold problem, rather than the minimum degree condition usually considered in the literature, the decisive factor is a clique density condition on co-neighborhoods of vertices. More precisely, we show that if an n-vertex K_r-free graph G satisfies that the common neighborhood of every pair of non-adjacent vertices induces a subgraph with K_{r-2}-density at least ε > 0, then G must be a blow-up of some K_r-free graph F on at most 2^O(r/ε log1/ε) vertices. Furthermore, this single exponential bound is optimal. We construct examples with no K_r-free homomorphic image of size smaller than 2^Ω_r(1/ε).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.71/LIPIcs.SoCG.2024.71.pdf
(p,q)-theorem
fractional Helly number
weak ε-net
chromatic threshold
VC dimension
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
72:1
72:11
10.4230/LIPIcs.SoCG.2024.72
article
A 1.9999-Approximation Algorithm for Vertex Cover on String Graphs
Lokshtanov, Daniel
1
https://orcid.org/0000-0002-3166-9212
Panolan, Fahad
2
https://orcid.org/0000-0001-6213-8687
Saurabh, Saket
3
https://orcid.org/0000-0001-7847-6402
Xue, Jie
4
https://orcid.org/0000-0001-7015-1988
Zehavi, Meirav
5
https://orcid.org/0000-0002-3636-5322
University of California, Santa Barbara, CA, USA
University of Leeds, UK
The Institute of Mathematical Sciences, HBNI, Chennai, India
New York University Shanghai, China
Ben-Gurion University of the Negev, Beer-Sheva, Israel
Vertex Cover is a fundamental optimization problem, and is among Karp’s 21 NP-complete problems. The problem aims to compute, for a given graph G, a minimum-size set S of vertices of G such that G - S contains no edge. Vertex Cover admits a simple polynomial-time 2-approximation algorithm, which is the best approximation ratio one can achieve in polynomial time, assuming the Unique Game Conjecture. However, on many restrictive graph classes, it is possible to obtain better-than-2 approximation in polynomial time (or even PTASes) for Vertex Cover. In the club of geometric intersection graphs, examples of such graph classes include unit-disk graphs, disk graphs, pseudo-disk graphs, rectangle graphs, etc.
In this paper, we study Vertex Cover on the broadest class of geometric intersection graphs in the plane, known as string graphs, which are intersection graphs of any connected geometric objects in the plane. Our main result is a polynomial-time 1.9999-approximation algorithm for Vertex Cover on string graphs, breaking the natural 2 barrier. Prior to this work, no better-than-2 approximation (in polynomial time) was known even for special cases of string graphs, such as intersection graphs of segments.
Our algorithm is simple, robust (in the sense that it does not require the geometric realization of the input string graph to be given), and also works for the weighted version of Vertex Cover. Due to a connection between approximation for Independent Set and approximation for Vertex Cover observed by Har-Peled, our result can be viewed as a first step towards obtaining constant-approximation algorithms for Independent Set on string graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.72/LIPIcs.SoCG.2024.72.pdf
vertex cover
geometric intersection graphs
approximation algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
73:1
73:16
10.4230/LIPIcs.SoCG.2024.73
article
Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data
Majhi, Sushovan
1
https://orcid.org/0000-0001-8689-4321
George Washington University, Washington D.C., USA
For a closed Riemannian manifold ℳ and a metric space S with a small Gromov-Hausdorff distance to it, Latschev’s theorem guarantees the existence of a sufficiently small scale β > 0 at which the Vietoris-Rips complex of S is homotopy equivalent to ℳ. Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from a noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale β in order to provide sampling conditions for S to be homotopy equivalent to ℳ. In this paper, we prove a stronger and pragmatic version of Latschev’s theorem, facilitating a simple description of β using the sectional curvatures and convexity radius of ℳ as the sampling parameters.
Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris-Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris-Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.73/LIPIcs.SoCG.2024.73.pdf
Vietoris-Rips complex
submanifold reconstruction
manifold reconstruction
Latschev’s theorem
homotopy Equivalence
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
74:1
74:14
10.4230/LIPIcs.SoCG.2024.74
article
Polychromatic Colorings of Geometric Hypergraphs via Shallow Hitting Sets
Planken, Tim
1
https://orcid.org/0009-0007-1659-9221
Ueckerdt, Torsten
2
https://orcid.org/0000-0002-0645-9715
University of Birmingham, UK
Karlsruhe Institute of Technology, Germany
A range family ℛ is a family of subsets of ℝ^d, like all halfplanes, or all unit disks. Given a range family ℛ, we consider the m-uniform range capturing hypergraphs ℋ(V,ℛ,m) whose vertex-sets V are finite sets of points in ℝ^d with any m vertices forming a hyperedge e whenever e = V ∩ R for some R ∈ ℛ. Given additionally an integer k ≥ 2, we seek to find the minimum m = m_ℛ(k) such that every ℋ(V,ℛ,m) admits a polychromatic k-coloring of its vertices, that is, where every hyperedge contains at least one point of each color. Clearly, m_ℛ(k) ≥ k and the gold standard is an upper bound m_ℛ(k) = O(k) that is linear in k.
A t-shallow hitting set in ℋ(V,ℛ,m) is a subset S ⊆ V such that 1 ≤ |e ∩ S| ≤ t for each hyperedge e; i.e., every hyperedge is hit at least once but at most t times by S. We show for several range families ℛ the existence of t-shallow hitting sets in every ℋ(V,ℛ,m) with t being a constant only depending on ℛ. This in particular proves that m_ℛ(k) ≤ tk = O(k) in such cases, improving previous polynomial bounds in k. Particularly, we prove this for the range families of all axis-aligned strips in ℝ^d, all bottomless and topless rectangles in ℝ², and for all unit-height axis-aligned rectangles in ℝ².
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.74/LIPIcs.SoCG.2024.74.pdf
geometric hypergraphs
range spaces
polychromatic coloring
shallow hitting sets
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
75:1
75:16
10.4230/LIPIcs.SoCG.2024.75
article
Morse Theory for the k-NN Distance Function
Reani, Yohai
1
https://orcid.org/0000-0002-0615-5789
Bobrowski, Omer
2
1
https://orcid.org/0000-0002-0860-7099
Viterbi Faculty of Electrical & Computer Engineering, Technion - Israel Institute of Technology, Haifa, Israel
School of Mathematical Sciences, Queen Mary University of London, UK
We study the k-th nearest neighbor distance function from a finite point-set in ℝ^d. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterization for critical points and their indices, along with detailed information about the possible changes in homology at the critical levels. We conclude by computing the expected number of critical points for a homogeneous Poisson process. Our results deliver significant insights and tools for the analysis of persistent homology in order-k Delaunay mosaics, and random k-fold coverage.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.75/LIPIcs.SoCG.2024.75.pdf
Applied topology
Morse theory
Distance function
k-nearest neighbor
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
76:1
76:18
10.4230/LIPIcs.SoCG.2024.76
article
Grid Peeling of Parabolas
Rote, Günter
1
https://orcid.org/0000-0002-0351-5945
Rüber, Moritz
2
Saghafian, Morteza
3
https://orcid.org/0000-0002-4201-5775
Institut für Informatik, Freie Universität Berlin, Germany
Freie Universität Berlin, Germany
Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria
Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.76/LIPIcs.SoCG.2024.76.pdf
grid polygons
curvature flow
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
77:1
77:16
10.4230/LIPIcs.SoCG.2024.77
article
A Topological Version of Schaefer’s Dichotomy Theorem
Schnider, Patrick
1
https://orcid.org/0000-0002-2172-9285
Weber, Simon
1
https://orcid.org/0000-0003-1901-3621
Department of Computer Science, ETH Zürich, Switzerland
Schaefer’s dichotomy theorem states that a Boolean constraint satisfaction problem (CSP) is polynomial-time solvable if one of four given conditions holds for every type of constraint allowed in its instances. Otherwise, it is NP-complete. In this paper, we analyze Boolean CSPs in terms of their topological complexity, instead of their computational complexity. Motivated by complexity and topological universality results in computational geometry, we attach a natural topological space to the set of solutions of a Boolean CSP and introduce the notion of projection-universality. We prove that a Boolean CSP is projection-universal if and only if it is categorized as NP-complete by Schaefer’s dichotomy theorem, showing that the dichotomy translates exactly from computational to topological complexity. We show a similar dichotomy for SAT variants and homotopy-universality.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.77/LIPIcs.SoCG.2024.77.pdf
Computational topology
Boolean CSP
satisfiability
computational complexity
solution space
homotopy universality
homological connectivity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
78:1
78:18
10.4230/LIPIcs.SoCG.2024.78
article
Pach’s Animal Problem Within the Bounding Box
Tancer, Martin
1
Department of Applied Mathematics, Charles University, Prague, Czech Republic
A collection of unit cubes with integer coordinates in ℝ³ is an animal if its union is homeomorphic to the 3-ball. Pach’s animal problem asks whether any animal can be transformed to a single cube by adding or removing cubes one by one in such a way that any intermediate step is an animal as well. Here we provide an example of an animal that cannot be transformed to a single cube this way within its bounding box.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.78/LIPIcs.SoCG.2024.78.pdf
Animal problem
bounding box
non-shellable balls
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
79:1
79:15
10.4230/LIPIcs.SoCG.2024.79
article
Algorithms for Halfplane Coverage and Related Problems
Wang, Haitao
1
https://orcid.org/0000-0001-8134-7409
Xue, Jie
2
https://orcid.org/0000-0001-7015-1988
Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Department of Computer Science, New York University Shanghai, China
Given in the plane a set of points and a set of halfplanes, we consider the problem of computing a smallest subset of halfplanes whose union covers all points. In this paper, we present an O(n^{4/3}log^{5/3}nlog^{O(1)}log n)-time algorithm for the problem, where n is the total number of all points and halfplanes. This improves the previously best algorithm of n^{10/3}2^{O(log^*n)} time by roughly a quadratic factor. For the special case where all halfplanes are lower ones, our algorithm runs in O(nlog n) time, which improves the previously best algorithm of n^{4/3}2^{O(log^*n)} time and matches an Ω(nlog n) lower bound. Further, our techniques can be extended to solve a star-shaped polygon coverage problem in O(nlog n) time, which in turn leads to an O(nlog n)-time algorithm for computing an instance-optimal ε-kernel of a set of n points in the plane. Agarwal and Har-Peled presented an O(nklog n)-time algorithm for this problem in SoCG 2023, where k is the size of the ε-kernel; they also raised an open question whether the problem can be solved in O(nlog n) time. Our result thus answers the open question affirmatively.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.79/LIPIcs.SoCG.2024.79.pdf
halfplane coverage
circular coverage
star-shaped polygon coverage
ε-kernels
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
80:1
80:18
10.4230/LIPIcs.SoCG.2024.80
article
Measure-Theoretic Reeb Graphs and Reeb Spaces
Wang, Qingsong
1
https://orcid.org/0000-0002-5106-2637
Ma, Guanqun
1
https://orcid.org/0000-0001-8102-3172
Sridharamurthy, Raghavendra
1
https://orcid.org/0000-0001-8463-0488
Wang, Bei
1
https://orcid.org/0000-0002-9240-0700
University of Utah, Salt Lake City, UT, USA
A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. A Reeb space is a generalization of the Reeb graph to a multiparameter function. In this paper, we propose novel constructions of Reeb graphs and Reeb spaces that incorporate the use of a measure. Specifically, we introduce measure-theoretic Reeb graphs and Reeb spaces when the domain or the range is modeled as a metric measure space (i.e., a metric space equipped with a measure). Our main goal is to enhance the robustness of the Reeb graph and Reeb space in representing the topological features of a scalar field while accounting for the distribution of the measure. We first introduce a Reeb graph with local smoothing and prove its stability with respect to the interleaving distance. We then prove the stability of a Reeb graph of a metric measure space with respect to the measure, defined using the distance to a measure or the kernel distance to a measure, respectively.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.80/LIPIcs.SoCG.2024.80.pdf
Reeb graph
Reeb space
metric measure space
topological data analysis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
81:1
81:12
10.4230/LIPIcs.SoCG.2024.81
article
Faster Approximation Scheme for Euclidean k-TSP
van Wijland, Ernest
1
Zhou, Hang
2
École Normale Supérieure Paris, France
École Polytechnique, Institut Polytechnique de Paris, France
In the Euclidean k-traveling salesman problem (k-TSP), we are given n points in the d-dimensional Euclidean space, for some fixed constant d ≥ 2, and a positive integer k. The goal is to find a shortest tour visiting at least k points.
We give an approximation scheme for the Euclidean k-TSP in time n⋅2^O(1/ε^{d-1})⋅(log n)^{2d²⋅2^d}. This improves Arora’s approximation scheme of running time n⋅k⋅(log n)^(O(√d/ε))^{d-1}} [J. ACM 1998]. Our algorithm is Gap-ETH tight and can be derandomized by increasing the running time by a factor O(n^d).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.81/LIPIcs.SoCG.2024.81.pdf
approximation algorithms
optimization
traveling salesman problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
82:1
82:16
10.4230/LIPIcs.SoCG.2024.82
article
Space Complexity of Euclidean Clustering
Zhu, Xiaoyi
1
https://orcid.org/0009-0002-5587-4752
Tian, Yuxiang
1
https://orcid.org/0009-0007-1933-1538
Huang, Lingxiao
2
https://orcid.org/0000-0001-7512-142X
Huang, Zengfeng
1
https://orcid.org/0000-0003-2671-7483
School of Data Science, Fudan University, Shanghai, China
State Key Laboratory of Novel Software Technology, Nanjing University, China
The (k, z)-Clustering problem in Euclidean space ℝ^d has been extensively studied. Given the scale of data involved, compression methods for the Euclidean (k, z)-Clustering problem, such as data compression and dimension reduction, have received significant attention in the literature. However, the space complexity of the clustering problem, specifically, the number of bits required to compress the cost function within a multiplicative error ε, remains unclear in existing literature.
This paper initiates the study of space complexity for Euclidean (k, z)-Clustering and offers both upper and lower bounds. Our space bounds are nearly tight when k is constant, indicating that storing a coreset, a well-known data compression approach, serves as the optimal compression scheme. Furthermore, our lower bound result for (k, z)-Clustering establishes a tight space bound of Θ(n d) for terminal embedding, where n represents the dataset size. Our technical approach leverages new geometric insights for principal angles and discrepancy methods, which may hold independent interest.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.82/LIPIcs.SoCG.2024.82.pdf
Space complexity
Euclidean clustering
coreset
terminal embedding
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
83:1
83:9
10.4230/LIPIcs.SoCG.2024.83
article
Computing Maximum Polygonal Packings in Convex Polygons Using Best-Fit, Genetic Algorithms and ILPs (CG Challenge)
Atak, Alkan
1
Buchin, Kevin
1
https://orcid.org/0000-0002-3022-7877
Hagedoorn, Mart
1
https://orcid.org/0000-0002-8591-3380
Heinrichs, Jona
1
Hogreve, Karsten
1
Li, Guangping
1
https://orcid.org/0000-0002-7966-076X
Pawelczyk, Patrick
1
TU Dortmund, Germany
Given a convex region P and a set of irregular polygons with associated profits, the Maximum Polygon Packing Problem seeks a non-overlapping packing of a subset of the polygons (without rotations) into P maximizing the profit of the packed polygons. Depending on the size of an instance, we use different algorithmic solutions: integer linear programs for small instances, genetic algorithms for medium-sized instances and a best-fit approach for large instances. For packing rectilinear polygons we provide a dedicated best-fit algorithm.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.83/LIPIcs.SoCG.2024.83.pdf
Polygon Packing
Nesting Problem
Genetic Algorithm
Integer Linear Programming
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
84:1
84:9
10.4230/LIPIcs.SoCG.2024.84
article
Shadoks Approach to Knapsack Polygonal Packing (CG Challenge)
da Fonseca, Guilherme D.
1
https://orcid.org/0000-0002-9807-028X
Gerard, Yan
2
https://orcid.org/0000-0002-2664-0650
LIS, Aix-Marseille Université, France
LIMOS, University Clermont Auvergne, Aubière, France
We describe the heuristics used by the Shadoks team in the CG:SHOP 2024 Challenge. Each instance consists of a convex polygon called container and a multiset of items, where each item is a simple polygon and has an associated value. The goal is to pack some of the items inside the container using translations, in order to maximize the sum of their values. Our strategy consists of obtaining good initial solutions and improving them with local search. To obtain the initial solutions we used integer programming and a carefully designed greedy approach.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.84/LIPIcs.SoCG.2024.84.pdf
Packing
polygons
heuristics
integer programming
computational geometry
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
85:1
85:6
10.4230/LIPIcs.SoCG.2024.85
article
Priority-Driven Nesting of Irregular Polygonal Shapes Within a Convex Polygonal Container Based on a Hierarchical Integer Grid (CG Challenge)
Held, Martin
1
https://orcid.org/0000-0003-0728-7545
FB Informatik, Universität Salzburg, Austria
Our work on nesting polygons is based on two key components: (1) a hierarchy of uniform integer grids for maintaining free space within the container during the nesting such that placement queries can be answered reasonably efficiently, and (2) priority heuristics for choosing the order in which the polygons are tested for placement. We discuss our approach and shed a light on the results obtained.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.85/LIPIcs.SoCG.2024.85.pdf
Computational Geometry
geometric optimization
nesting
packing
algorithm engineering
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
86:1
86:9
10.4230/LIPIcs.SoCG.2024.86
article
A General Heuristic Approach for Maximum Polygon Packing (CG Challenge)
Luo, Canhui
1
https://orcid.org/0000-0003-4753-3843
Su, Zhouxing
1
https://orcid.org/0000-0002-4794-9833
Lü, Zhipeng
1
https://orcid.org/0000-0001-9185-3233
Huazhong University of Science and Technology, Wuhan, China
This work proposes a general heuristic packing approach to address the Maximum Polygon Packing Problem introduced by the CG:SHOP 2024 Challenge. Our solver primarily consists of two steps: (1) Partitioning the container and polygons to form a series of small-scale subproblems; (2) For each subproblem, sequentially placing polygons into the container and attempting to eliminate overlaps.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.86/LIPIcs.SoCG.2024.86.pdf
packing
polygon
heuristic
differential evolution
local search
tabu search
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
87:1
87:6
10.4230/LIPIcs.SoCG.2024.87
article
The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition)
Attali, Dominique
1
https://orcid.org/0000-0003-4808-6301
Dal Poz Kouřimská, Hana
2
https://orcid.org/0000-0001-7841-0091
Fillmore, Christopher
2
https://orcid.org/0000-0001-7631-2885
Ghosh, Ishika
2
3
https://orcid.org/0000-0002-7901-5912
Lieutier, André
4
Stephenson, Elizabeth
2
https://orcid.org/0000-0002-6862-208X
Wintraecken, Mathijs
5
https://orcid.org/0000-0002-7472-2220
Université Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France
IST Austria, Klosterneuburg, Austria
Michigan State University, East Lansing, MI, USA
Aix-en-Provence, France
Inria Sophia Antipolis, Université Côte d'Azur, Sophia Antipolis, France
In our companion paper "Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.87/LIPIcs.SoCG.2024.87.pdf
Homotopy
Inference
Sets of positive reach
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
88:1
88:4
10.4230/LIPIcs.SoCG.2024.88
article
Computational Geometry Concept Videos: A Dual-Use Project in Education and Outreach (Media Exposition)
Haagsman, Marjolein
1
https://orcid.org/0000-0003-4000-1959
Löffler, Maarten
1
https://orcid.org/0009-0001-9403-8856
Wenk, Carola
2
https://orcid.org/0000-0001-9275-5336
Utrecht University, The Netherlands
Tulane University, New Orleans, LA, USA
We present a series of nine Computational Geometry Concept Videos, available on Youtube. The videos are aimed at a general audience and introduce concepts ranging from closest and farthest pairs to data structures for range searching and for point location. The video series grew out of the development of an online graduate course on computational geometry, and the beginning portions of the videos are used in the course to motivate the concept and to tie it to a "real" problem in New Orleans. Thus our videos serve a dual purpose of outreach and education.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.88/LIPIcs.SoCG.2024.88.pdf
Computational geometry concepts
videos
online education
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
89:1
89:4
10.4230/LIPIcs.SoCG.2024.89
article
Optimal In-Place Compaction of Sliding Cubes (Media Exposition)
Kostitsyna, Irina
1
https://orcid.org/0000-0003-0544-2257
Ophelders, Tim
2
1
https://orcid.org/0000-0002-9570-024X
Parada, Irene
3
https://orcid.org/0000-0003-2401-8670
Peters, Tom
1
https://orcid.org/0000-0002-2702-7532
Sonke, Willem
1
https://orcid.org/0000-0001-9553-7385
Speckmann, Bettina
1
https://orcid.org/0000-0002-8514-7858
TU Eindhoven, The Netherlands
Utrecht University, The Netherlands
Universitat Politècnica de Catalunya, Barcelona, Spain
The sliding cubes model is a well-established theoretical framework that supports the analysis of reconfiguration algorithms for modular robots consisting of face-connected cubes. This note accompanies a video that explains our in-place algorithm for reconfiguration in the sliding cubes model. Specifically, our algorithm [Irina Kostitsyna et al., 2023] reconfigures any n-cube configuration into a compact canonical shape using a number of moves proportional to the sum of coordinates of the input cubes. As is common in the literature, we can then reconfigure between two arbitrary shapes via their canonical configurations. The number of moves performed by our algorithm is asymptotically worst-case optimal and strictly improves upon the current state-of-the-art.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.89/LIPIcs.SoCG.2024.89.pdf
Sliding cubes
Reconfiguration algorithm
Modular robots
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
90:1
90:6
10.4230/LIPIcs.SoCG.2024.90
article
Visualizing Lucas’s Hamiltonian Paths Through the Associahedron 1-Skeleton (Media Exposition)
La, Kacey Thien-Huu
1
Arbelo, Jose E.
1
Tralie, Christopher J.
1
https://orcid.org/0000-0003-4206-1963
Ursinus College Mathematics And Computer Science, Collegeville, PA, USA
We re-examine the 1987 paper by Joan Lucas[Lucas, 1987], who showed that the edge-flip graph of convex polygon triangulations is Hamiltonian. We focus specifically on the first part of her paper on Hamiltonian paths, and we provide a simplified algorithm for that case which elucidates how to assemble a recursive subdivision that she refers to as "stacks." Finally, we provide an interactive web-based visualization of Hamiltonian paths through the stacks.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.90/LIPIcs.SoCG.2024.90.pdf
associahedron
hamiltonian paths
visualization
tree rotations
convex polygons
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
91:1
91:7
10.4230/LIPIcs.SoCG.2024.91
article
Image Triangulation Using the Sobel Operator for Vertex Selection (Media Exposition)
Laske, Olivia X.
1
Ziegelmeier, Lori
1
Department of Mathematics, Statistics, and Computer Science, Macalester College, St. Paul, MN, USA
Image triangulation, the practice of decomposing images into triangles, deliberately employs simplification to create an abstracted representation. While triangulating an image is a relatively simple process, difficulties arise when determining which vertices produce recognizable and visually pleasing output images. With the goal of producing art, we discuss an image triangulation algorithm in Python that utilizes Sobel edge detection and point cloud sparsification to determine final vertices for a triangulation, resulting in the creation of artistic triangulated compositions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.91/LIPIcs.SoCG.2024.91.pdf
Image Triangulation
Sharpening
Sobel Edge Detection
Delaunay Triangulation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
293
92:1
92:7
10.4230/LIPIcs.SoCG.2024.92
article
Ipelets for the Convex Polygonal Geometry (Media Exposition)
Parepally, Nithin
1
Chatterjee, Ainesh
1
Gezalyan, Auguste H.
1
https://orcid.org/0000-0002-5704-312X
Du, Hongyang
1
Mangla, Sukrit
1
Wu, Kenny
1
Hwang, Sarah
1
Mount, David M.
1
https://orcid.org/0000-0002-3290-8932
Department of Computer Science, University of Maryland, College Park, MD, USA
There are many structures, both classical and modern, involving convex polygonal geometries whose deeper understanding would be facilitated through interactive visualizations. The Ipe extensible drawing editor, developed by Otfried Cheong, is a widely used software system for generating geometric figures. One of its features is the capability to extend its functionality through programs called Ipelets. In this media submission, we showcase a collection of new Ipelets that construct a variety of geometric objects based on polygonal geometries. These include Macbeath regions, metric balls in the forward and reverse Funk distance, metric balls in the Hilbert metric, polar bodies, the minimum enclosing ball of a point set, and minimum spanning trees in both the Funk and Hilbert metrics. We also include a number of utilities on convex polygons, including union, intersection, subtraction, and Minkowski sum (previously implemented as a CGAL Ipelet).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.92/LIPIcs.SoCG.2024.92.pdf
Hilbert metric
Macbeath Regions
Polar Bodies
Convexity