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Documents authored by Toth, Csaba D.

Found 3 Possible Name Variants:

Tóth, Csaba D.

Document
DOI: 10.4230/LIPIcs.ISAAC.2024.4
Minimum Plane Bichromatic Spanning Trees

Authors: Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, and Csaba D. Tóth

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in O(n log n) time where n is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of O(√n). It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an O(log n)-factor approximation algorithm for the general case.
Cite as

Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, and Csaba D. Tóth. Minimum Plane Bichromatic Spanning Trees. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{a.akitaya_et_al:LIPIcs.ISAAC.2024.4,
  author =	{A. Akitaya, Hugo and Biniaz, Ahmad and Demaine, Erik D. and Kleist, Linda and Stock, Frederick and T\'{o}th, Csaba D.},
  title =	{{Minimum Plane Bichromatic Spanning Trees}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.4},
  URN =		{urn:nbn:de:0030-drops-221319},
  doi =		{10.4230/LIPIcs.ISAAC.2024.4},
  annote =	{Keywords: Bichromatic Spanning Tree, Minimum Spanning Tree, Plane Tree}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.36
Noncrossing Longest Paths and Cycles

Authors: Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, David Eppstein, Anil Maheshwari, Saeed Odak, Michiel Smid, Csaba D. Tóth, and Pavel Valtr

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.
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Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, David Eppstein, Anil Maheshwari, Saeed Odak, Michiel Smid, Csaba D. Tóth, and Pavel Valtr. Noncrossing Longest Paths and Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aloupis_et_al:LIPIcs.GD.2024.36,
  author =	{Aloupis, Greg and Biniaz, Ahmad and Bose, Prosenjit and De Carufel, Jean-Lou and Eppstein, David and Maheshwari, Anil and Odak, Saeed and Smid, Michiel and T\'{o}th, Csaba D. and Valtr, Pavel},
  title =	{{Noncrossing Longest Paths and Cycles}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{36:1--36:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.36},
  URN =		{urn:nbn:de:0030-drops-213203},
  doi =		{10.4230/LIPIcs.GD.2024.36},
  annote =	{Keywords: Longest Paths, Longest Cycles, Noncrossing Paths, Noncrossing Cycles}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.19
Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

Authors: Sujoy Bhore, Martin Nöllenburg, Csaba D. Tóth, and Jules Wulms

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
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.
Cite as

Sujoy Bhore, Martin Nöllenburg, Csaba D. Tóth, and Jules Wulms. Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2024.19,
  author =	{Bhore, Sujoy and N\"{o}llenburg, Martin and T\'{o}th, Csaba D. and Wulms, Jules},
  title =	{{Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.19},
  URN =		{urn:nbn:de:0030-drops-199649},
  doi =		{10.4230/LIPIcs.SoCG.2024.19},
  annote =	{Keywords: Dynamic algorithm, Independent set, Geometric intersection graph}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.6
Reconfiguration of Polygonal Subdivisions via Recombination

Authors: Hugo A. Akitaya, Andrei Gonczi, Diane L. Souvaine, Csaba D. Tóth, and Thomas Weighill

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
Motivated by the problem of redistricting, we study area-preserving reconfigurations of connected subdivisions of a simple polygon. A connected subdivision of a polygon ℛ, called a district map, is a set of interior disjoint connected polygons called districts whose union equals ℛ. We consider the recombination as the reconfiguration move which takes a subdivision and produces another by merging two adjacent districts, and by splitting them into two connected polygons of the same area as the original districts. The complexity of a map is the number of vertices in the boundaries of its districts. Given two maps with k districts, with complexity O(n), and a perfect matching between districts of the same area in the two maps, we show constructively that (log n)^O(log k) recombination moves are sufficient to reconfigure one into the other. We also show that Ω(log n) recombination moves are sometimes necessary even when k = 3, thus providing a tight bound when k = 3.
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Hugo A. Akitaya, Andrei Gonczi, Diane L. Souvaine, Csaba D. Tóth, and Thomas Weighill. Reconfiguration of Polygonal Subdivisions via Recombination. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{a.akitaya_et_al:LIPIcs.ESA.2023.6,
  author =	{A. Akitaya, Hugo and Gonczi, Andrei and Souvaine, Diane L. and T\'{o}th, Csaba D. and Weighill, Thomas},
  title =	{{Reconfiguration of Polygonal Subdivisions via Recombination}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.6},
  URN =		{urn:nbn:de:0030-drops-186598},
  doi =		{10.4230/LIPIcs.ESA.2023.6},
  annote =	{Keywords: configuration space, gerrymandering, polygonal subdivision, recombination}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.18
Online Spanners in Metric Spaces

Authors: Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, and Csaba D. Tóth

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness.
Cite as

Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, and Csaba D. Tóth. Online Spanners in Metric Spaces. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhore_et_al:LIPIcs.ESA.2022.18,
  author =	{Bhore, Sujoy and Filtser, Arnold and Khodabandeh, Hadi and T\'{o}th, Csaba D.},
  title =	{{Online Spanners in Metric Spaces}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{18:1--18:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.18},
  URN =		{urn:nbn:de:0030-drops-169564},
  doi =		{10.4230/LIPIcs.ESA.2022.18},
  annote =	{Keywords: spanner, online algorithm, lightness, sparsity, minimum weight}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.30
Hop-Spanners for Geometric Intersection Graphs

Authors: Jonathan B. Conroy and Csaba D. Tóth

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
A t-spanner of a graph G = (V,E) is a subgraph H = (V,E') that contains a uv-path of length at most t for every uv ∈ E. It is known that every n-vertex graph admits a (2k-1)-spanner with O(n^{1+1/k}) edges for k ≥ 1. This bound is the best possible for 1 ≤ k ≤ 9 and is conjectured to be optimal due to Erdős' girth conjecture. We study t-spanners for t ∈ {2,3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O(nlog n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(nlog n) edges, and this bound is the best possible. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(nlog n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(nlog² n) edges.
Cite as

Jonathan B. Conroy and Csaba D. Tóth. Hop-Spanners for Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{conroy_et_al:LIPIcs.SoCG.2022.30,
  author =	{Conroy, Jonathan B. and T\'{o}th, Csaba D.},
  title =	{{Hop-Spanners for Geometric Intersection Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{30:1--30:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.30},
  URN =		{urn:nbn:de:0030-drops-160381},
  doi =		{10.4230/LIPIcs.SoCG.2022.30},
  annote =	{Keywords: geometric intersection graph, unit disk graph, hop-spanner}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.16
Online Euclidean Spanners

Authors: Sujoy Bhore and Csaba D. Tóth

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
In this paper, we study the online Euclidean spanners problem for points in ℝ^d. Given a set S of n points in ℝ^d, a t-spanner on S is a subgraph of the underlying complete graph G = (S,binom(S,2)), that preserves the pairwise Euclidean distances between points in S to within a factor of t, that is the stretch factor. Suppose we are given a sequence of n points (s₁,s₂,…, s_n) in ℝ^d, where point s_i is presented in step i for i = 1,…, n. The objective of an online algorithm is to maintain a geometric t-spanner on S_i = {s₁,…, s_i} for each step i. The algorithm is allowed to add new edges to the spanner when a new point is presented, but cannot remove any edge from the spanner. The performance of an online algorithm is measured by its competitive ratio, which is the supremum, over all sequences of points, of the ratio between the weight of the spanner constructed by the algorithm and the weight of an optimum spanner. Here the weight of a spanner is the sum of all edge weights. First, we establish a lower bound of Ω(ε^{-1}log n / log ε^{-1}) for the competitive ratio of any online (1+ε)-spanner algorithm, for a sequence of n points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a (1+ε)-spanner with competitive ratio O(ε^{-1}log n / log ε^{-1}). Next, we design online algorithms for sequences of points in ℝ^d, for any constant d ≥ 2, under the L₂ norm. We show that previously known incremental algorithms achieve a competitive ratio O(ε^{-(d+1)}log n). However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of ε. We describe an online Steiner (1+ε)-spanner algorithm with competitive ratio O(ε^{(1-d)/2} log n). As a counterpart, we show that the dependence on n cannot be eliminated in dimensions d ≥ 2. In particular, we prove that any online spanner algorithm for a sequence of n points in ℝ^d under the L₂ norm has competitive ratio Ω(f(n)), where lim_{n → ∞}f(n) = ∞. Finally, we provide improved lower bounds under the L₁ norm: Ω(ε^{-2}/log ε^{-1}) in the plane and Ω(ε^{-d}) in ℝ^d for d ≥ 3.
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Sujoy Bhore and Csaba D. Tóth. Online Euclidean Spanners. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhore_et_al:LIPIcs.ESA.2021.16,
  author =	{Bhore, Sujoy and T\'{o}th, Csaba D.},
  title =	{{Online Euclidean Spanners}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{16:1--16:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.16},
  URN =		{urn:nbn:de:0030-drops-145974},
  doi =		{10.4230/LIPIcs.ESA.2021.16},
  annote =	{Keywords: Geometric spanner, (1+\epsilon)-spanner, minimum weight, online algorithm}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.15
Light Euclidean Steiner Spanners in the Plane

Authors: Sujoy Bhore and Csaba D. Tóth

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in ℝ^d. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on ε > 0 and d ∈ ℕ of the minimum lightness of a (1+ε)-spanner, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1+ε)-spanners of lightness O(ε^{-1}logΔ) in the plane, where Δ ≥ Ω(√n) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness Õ(ε^{-(d+1)/2}) in dimensions d ≥ 3. Recently, Bhore and Tóth (2020) established a lower bound of Ω(ε^{-d/2}) for the lightness of Steiner (1+ε)-spanners in ℝ^d, for d ≥ 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d ≥ 2. In this work, we show that for every finite set of points in the plane and every ε > 0, there exists a Euclidean Steiner (1+ε)-spanner of lightness O(ε^{-1}); this matches the lower bound for d = 2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.
Cite as

Sujoy Bhore and Csaba D. Tóth. Light Euclidean Steiner Spanners in the Plane. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2021.15,
  author =	{Bhore, Sujoy and T\'{o}th, Csaba D.},
  title =	{{Light Euclidean Steiner Spanners in the Plane}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.15},
  URN =		{urn:nbn:de:0030-drops-138145},
  doi =		{10.4230/LIPIcs.SoCG.2021.15},
  annote =	{Keywords: Geometric spanner, lightness, minimum weight}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.13
On Euclidean Steiner (1+ε)-Spanners

Authors: Sujoy Bhore and Csaba D. Tóth

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
Lightness and sparsity are two natural parameters for Euclidean (1+ε)-spanners. Classical results show that, when the dimension d ∈ ℕ and ε > 0 are constant, every set S of n points in d-space admits an (1+ε)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ε > 0 for constant d ∈ ℕ have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+ε)-spanner. They gave upper bounds of Õ(ε^{-(d+1)/2}) for the minimum lightness in dimensions d ≥ 3, and Õ(ε^{-(d-1))/2}) for the minimum sparsity in d-space for all d ≥ 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+ε)-spanners of lightness O(ε^{-1}logΔ) in the plane, where Δ ∈ Ω(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+ε)-spanners. Using a new geometric analysis, we establish lower bounds of Ω(ε^{-d/2}) for the lightness and Ω(ε^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ≥ 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+ε)-spanners of lightness O(ε^{-1}log n) for n points in Euclidean plane.
Cite as

Sujoy Bhore and Csaba D. Tóth. On Euclidean Steiner (1+ε)-Spanners. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhore_et_al:LIPIcs.STACS.2021.13,
  author =	{Bhore, Sujoy and T\'{o}th, Csaba D.},
  title =	{{On Euclidean Steiner (1+\epsilon)-Spanners}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.13},
  URN =		{urn:nbn:de:0030-drops-136586},
  doi =		{10.4230/LIPIcs.STACS.2021.13},
  annote =	{Keywords: Geometric spanner, (1+\epsilon)-spanner, lightness, sparsity, minimum weight}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.57
Sparse Hop Spanners for Unit Disk Graphs

Authors: Adrian Dumitrescu, Anirban Ghosh, and Csaba D. Tóth

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
A unit disk graph G on a given set of points P in the plane is a geometric graph where an edge exists between two points p,q ∈ P if and only if |pq| ≤ 1. A subgraph G' of G is a k-hop spanner if and only if for every edge pq ∈ G, the topological shortest path between p,q in G' has at most k edges. We obtain the following results for unit disk graphs. 1) Every n-vertex unit disk graph has a 5-hop spanner with at most 5.5n edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from 9n to 5.5n. 2) Using a new construction, we show that every n-vertex unit disk graph has a 3-hop spanner with at most 11n edges. 3) Every n-vertex unit disk graph has a 2-hop spanner with O(nlog n) edges. This is the first nontrivial construction of 2-hop spanners. 4) For every sufficiently large n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known. 5) For every point set on a circle, there exists a plane 4-hop spanner. As such, this provides a tight bound for points on a circle. 6) The maximum degree of k-hop spanners cannot be bounded from above by a function of k.
Cite as

Adrian Dumitrescu, Anirban Ghosh, and Csaba D. Tóth. Sparse Hop Spanners for Unit Disk Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dumitrescu_et_al:LIPIcs.ISAAC.2020.57,
  author =	{Dumitrescu, Adrian and Ghosh, Anirban and T\'{o}th, Csaba D.},
  title =	{{Sparse Hop Spanners for Unit Disk Graphs}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{57:1--57:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.57},
  URN =		{urn:nbn:de:0030-drops-134018},
  doi =		{10.4230/LIPIcs.ISAAC.2020.57},
  annote =	{Keywords: graph approximation, \epsilon-net, hop-spanner, unit disk graph, lower bound}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.7
Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

Authors: Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.
Cite as

Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth. Cutting Polygons into Small Pieces with Chords: Laser-Based Localization. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arkin_et_al:LIPIcs.ESA.2020.7,
  author =	{Arkin, Esther M. and Das, Rathish and Gao, Jie and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin and T\'{o}th, Csaba D.},
  title =	{{Cutting Polygons into Small Pieces with Chords: Laser-Based Localization}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.7},
  URN =		{urn:nbn:de:0030-drops-128736},
  doi =		{10.4230/LIPIcs.ESA.2020.7},
  annote =	{Keywords: Polygon partition, Arrangements, Visibility, Localization}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.56
On the Stretch Factor of Polygonal Chains

Authors: Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i<j<k <= n. The stretch factor is a global property: it measures how close P is to a straight line, and it involves all the vertices of P; being a c-chain, on the other hand, is a fingerprint-property: it only depends on subsets of O(1) vertices of the chain. We investigate how the c-chain property influences the stretch factor in the plane: (i) we show that for every epsilon > 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space.
Cite as

Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth. On the Stretch Factor of Polygonal Chains. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chen_et_al:LIPIcs.MFCS.2019.56,
  author =	{Chen, Ke and Dumitrescu, Adrian and Mulzer, Wolfgang and T\'{o}th, Csaba D.},
  title =	{{On the Stretch Factor of Polygonal Chains}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{56:1--56:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.56},
  URN =		{urn:nbn:de:0030-drops-110005},
  doi =		{10.4230/LIPIcs.MFCS.2019.56},
  annote =	{Keywords: polygonal chain, vertex dilation, Koch curve, recursive construction}
}
Document
DOI: 10.4230/DagRep.9.2.123
Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092)

Authors: Seok-Hee Hong, Michael Kaufmann, János Pach, and Csaba D. Tóth

Published in: Dagstuhl Reports, Volume 9, Issue 2 (2019)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 19092 "Beyond-Planar Graphs: Combinatorics, Models and Algorithms" which brought together 36 researchers in the areas of graph theory, combinatorics, computational geometry, and graph drawing. This seminar continued the work initiated in Dagstuhl Seminar 16452 "Beyond-Planar Graphs: Algorithmics and Combinatorics" and focused on the exploration of structural properties and the development of algorithms for so-called beyond-planar graphs, i.e., non-planar graphs that admit a drawing with topological constraints such as specific types of crossings, or with some forbidden crossing patterns. The seminar began with four talks about the results of scientific collaborations originating from the previous Dagstuhl seminar. Next we discussed open research problems about beyond planar graphs, such as their combinatorial structures (e.g., book thickness, queue number), their topology (e.g., simultaneous embeddability, gap planarity, quasi-quasiplanarity), their geometric representations (e.g., representations on few segments or arcs), and applications (e.g., manipulation of graph drawings by untangling operations). Six working groups were formed that investigated several of the open research questions. In addition, talks on related subjects and recent conference contributions were presented in the morning opening sessions. Abstracts of all talks and a report from each working group are included in this report.
Cite as

Seok-Hee Hong, Michael Kaufmann, János Pach, and Csaba D. Tóth. Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092). In Dagstuhl Reports, Volume 9, Issue 2, pp. 123-156, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{hong_et_al:DagRep.9.2.123,
  author =	{Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and T\'{o}th, Csaba D.},
  title =	{{Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092)}},
  pages =	{123--156},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2019},
  volume =	{9},
  number =	{2},
  editor =	{Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.2.123},
  URN =		{urn:nbn:de:0030-drops-108634},
  doi =		{10.4230/DagRep.9.2.123},
  annote =	{Keywords: combinatorial geometry, geometric algorithms, graph algorithms, graph drawing, graph theory, network visualization}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.9
Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Authors: Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, and Csaba D. Tóth

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
Given a planar straight-line graph G=(V,E) in R^2, a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P. We prove that every arrangement of n disjoint line segments in the plane has a subset of size Omega(sqrt{n}) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R^3. We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.
Cite as

Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, and Csaba D. Tóth. Circumscribing Polygons and Polygonizations for Disjoint Line Segments. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akitaya_et_al:LIPIcs.SoCG.2019.9,
  author =	{Akitaya, Hugo A. and Korman, Matias and Rudoy, Mikhail and Souvaine, Diane L. and T\'{o}th, Csaba D.},
  title =	{{Circumscribing Polygons and Polygonizations for Disjoint Line Segments}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.9},
  URN =		{urn:nbn:de:0030-drops-104136},
  doi =		{10.4230/LIPIcs.SoCG.2019.9},
  annote =	{Keywords: circumscribing polygon, Hamiltonicity, extremal combinatorics}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.22
Convex Polygons in Cartesian Products

Authors: Jean-Lou De Carufel, Adrian Dumitrescu, Wouter Meulemans, Tim Ophelders, Claire Pennarun, Csaba D. Tóth, and Sander Verdonschot

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.
Cite as

Jean-Lou De Carufel, Adrian Dumitrescu, Wouter Meulemans, Tim Ophelders, Claire Pennarun, Csaba D. Tóth, and Sander Verdonschot. Convex Polygons in Cartesian Products. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{decarufel_et_al:LIPIcs.SoCG.2019.22,
  author =	{De Carufel, Jean-Lou and Dumitrescu, Adrian and Meulemans, Wouter and Ophelders, Tim and Pennarun, Claire and T\'{o}th, Csaba D. and Verdonschot, Sander},
  title =	{{Convex Polygons in Cartesian Products}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.22},
  URN =		{urn:nbn:de:0030-drops-104267},
  doi =		{10.4230/LIPIcs.SoCG.2019.22},
  annote =	{Keywords: Erd\H{o}s-Szekeres theorem, Cartesian product, convexity, polyhedron, recursive construction, approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.77
Maximum Area Axis-Aligned Square Packings

Authors: Hugo A. Akitaya, Matthew D. Jones, David Stalfa, and Csaba D. Tóth

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
Given a point set S={s_1,... , s_n} in the unit square U=[0,1]^2, an anchored square packing is a set of n interior-disjoint empty squares in U such that s_i is a corner of the ith square. The reach R(S) of S is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in S. It is shown that area(R(S))>= 1/2 for every finite set S subset U, and this bound is the best possible. The region R(S) can be computed in O(n log n) time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted.
Cite as

Hugo A. Akitaya, Matthew D. Jones, David Stalfa, and Csaba D. Tóth. Maximum Area Axis-Aligned Square Packings. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 77:1-77:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{akitaya_et_al:LIPIcs.MFCS.2018.77,
  author =	{Akitaya, Hugo A. and Jones, Matthew D. and Stalfa, David and T\'{o}th, Csaba D.},
  title =	{{Maximum Area Axis-Aligned Square Packings}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{77:1--77:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.77},
  URN =		{urn:nbn:de:0030-drops-96594},
  doi =		{10.4230/LIPIcs.MFCS.2018.77},
  annote =	{Keywords: square packing, geometric optimization}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.SoCG.2018
LIPIcs, Volume 99, SoCG'18, Complete Volume

Authors: Bettina Speckmann and Csaba D. Tóth

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
LIPIcs, Volume 99, SoCG'18, Complete Volume
Cite as

34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Proceedings{speckmann_et_al:LIPIcs.SoCG.2018,
  title =	{{LIPIcs, Volume 99, SoCG'18, Complete Volume}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018},
  URN =		{urn:nbn:de:0030-drops-89308},
  doi =		{10.4230/LIPIcs.SoCG.2018},
  annote =	{Keywords: Theory of computation, Computational geometry, Mathematics of computing, Combinatorics, Theory of computation, Design and analysis of algorithms}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.SoCG.2018.0
Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks

Authors: Bettina Speckmann and Csaba D. Tóth

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks
Cite as

34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 0:i-0:xi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{speckmann_et_al:LIPIcs.SoCG.2018.0,
  author =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  title =	{{Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{0:i--0:xi},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.0},
  URN =		{urn:nbn:de:0030-drops-87136},
  doi =		{10.4230/LIPIcs.SoCG.2018.0},
  annote =	{Keywords: Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2017.47
Two-Planar Graphs Are Quasiplanar

Authors: Michael Hoffmann and Csaba D. Tóth

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract
It is shown that every 2-planar graph is quasiplanar, that is, if a simple graph admits a drawing in the plane such that every edge is crossed at most twice, then it also admits a drawing in which no three edges pairwise cross. We further show that quasiplanarity is witnessed by a simple topological drawing, that is, any two edges cross at most once and adjacent edges do not cross.
Cite as

Michael Hoffmann and Csaba D. Tóth. Two-Planar Graphs Are Quasiplanar. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{hoffmann_et_al:LIPIcs.MFCS.2017.47,
  author =	{Hoffmann, Michael and T\'{o}th, Csaba D.},
  title =	{{Two-Planar Graphs Are Quasiplanar}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.47},
  URN =		{urn:nbn:de:0030-drops-80811},
  doi =		{10.4230/LIPIcs.MFCS.2017.47},
  annote =	{Keywords: graph drawing, near-planar graph, simple topological plane graph}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2016.10
Reconstruction of Weakly Simple Polygons from their Edges

Authors: Hugo A. Akitaya and Csaba D. Tóth

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract
Given n line segments in the plane, do they form the edge set of a weakly simple polygon; that is, can the segment endpoints be perturbed by at most epsilon, for any epsilon > 0, to obtain a simple polygon? While the analogous question for simple polygons can easily be answered in O(n log n) time, we show that it is NP-complete for weakly simple polygons. We give O(n)-time algorithms in two special cases: when all segments are collinear, or the segment endpoints are in general position. These results extend to the variant in which the segments are directed, and the counterclockwise traversal of a polygon should follow the orientation. We study related problems for the case that the union of the n input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems.
Cite as

Hugo A. Akitaya and Csaba D. Tóth. Reconstruction of Weakly Simple Polygons from their Edges. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{akitaya_et_al:LIPIcs.ISAAC.2016.10,
  author =	{Akitaya, Hugo A. and T\'{o}th, Csaba D.},
  title =	{{Reconstruction of Weakly Simple Polygons from their Edges}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{10:1--10:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.10},
  URN =		{urn:nbn:de:0030-drops-67795},
  doi =		{10.4230/LIPIcs.ISAAC.2016.10},
  annote =	{Keywords: simple polygon, line segment, geometric graph}
}
Document
DOI: 10.4230/LIPIcs.STACS.2015.197
Arc Diagrams, Flip Distances, and Hamiltonian Triangulations

Authors: Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract
We show that every triangulation (maximal planar graph) on n\ge 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n-33.6 to 5n-23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting graph admits a 2-page book embedding.
Cite as

Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein. Arc Diagrams, Flip Distances, and Hamiltonian Triangulations. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 197-210, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cardinal_et_al:LIPIcs.STACS.2015.197,
  author =	{Cardinal, Jean and Hoffmann, Michael and Kusters, Vincent and T\'{o}th, Csaba D. and Wettstein, Manuel},
  title =	{{Arc Diagrams, Flip Distances, and Hamiltonian Triangulations}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{197--210},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.197},
  URN =		{urn:nbn:de:0030-drops-49141},
  doi =		{10.4230/LIPIcs.STACS.2015.197},
  annote =	{Keywords: graph embeddings, edge flips, flip graph, separating triangles}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.128
Computing Opaque Interior Barriers à la Shermer

Authors: Adrian Dumitrescu, Minghui Jiang, and Csaba D. Tóth

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
The problem of finding a collection of curves of minimum total length that meet all the lines intersecting a given polygon was initiated by Mazurkiewicz in 1916. Such a collection forms an opaque barrier for the polygon. In 1991 Shermer proposed an exponential-time algorithm that computes an interior-restricted barrier made of segments for any given convex n-gon. He conjectured that the barrier found by his algorithm is optimal, however this was refuted recently by Provan et al. Here we give a Shermer like algorithm that computes an interior polygonal barrier whose length is at most 1.7168 times the optimal and that runs in O(n) time. As a byproduct, we also deduce upper and lower bounds on the approximation ratio of Shermer's algorithm.
Cite as

Adrian Dumitrescu, Minghui Jiang, and Csaba D. Tóth. Computing Opaque Interior Barriers à la Shermer. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 128-143, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dumitrescu_et_al:LIPIcs.APPROX-RANDOM.2014.128,
  author =	{Dumitrescu, Adrian and Jiang, Minghui and T\'{o}th, Csaba D.},
  title =	{{Computing Opaque Interior Barriers \`{a} la Shermer}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{128--143},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.128},
  URN =		{urn:nbn:de:0030-drops-46938},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.128},
  annote =	{Keywords: Opaque barrier, approximation algorithm, isoperimetric inequality}
}
Document
DOI: 10.4230/LIPIcs.STACS.2011.637
Bounds on the maximum multiplicity of some common geometric graphs

Authors: Adrian Dumitrescu, Andre Schulz, Adam Sheffer, and Csaba D. Toth

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Abstract
We obtain new lower and upper bounds for the maximum multiplicity of some weighted, and respectively non-weighted, common geometric graphs drawn on $n$ points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of $n$ points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits Omega (8.65^n) different triangulations. This improves the bound Omega (8.48^n) achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We present a new lower bound of Omega(11.97^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, Omega(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.664^n) non-crossing spanning cycles. (iv) We derive exponential lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). It was known that the number of longest non-crossing spanning trees of a point set can be exponentially large, and here we show that this can be also realized with points in convex position. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(n log n) time algorithm for computing them.
Cite as

Adrian Dumitrescu, Andre Schulz, Adam Sheffer, and Csaba D. Toth. Bounds on the maximum multiplicity of some common geometric graphs. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 637-648, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{dumitrescu_et_al:LIPIcs.STACS.2011.637,
  author =	{Dumitrescu, Adrian and Schulz, Andre and Sheffer, Adam and Toth, Csaba D.},
  title =	{{Bounds on the maximum multiplicity of some common geometric graphs}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{637--648},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.637},
  URN =		{urn:nbn:de:0030-drops-30505},
  doi =		{10.4230/LIPIcs.STACS.2011.637},
  annote =	{Keywords: combinatorial geometry, matching, triangulation, spanning tree, spanning cycle, weighted structure, non-crossing condition}
}
Document
DOI: 10.4230/LIPIcs.STACS.2010.2465
Long Non-crossing Configurations in the Plane

Authors: Adrian Dumitrescu and Csaba D. Tóth

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract
We revisit several maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1993). Given a set of $n$ points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching (b) a Hamiltonian path (c) a spanning tree. Here we obtain new results for (b) and (c), as well as for the Hamiltonian cycle problem: (i) For the longest non-crossing Hamiltonian path problem, we give an approximation algorithm with ratio $\frac{2}{\pi+1} \approx 0.4829$. The previous best ratio, due to Alon et al., was $1/\pi \approx 0.3183$. Moreover, the ratio of our algorithm is close to $2/\pi$ on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time. (ii) For the longest non-crossing spanning tree problem, we give an approximation algorithm with ratio $0.502$ which runs in $O(n \log{n})$ time. The previous ratio, $1/2$, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first re-derive the result of Alon et al. with a faster $O(n \log{n})$-time algorithm and a very simple analysis. (iii) For the longest non-crossing Hamiltonian cycle problem, we give an approximation algorithm whose ratio is close to $2/\pi$ on a relatively broad class of instances: for point sets with the product $\bf{\langle}$~diameter~$\times$ ~convex hull size $\bf{\rangle}$ much smaller than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time. No previous approximation results were known for this problem.
Cite as

Adrian Dumitrescu and Csaba D. Tóth. Long Non-crossing Configurations in the Plane. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 311-322, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dumitrescu_et_al:LIPIcs.STACS.2010.2465,
  author =	{Dumitrescu, Adrian and T\'{o}th, Csaba D.},
  title =	{{Long Non-crossing Configurations in the Plane}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{311--322},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2465},
  URN =		{urn:nbn:de:0030-drops-24655},
  doi =		{10.4230/LIPIcs.STACS.2010.2465},
  annote =	{Keywords: Longest non-crossing Hamiltonian path, longest non-crossing Hamiltonian cycle, longest non-crossing spanning tree, approximation algorithm.}
}

Toth, Csaba D.

Document
DOI: 10.4230/LIPIcs.ISAAC.2024.4
Minimum Plane Bichromatic Spanning Trees

Authors: Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, and Csaba D. Tóth

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in O(n log n) time where n is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of O(√n). It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an O(log n)-factor approximation algorithm for the general case.
Cite as

Hugo A. Akitaya, Ahmad Biniaz, Erik D. Demaine, Linda Kleist, Frederick Stock, and Csaba D. Tóth. Minimum Plane Bichromatic Spanning Trees. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{a.akitaya_et_al:LIPIcs.ISAAC.2024.4,
  author =	{A. Akitaya, Hugo and Biniaz, Ahmad and Demaine, Erik D. and Kleist, Linda and Stock, Frederick and T\'{o}th, Csaba D.},
  title =	{{Minimum Plane Bichromatic Spanning Trees}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{4:1--4:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.4},
  URN =		{urn:nbn:de:0030-drops-221319},
  doi =		{10.4230/LIPIcs.ISAAC.2024.4},
  annote =	{Keywords: Bichromatic Spanning Tree, Minimum Spanning Tree, Plane Tree}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.36
Noncrossing Longest Paths and Cycles

Authors: Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, David Eppstein, Anil Maheshwari, Saeed Odak, Michiel Smid, Csaba D. Tóth, and Pavel Valtr

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.
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Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, David Eppstein, Anil Maheshwari, Saeed Odak, Michiel Smid, Csaba D. Tóth, and Pavel Valtr. Noncrossing Longest Paths and Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aloupis_et_al:LIPIcs.GD.2024.36,
  author =	{Aloupis, Greg and Biniaz, Ahmad and Bose, Prosenjit and De Carufel, Jean-Lou and Eppstein, David and Maheshwari, Anil and Odak, Saeed and Smid, Michiel and T\'{o}th, Csaba D. and Valtr, Pavel},
  title =	{{Noncrossing Longest Paths and Cycles}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{36:1--36:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.36},
  URN =		{urn:nbn:de:0030-drops-213203},
  doi =		{10.4230/LIPIcs.GD.2024.36},
  annote =	{Keywords: Longest Paths, Longest Cycles, Noncrossing Paths, Noncrossing Cycles}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.19
Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time

Authors: Sujoy Bhore, Martin Nöllenburg, Csaba D. Tóth, and Jules Wulms

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
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.
Cite as

Sujoy Bhore, Martin Nöllenburg, Csaba D. Tóth, and Jules Wulms. Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2024.19,
  author =	{Bhore, Sujoy and N\"{o}llenburg, Martin and T\'{o}th, Csaba D. and Wulms, Jules},
  title =	{{Fully Dynamic Maximum Independent Sets of Disks in Polylogarithmic Update Time}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.19},
  URN =		{urn:nbn:de:0030-drops-199649},
  doi =		{10.4230/LIPIcs.SoCG.2024.19},
  annote =	{Keywords: Dynamic algorithm, Independent set, Geometric intersection graph}
}
Document
DOI: 10.4230/LIPIcs.ESA.2023.6
Reconfiguration of Polygonal Subdivisions via Recombination

Authors: Hugo A. Akitaya, Andrei Gonczi, Diane L. Souvaine, Csaba D. Tóth, and Thomas Weighill

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
Motivated by the problem of redistricting, we study area-preserving reconfigurations of connected subdivisions of a simple polygon. A connected subdivision of a polygon ℛ, called a district map, is a set of interior disjoint connected polygons called districts whose union equals ℛ. We consider the recombination as the reconfiguration move which takes a subdivision and produces another by merging two adjacent districts, and by splitting them into two connected polygons of the same area as the original districts. The complexity of a map is the number of vertices in the boundaries of its districts. Given two maps with k districts, with complexity O(n), and a perfect matching between districts of the same area in the two maps, we show constructively that (log n)^O(log k) recombination moves are sufficient to reconfigure one into the other. We also show that Ω(log n) recombination moves are sometimes necessary even when k = 3, thus providing a tight bound when k = 3.
Cite as

Hugo A. Akitaya, Andrei Gonczi, Diane L. Souvaine, Csaba D. Tóth, and Thomas Weighill. Reconfiguration of Polygonal Subdivisions via Recombination. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{a.akitaya_et_al:LIPIcs.ESA.2023.6,
  author =	{A. Akitaya, Hugo and Gonczi, Andrei and Souvaine, Diane L. and T\'{o}th, Csaba D. and Weighill, Thomas},
  title =	{{Reconfiguration of Polygonal Subdivisions via Recombination}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.6},
  URN =		{urn:nbn:de:0030-drops-186598},
  doi =		{10.4230/LIPIcs.ESA.2023.6},
  annote =	{Keywords: configuration space, gerrymandering, polygonal subdivision, recombination}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.18
Online Spanners in Metric Spaces

Authors: Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, and Csaba D. Tóth

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness.
Cite as

Sujoy Bhore, Arnold Filtser, Hadi Khodabandeh, and Csaba D. Tóth. Online Spanners in Metric Spaces. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bhore_et_al:LIPIcs.ESA.2022.18,
  author =	{Bhore, Sujoy and Filtser, Arnold and Khodabandeh, Hadi and T\'{o}th, Csaba D.},
  title =	{{Online Spanners in Metric Spaces}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{18:1--18:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.18},
  URN =		{urn:nbn:de:0030-drops-169564},
  doi =		{10.4230/LIPIcs.ESA.2022.18},
  annote =	{Keywords: spanner, online algorithm, lightness, sparsity, minimum weight}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.30
Hop-Spanners for Geometric Intersection Graphs

Authors: Jonathan B. Conroy and Csaba D. Tóth

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
A t-spanner of a graph G = (V,E) is a subgraph H = (V,E') that contains a uv-path of length at most t for every uv ∈ E. It is known that every n-vertex graph admits a (2k-1)-spanner with O(n^{1+1/k}) edges for k ≥ 1. This bound is the best possible for 1 ≤ k ≤ 9 and is conjectured to be optimal due to Erdős' girth conjecture. We study t-spanners for t ∈ {2,3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every n-vertex unit disk graph (UDG) admits a 2-hop spanner with O(n) edges; improving upon the previous bound of O(nlog n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(nlog n) edges, and this bound is the best possible. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(nlog n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(nlog² n) edges.
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Jonathan B. Conroy and Csaba D. Tóth. Hop-Spanners for Geometric Intersection Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{conroy_et_al:LIPIcs.SoCG.2022.30,
  author =	{Conroy, Jonathan B. and T\'{o}th, Csaba D.},
  title =	{{Hop-Spanners for Geometric Intersection Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{30:1--30:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.30},
  URN =		{urn:nbn:de:0030-drops-160381},
  doi =		{10.4230/LIPIcs.SoCG.2022.30},
  annote =	{Keywords: geometric intersection graph, unit disk graph, hop-spanner}
}
Document
DOI: 10.4230/LIPIcs.ESA.2021.16
Online Euclidean Spanners

Authors: Sujoy Bhore and Csaba D. Tóth

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
In this paper, we study the online Euclidean spanners problem for points in ℝ^d. Given a set S of n points in ℝ^d, a t-spanner on S is a subgraph of the underlying complete graph G = (S,binom(S,2)), that preserves the pairwise Euclidean distances between points in S to within a factor of t, that is the stretch factor. Suppose we are given a sequence of n points (s₁,s₂,…, s_n) in ℝ^d, where point s_i is presented in step i for i = 1,…, n. The objective of an online algorithm is to maintain a geometric t-spanner on S_i = {s₁,…, s_i} for each step i. The algorithm is allowed to add new edges to the spanner when a new point is presented, but cannot remove any edge from the spanner. The performance of an online algorithm is measured by its competitive ratio, which is the supremum, over all sequences of points, of the ratio between the weight of the spanner constructed by the algorithm and the weight of an optimum spanner. Here the weight of a spanner is the sum of all edge weights. First, we establish a lower bound of Ω(ε^{-1}log n / log ε^{-1}) for the competitive ratio of any online (1+ε)-spanner algorithm, for a sequence of n points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a (1+ε)-spanner with competitive ratio O(ε^{-1}log n / log ε^{-1}). Next, we design online algorithms for sequences of points in ℝ^d, for any constant d ≥ 2, under the L₂ norm. We show that previously known incremental algorithms achieve a competitive ratio O(ε^{-(d+1)}log n). However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of ε. We describe an online Steiner (1+ε)-spanner algorithm with competitive ratio O(ε^{(1-d)/2} log n). As a counterpart, we show that the dependence on n cannot be eliminated in dimensions d ≥ 2. In particular, we prove that any online spanner algorithm for a sequence of n points in ℝ^d under the L₂ norm has competitive ratio Ω(f(n)), where lim_{n → ∞}f(n) = ∞. Finally, we provide improved lower bounds under the L₁ norm: Ω(ε^{-2}/log ε^{-1}) in the plane and Ω(ε^{-d}) in ℝ^d for d ≥ 3.
Cite as

Sujoy Bhore and Csaba D. Tóth. Online Euclidean Spanners. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhore_et_al:LIPIcs.ESA.2021.16,
  author =	{Bhore, Sujoy and T\'{o}th, Csaba D.},
  title =	{{Online Euclidean Spanners}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{16:1--16:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.16},
  URN =		{urn:nbn:de:0030-drops-145974},
  doi =		{10.4230/LIPIcs.ESA.2021.16},
  annote =	{Keywords: Geometric spanner, (1+\epsilon)-spanner, minimum weight, online algorithm}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.15
Light Euclidean Steiner Spanners in the Plane

Authors: Sujoy Bhore and Csaba D. Tóth

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in ℝ^d. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on ε > 0 and d ∈ ℕ of the minimum lightness of a (1+ε)-spanner, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1+ε)-spanners of lightness O(ε^{-1}logΔ) in the plane, where Δ ≥ Ω(√n) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness Õ(ε^{-(d+1)/2}) in dimensions d ≥ 3. Recently, Bhore and Tóth (2020) established a lower bound of Ω(ε^{-d/2}) for the lightness of Steiner (1+ε)-spanners in ℝ^d, for d ≥ 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d ≥ 2. In this work, we show that for every finite set of points in the plane and every ε > 0, there exists a Euclidean Steiner (1+ε)-spanner of lightness O(ε^{-1}); this matches the lower bound for d = 2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.
Cite as

Sujoy Bhore and Csaba D. Tóth. Light Euclidean Steiner Spanners in the Plane. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2021.15,
  author =	{Bhore, Sujoy and T\'{o}th, Csaba D.},
  title =	{{Light Euclidean Steiner Spanners in the Plane}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.15},
  URN =		{urn:nbn:de:0030-drops-138145},
  doi =		{10.4230/LIPIcs.SoCG.2021.15},
  annote =	{Keywords: Geometric spanner, lightness, minimum weight}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.13
On Euclidean Steiner (1+ε)-Spanners

Authors: Sujoy Bhore and Csaba D. Tóth

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
Lightness and sparsity are two natural parameters for Euclidean (1+ε)-spanners. Classical results show that, when the dimension d ∈ ℕ and ε > 0 are constant, every set S of n points in d-space admits an (1+ε)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ε > 0 for constant d ∈ ℕ have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1+ε)-spanner. They gave upper bounds of Õ(ε^{-(d+1)/2}) for the minimum lightness in dimensions d ≥ 3, and Õ(ε^{-(d-1))/2}) for the minimum sparsity in d-space for all d ≥ 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1+ε)-spanners of lightness O(ε^{-1}logΔ) in the plane, where Δ ∈ Ω(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1+ε)-spanners. Using a new geometric analysis, we establish lower bounds of Ω(ε^{-d/2}) for the lightness and Ω(ε^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d ≥ 2. We use the geometric insight from our lower bound analysis to construct Steiner (1+ε)-spanners of lightness O(ε^{-1}log n) for n points in Euclidean plane.
Cite as

Sujoy Bhore and Csaba D. Tóth. On Euclidean Steiner (1+ε)-Spanners. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhore_et_al:LIPIcs.STACS.2021.13,
  author =	{Bhore, Sujoy and T\'{o}th, Csaba D.},
  title =	{{On Euclidean Steiner (1+\epsilon)-Spanners}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.13},
  URN =		{urn:nbn:de:0030-drops-136586},
  doi =		{10.4230/LIPIcs.STACS.2021.13},
  annote =	{Keywords: Geometric spanner, (1+\epsilon)-spanner, lightness, sparsity, minimum weight}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.57
Sparse Hop Spanners for Unit Disk Graphs

Authors: Adrian Dumitrescu, Anirban Ghosh, and Csaba D. Tóth

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
A unit disk graph G on a given set of points P in the plane is a geometric graph where an edge exists between two points p,q ∈ P if and only if |pq| ≤ 1. A subgraph G' of G is a k-hop spanner if and only if for every edge pq ∈ G, the topological shortest path between p,q in G' has at most k edges. We obtain the following results for unit disk graphs. 1) Every n-vertex unit disk graph has a 5-hop spanner with at most 5.5n edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from 9n to 5.5n. 2) Using a new construction, we show that every n-vertex unit disk graph has a 3-hop spanner with at most 11n edges. 3) Every n-vertex unit disk graph has a 2-hop spanner with O(nlog n) edges. This is the first nontrivial construction of 2-hop spanners. 4) For every sufficiently large n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known. 5) For every point set on a circle, there exists a plane 4-hop spanner. As such, this provides a tight bound for points on a circle. 6) The maximum degree of k-hop spanners cannot be bounded from above by a function of k.
Cite as

Adrian Dumitrescu, Anirban Ghosh, and Csaba D. Tóth. Sparse Hop Spanners for Unit Disk Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dumitrescu_et_al:LIPIcs.ISAAC.2020.57,
  author =	{Dumitrescu, Adrian and Ghosh, Anirban and T\'{o}th, Csaba D.},
  title =	{{Sparse Hop Spanners for Unit Disk Graphs}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{57:1--57:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.57},
  URN =		{urn:nbn:de:0030-drops-134018},
  doi =		{10.4230/LIPIcs.ISAAC.2020.57},
  annote =	{Keywords: graph approximation, \epsilon-net, hop-spanner, unit disk graph, lower bound}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.7
Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

Authors: Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different objectives, either minimizing the maximum size of a piece for a given number of chords, or minimizing the number of chords that achieve a given size threshold for the pieces. We give hardness results for polygons with holes and approximation algorithms for multiple variants of the problem.
Cite as

Esther M. Arkin, Rathish Das, Jie Gao, Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk, and Csaba D. Tóth. Cutting Polygons into Small Pieces with Chords: Laser-Based Localization. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arkin_et_al:LIPIcs.ESA.2020.7,
  author =	{Arkin, Esther M. and Das, Rathish and Gao, Jie and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin and T\'{o}th, Csaba D.},
  title =	{{Cutting Polygons into Small Pieces with Chords: Laser-Based Localization}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{7:1--7:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.7},
  URN =		{urn:nbn:de:0030-drops-128736},
  doi =		{10.4230/LIPIcs.ESA.2020.7},
  annote =	{Keywords: Polygon partition, Arrangements, Visibility, Localization}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.56
On the Stretch Factor of Polygonal Chains

Authors: Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
Let P=(p_1, p_2, ..., p_n) be a polygonal chain. The stretch factor of P is the ratio between the total length of P and the distance of its endpoints, sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|. For a parameter c >= 1, we call P a c-chain if |p_ip_j|+|p_jp_k| <= c|p_ip_k|, for every triple (i,j,k), 1 <= i<j<k <= n. The stretch factor is a global property: it measures how close P is to a straight line, and it involves all the vertices of P; being a c-chain, on the other hand, is a fingerprint-property: it only depends on subsets of O(1) vertices of the chain. We investigate how the c-chain property influences the stretch factor in the plane: (i) we show that for every epsilon > 0, there is a noncrossing c-chain that has stretch factor Omega(n^{1/2-epsilon}), for sufficiently large constant c=c(epsilon); (ii) on the other hand, the stretch factor of a c-chain P is O(n^{1/2}), for every constant c >= 1, regardless of whether P is crossing or noncrossing; and (iii) we give a randomized algorithm that can determine, for a polygonal chain P in R^2 with n vertices, the minimum c >= 1 for which P is a c-chain in O(n^{2.5} polylog n) expected time and O(n log n) space.
Cite as

Ke Chen, Adrian Dumitrescu, Wolfgang Mulzer, and Csaba D. Tóth. On the Stretch Factor of Polygonal Chains. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chen_et_al:LIPIcs.MFCS.2019.56,
  author =	{Chen, Ke and Dumitrescu, Adrian and Mulzer, Wolfgang and T\'{o}th, Csaba D.},
  title =	{{On the Stretch Factor of Polygonal Chains}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{56:1--56:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.56},
  URN =		{urn:nbn:de:0030-drops-110005},
  doi =		{10.4230/LIPIcs.MFCS.2019.56},
  annote =	{Keywords: polygonal chain, vertex dilation, Koch curve, recursive construction}
}
Document
DOI: 10.4230/DagRep.9.2.123
Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092)

Authors: Seok-Hee Hong, Michael Kaufmann, János Pach, and Csaba D. Tóth

Published in: Dagstuhl Reports, Volume 9, Issue 2 (2019)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 19092 "Beyond-Planar Graphs: Combinatorics, Models and Algorithms" which brought together 36 researchers in the areas of graph theory, combinatorics, computational geometry, and graph drawing. This seminar continued the work initiated in Dagstuhl Seminar 16452 "Beyond-Planar Graphs: Algorithmics and Combinatorics" and focused on the exploration of structural properties and the development of algorithms for so-called beyond-planar graphs, i.e., non-planar graphs that admit a drawing with topological constraints such as specific types of crossings, or with some forbidden crossing patterns. The seminar began with four talks about the results of scientific collaborations originating from the previous Dagstuhl seminar. Next we discussed open research problems about beyond planar graphs, such as their combinatorial structures (e.g., book thickness, queue number), their topology (e.g., simultaneous embeddability, gap planarity, quasi-quasiplanarity), their geometric representations (e.g., representations on few segments or arcs), and applications (e.g., manipulation of graph drawings by untangling operations). Six working groups were formed that investigated several of the open research questions. In addition, talks on related subjects and recent conference contributions were presented in the morning opening sessions. Abstracts of all talks and a report from each working group are included in this report.
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Seok-Hee Hong, Michael Kaufmann, János Pach, and Csaba D. Tóth. Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092). In Dagstuhl Reports, Volume 9, Issue 2, pp. 123-156, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{hong_et_al:DagRep.9.2.123,
  author =	{Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and T\'{o}th, Csaba D.},
  title =	{{Beyond-Planar Graphs: Combinatorics, Models and Algorithms (Dagstuhl Seminar 19092)}},
  pages =	{123--156},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2019},
  volume =	{9},
  number =	{2},
  editor =	{Hong, Seok-Hee and Kaufmann, Michael and Pach, J\'{a}nos and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.2.123},
  URN =		{urn:nbn:de:0030-drops-108634},
  doi =		{10.4230/DagRep.9.2.123},
  annote =	{Keywords: combinatorial geometry, geometric algorithms, graph algorithms, graph drawing, graph theory, network visualization}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.9
Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Authors: Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, and Csaba D. Tóth

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
Given a planar straight-line graph G=(V,E) in R^2, a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P. We prove that every arrangement of n disjoint line segments in the plane has a subset of size Omega(sqrt{n}) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R^3. We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.
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Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, and Csaba D. Tóth. Circumscribing Polygons and Polygonizations for Disjoint Line Segments. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akitaya_et_al:LIPIcs.SoCG.2019.9,
  author =	{Akitaya, Hugo A. and Korman, Matias and Rudoy, Mikhail and Souvaine, Diane L. and T\'{o}th, Csaba D.},
  title =	{{Circumscribing Polygons and Polygonizations for Disjoint Line Segments}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.9},
  URN =		{urn:nbn:de:0030-drops-104136},
  doi =		{10.4230/LIPIcs.SoCG.2019.9},
  annote =	{Keywords: circumscribing polygon, Hamiltonicity, extremal combinatorics}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.22
Convex Polygons in Cartesian Products

Authors: Jean-Lou De Carufel, Adrian Dumitrescu, Wouter Meulemans, Tim Ophelders, Claire Pennarun, Csaba D. Tóth, and Sander Verdonschot

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.
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Jean-Lou De Carufel, Adrian Dumitrescu, Wouter Meulemans, Tim Ophelders, Claire Pennarun, Csaba D. Tóth, and Sander Verdonschot. Convex Polygons in Cartesian Products. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{decarufel_et_al:LIPIcs.SoCG.2019.22,
  author =	{De Carufel, Jean-Lou and Dumitrescu, Adrian and Meulemans, Wouter and Ophelders, Tim and Pennarun, Claire and T\'{o}th, Csaba D. and Verdonschot, Sander},
  title =	{{Convex Polygons in Cartesian Products}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.22},
  URN =		{urn:nbn:de:0030-drops-104267},
  doi =		{10.4230/LIPIcs.SoCG.2019.22},
  annote =	{Keywords: Erd\H{o}s-Szekeres theorem, Cartesian product, convexity, polyhedron, recursive construction, approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.77
Maximum Area Axis-Aligned Square Packings

Authors: Hugo A. Akitaya, Matthew D. Jones, David Stalfa, and Csaba D. Tóth

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
Given a point set S={s_1,... , s_n} in the unit square U=[0,1]^2, an anchored square packing is a set of n interior-disjoint empty squares in U such that s_i is a corner of the ith square. The reach R(S) of S is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in S. It is shown that area(R(S))>= 1/2 for every finite set S subset U, and this bound is the best possible. The region R(S) can be computed in O(n log n) time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted.
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Hugo A. Akitaya, Matthew D. Jones, David Stalfa, and Csaba D. Tóth. Maximum Area Axis-Aligned Square Packings. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 77:1-77:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{akitaya_et_al:LIPIcs.MFCS.2018.77,
  author =	{Akitaya, Hugo A. and Jones, Matthew D. and Stalfa, David and T\'{o}th, Csaba D.},
  title =	{{Maximum Area Axis-Aligned Square Packings}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{77:1--77:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.77},
  URN =		{urn:nbn:de:0030-drops-96594},
  doi =		{10.4230/LIPIcs.MFCS.2018.77},
  annote =	{Keywords: square packing, geometric optimization}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.SoCG.2018
LIPIcs, Volume 99, SoCG'18, Complete Volume

Authors: Bettina Speckmann and Csaba D. Tóth

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
LIPIcs, Volume 99, SoCG'18, Complete Volume
Cite as

34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Proceedings{speckmann_et_al:LIPIcs.SoCG.2018,
  title =	{{LIPIcs, Volume 99, SoCG'18, Complete Volume}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018},
  URN =		{urn:nbn:de:0030-drops-89308},
  doi =		{10.4230/LIPIcs.SoCG.2018},
  annote =	{Keywords: Theory of computation, Computational geometry, Mathematics of computing, Combinatorics, Theory of computation, Design and analysis of algorithms}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.SoCG.2018.0
Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks

Authors: Bettina Speckmann and Csaba D. Tóth

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks
Cite as

34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 0:i-0:xi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{speckmann_et_al:LIPIcs.SoCG.2018.0,
  author =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  title =	{{Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{0:i--0:xi},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.0},
  URN =		{urn:nbn:de:0030-drops-87136},
  doi =		{10.4230/LIPIcs.SoCG.2018.0},
  annote =	{Keywords: Front Matter, Table of Contents, Foreword, Conference Organization, Additional Reviewers, Acknowledgement of Support, Invited Talks}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2017.47
Two-Planar Graphs Are Quasiplanar

Authors: Michael Hoffmann and Csaba D. Tóth

Published in: LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Abstract
It is shown that every 2-planar graph is quasiplanar, that is, if a simple graph admits a drawing in the plane such that every edge is crossed at most twice, then it also admits a drawing in which no three edges pairwise cross. We further show that quasiplanarity is witnessed by a simple topological drawing, that is, any two edges cross at most once and adjacent edges do not cross.
Cite as

Michael Hoffmann and Csaba D. Tóth. Two-Planar Graphs Are Quasiplanar. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{hoffmann_et_al:LIPIcs.MFCS.2017.47,
  author =	{Hoffmann, Michael and T\'{o}th, Csaba D.},
  title =	{{Two-Planar Graphs Are Quasiplanar}},
  booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-046-0},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{83},
  editor =	{Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.47},
  URN =		{urn:nbn:de:0030-drops-80811},
  doi =		{10.4230/LIPIcs.MFCS.2017.47},
  annote =	{Keywords: graph drawing, near-planar graph, simple topological plane graph}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2016.10
Reconstruction of Weakly Simple Polygons from their Edges

Authors: Hugo A. Akitaya and Csaba D. Tóth

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract
Given n line segments in the plane, do they form the edge set of a weakly simple polygon; that is, can the segment endpoints be perturbed by at most epsilon, for any epsilon > 0, to obtain a simple polygon? While the analogous question for simple polygons can easily be answered in O(n log n) time, we show that it is NP-complete for weakly simple polygons. We give O(n)-time algorithms in two special cases: when all segments are collinear, or the segment endpoints are in general position. These results extend to the variant in which the segments are directed, and the counterclockwise traversal of a polygon should follow the orientation. We study related problems for the case that the union of the n input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems.
Cite as

Hugo A. Akitaya and Csaba D. Tóth. Reconstruction of Weakly Simple Polygons from their Edges. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{akitaya_et_al:LIPIcs.ISAAC.2016.10,
  author =	{Akitaya, Hugo A. and T\'{o}th, Csaba D.},
  title =	{{Reconstruction of Weakly Simple Polygons from their Edges}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{10:1--10:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.10},
  URN =		{urn:nbn:de:0030-drops-67795},
  doi =		{10.4230/LIPIcs.ISAAC.2016.10},
  annote =	{Keywords: simple polygon, line segment, geometric graph}
}
Document
DOI: 10.4230/LIPIcs.STACS.2015.197
Arc Diagrams, Flip Distances, and Hamiltonian Triangulations

Authors: Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

Abstract
We show that every triangulation (maximal planar graph) on n\ge 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4-connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4-connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n-33.6 to 5n-23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4-connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n-6) edges the resulting graph admits a 2-page book embedding.
Cite as

Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein. Arc Diagrams, Flip Distances, and Hamiltonian Triangulations. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 197-210, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{cardinal_et_al:LIPIcs.STACS.2015.197,
  author =	{Cardinal, Jean and Hoffmann, Michael and Kusters, Vincent and T\'{o}th, Csaba D. and Wettstein, Manuel},
  title =	{{Arc Diagrams, Flip Distances, and Hamiltonian Triangulations}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{197--210},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.197},
  URN =		{urn:nbn:de:0030-drops-49141},
  doi =		{10.4230/LIPIcs.STACS.2015.197},
  annote =	{Keywords: graph embeddings, edge flips, flip graph, separating triangles}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.128
Computing Opaque Interior Barriers à la Shermer

Authors: Adrian Dumitrescu, Minghui Jiang, and Csaba D. Tóth

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
The problem of finding a collection of curves of minimum total length that meet all the lines intersecting a given polygon was initiated by Mazurkiewicz in 1916. Such a collection forms an opaque barrier for the polygon. In 1991 Shermer proposed an exponential-time algorithm that computes an interior-restricted barrier made of segments for any given convex n-gon. He conjectured that the barrier found by his algorithm is optimal, however this was refuted recently by Provan et al. Here we give a Shermer like algorithm that computes an interior polygonal barrier whose length is at most 1.7168 times the optimal and that runs in O(n) time. As a byproduct, we also deduce upper and lower bounds on the approximation ratio of Shermer's algorithm.
Cite as

Adrian Dumitrescu, Minghui Jiang, and Csaba D. Tóth. Computing Opaque Interior Barriers à la Shermer. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 128-143, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dumitrescu_et_al:LIPIcs.APPROX-RANDOM.2014.128,
  author =	{Dumitrescu, Adrian and Jiang, Minghui and T\'{o}th, Csaba D.},
  title =	{{Computing Opaque Interior Barriers \`{a} la Shermer}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{128--143},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.128},
  URN =		{urn:nbn:de:0030-drops-46938},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.128},
  annote =	{Keywords: Opaque barrier, approximation algorithm, isoperimetric inequality}
}
Document
DOI: 10.4230/LIPIcs.STACS.2011.637
Bounds on the maximum multiplicity of some common geometric graphs

Authors: Adrian Dumitrescu, Andre Schulz, Adam Sheffer, and Csaba D. Toth

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Abstract
We obtain new lower and upper bounds for the maximum multiplicity of some weighted, and respectively non-weighted, common geometric graphs drawn on $n$ points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of $n$ points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits Omega (8.65^n) different triangulations. This improves the bound Omega (8.48^n) achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We present a new lower bound of Omega(11.97^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, Omega(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.664^n) non-crossing spanning cycles. (iv) We derive exponential lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). It was known that the number of longest non-crossing spanning trees of a point set can be exponentially large, and here we show that this can be also realized with points in convex position. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(n log n) time algorithm for computing them.
Cite as

Adrian Dumitrescu, Andre Schulz, Adam Sheffer, and Csaba D. Toth. Bounds on the maximum multiplicity of some common geometric graphs. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 637-648, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{dumitrescu_et_al:LIPIcs.STACS.2011.637,
  author =	{Dumitrescu, Adrian and Schulz, Andre and Sheffer, Adam and Toth, Csaba D.},
  title =	{{Bounds on the maximum multiplicity of some common geometric graphs}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{637--648},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.637},
  URN =		{urn:nbn:de:0030-drops-30505},
  doi =		{10.4230/LIPIcs.STACS.2011.637},
  annote =	{Keywords: combinatorial geometry, matching, triangulation, spanning tree, spanning cycle, weighted structure, non-crossing condition}
}
Document
DOI: 10.4230/LIPIcs.STACS.2010.2465
Long Non-crossing Configurations in the Plane

Authors: Adrian Dumitrescu and Csaba D. Tóth

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract
We revisit several maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1993). Given a set of $n$ points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching (b) a Hamiltonian path (c) a spanning tree. Here we obtain new results for (b) and (c), as well as for the Hamiltonian cycle problem: (i) For the longest non-crossing Hamiltonian path problem, we give an approximation algorithm with ratio $\frac{2}{\pi+1} \approx 0.4829$. The previous best ratio, due to Alon et al., was $1/\pi \approx 0.3183$. Moreover, the ratio of our algorithm is close to $2/\pi$ on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time. (ii) For the longest non-crossing spanning tree problem, we give an approximation algorithm with ratio $0.502$ which runs in $O(n \log{n})$ time. The previous ratio, $1/2$, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first re-derive the result of Alon et al. with a faster $O(n \log{n})$-time algorithm and a very simple analysis. (iii) For the longest non-crossing Hamiltonian cycle problem, we give an approximation algorithm whose ratio is close to $2/\pi$ on a relatively broad class of instances: for point sets with the product $\bf{\langle}$~diameter~$\times$ ~convex hull size $\bf{\rangle}$ much smaller than the maximum length matching. The algorithm runs in $O(n^{7/3}\log{n})$ time. No previous approximation results were known for this problem.
Cite as

Adrian Dumitrescu and Csaba D. Tóth. Long Non-crossing Configurations in the Plane. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 311-322, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dumitrescu_et_al:LIPIcs.STACS.2010.2465,
  author =	{Dumitrescu, Adrian and T\'{o}th, Csaba D.},
  title =	{{Long Non-crossing Configurations in the Plane}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{311--322},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2465},
  URN =		{urn:nbn:de:0030-drops-24655},
  doi =		{10.4230/LIPIcs.STACS.2010.2465},
  annote =	{Keywords: Longest non-crossing Hamiltonian path, longest non-crossing Hamiltonian cycle, longest non-crossing spanning tree, approximation algorithm.}
}

Tóth, Csaba

Document
DOI: 10.4230/LIPIcs.SoCG.2016.8
Recognizing Weakly Simple Polygons

Authors: Hugo A. Akitaya, Greg Aloupis, Jeff Erickson, and Csaba Tóth

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
We present an O(n log n)-time algorithm that determines whether a given planar n-gon is weakly simple. This improves upon an O(n^2 log n)-time algorithm by [Chang, Erickson, and Xu, SODA, 2015]. Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question.
Cite as

Hugo A. Akitaya, Greg Aloupis, Jeff Erickson, and Csaba Tóth. Recognizing Weakly Simple Polygons. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{akitaya_et_al:LIPIcs.SoCG.2016.8,
  author =	{Akitaya, Hugo A. and Aloupis, Greg and Erickson, Jeff and T\'{o}th, Csaba},
  title =	{{Recognizing Weakly Simple Polygons}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.8},
  URN =		{urn:nbn:de:0030-drops-59003},
  doi =		{10.4230/LIPIcs.SoCG.2016.8},
  annote =	{Keywords: weakly simple polygon, crossing}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.13
Anchored Rectangle and Square Packings

Authors: Kevin Balas, Adrian Dumitrescu, and Csaba Tóth

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
For points p_1,...,p_n in the unit square [0,1]^2, an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r_1,...,r_n in [0,1]^2 such that point p_i is a corner of the rectangle r_i (that is, r_i is anchored at p_i) for i=1,...,n. We show that for every set of n points in [0,1]^2, there is an anchored rectangle packing of area at least 7/12-O(1/n), and for every n, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27. The above constructive lower bounds immediately yield constant-factor approximations, of 7/12 -epsilon for rectangles and 5/32 for squares, for computing anchored packings of maximum area in O(n log n) time. We prove that a simple greedy strategy achieves a 9/47-approximation for anchored square packings, and 1/3 for lower-left anchored square packings. Reductions to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n^{O(1/epsilon)} and exp(poly(log (n/epsilon))) time, respectively.
Cite as

Kevin Balas, Adrian Dumitrescu, and Csaba Tóth. Anchored Rectangle and Square Packings. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{balas_et_al:LIPIcs.SoCG.2016.13,
  author =	{Balas, Kevin and Dumitrescu, Adrian and T\'{o}th, Csaba},
  title =	{{Anchored Rectangle and Square Packings}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.13},
  URN =		{urn:nbn:de:0030-drops-59054},
  doi =		{10.4230/LIPIcs.SoCG.2016.13},
  annote =	{Keywords: Rectangle packing, anchored rectangle, greedy algorithm, charging scheme, approximation algorithm.}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.41
The Planar Tree Packing Theorem

Authors: Markus Geyer, Michael Hoffmann, Michael Kaufmann, Vincent Kusters, and Csaba Tóth

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of Garcia et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.
Cite as

Markus Geyer, Michael Hoffmann, Michael Kaufmann, Vincent Kusters, and Csaba Tóth. The Planar Tree Packing Theorem. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{geyer_et_al:LIPIcs.SoCG.2016.41,
  author =	{Geyer, Markus and Hoffmann, Michael and Kaufmann, Michael and Kusters, Vincent and T\'{o}th, Csaba},
  title =	{{The Planar Tree Packing Theorem}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{41:1--41:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.41},
  URN =		{urn:nbn:de:0030-drops-59337},
  doi =		{10.4230/LIPIcs.SoCG.2016.41},
  annote =	{Keywords: graph drawing, simultaneous embedding, planar graph, graph packin}
}
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