24 Search Results for "Tóth, Csaba D."


Volume

LIPIcs, Volume 99

34th International Symposium on Computational Geometry (SoCG 2018)

SoCG 2018, June 11-14, 2018, Budapest, Hungary

Editors: Bettina Speckmann and Csaba D. Tóth

Document
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
Track A: Algorithms, Complexity and Games
On Greedily Packing Anchored Rectangles

Authors: Christoph Damerius, Dominik Kaaser, Peter Kling, and Florian Schneider

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)


Abstract
Consider a set P of points in the unit square U = [1,0), one of them being the origin. For each point p ∈ P you may draw an axis-aligned rectangle in U with its lower-left corner being p. What is the maximum area such rectangles can cover without overlapping each other? Freedman posed this problem in 1969, asking whether one can always cover at least 50% of U. Over 40 years later, Dumitrescu and Tóth [Adrian Dumitrescu and Csaba D. Tóth, 2015] achieved the first constant coverage of 9.1%; since then, no significant progress was made. While 9.1% might seem low, the authors could not find any instance where their algorithm covers less than 50%, nourishing the hope to eventually prove a 50% bound. While we indeed significantly raise the algorithm’s coverage to 39%, we extinguish the hope of reaching 50% by giving points for which its coverage stays below 43.3%. Our analysis studies the algorithm’s average and worst-case density of so-called tiles, which represent the staircase polygons in which a point can freely choose its maximum-area rectangle. Our approach is comparatively general and may potentially help in analyzing related algorithms.

Cite as

Christoph Damerius, Dominik Kaaser, Peter Kling, and Florian Schneider. On Greedily Packing Anchored Rectangles. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 61:1-61:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Copy BibTex To Clipboard

@InProceedings{damerius_et_al:LIPIcs.ICALP.2021.61,
  author =	{Damerius, Christoph and Kaaser, Dominik and Kling, Peter and Schneider, Florian},
  title =	{{On Greedily Packing Anchored Rectangles}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{61:1--61:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.61},
  URN =		{urn:nbn:de:0030-drops-141306},
  doi =		{10.4230/LIPIcs.ICALP.2021.61},
  annote =	{Keywords: lower-left anchored rectangle packing, greedy algorithm, charging scheme}
}
Document
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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
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)


Copy BibTex To Clipboard

@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-dev.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}
}
  • Refine by Author
  • 20 Tóth, Csaba D.
  • 6 Dumitrescu, Adrian
  • 4 Bhore, Sujoy
  • 3 Akitaya, Hugo A.
  • 2 Hoffmann, Michael
  • Show More...

  • Refine by Classification
  • 12 Theory of computation → Computational geometry
  • 6 Mathematics of computing → Paths and connectivity problems
  • 4 Mathematics of computing → Approximation algorithms
  • 1 Mathematics of computing → Combinatorial algorithms
  • 1 Mathematics of computing → Combinatorial optimization
  • Show More...

  • Refine by Keyword
  • 4 minimum weight
  • 3 Geometric spanner
  • 3 lightness
  • 2 (1+ε)-spanner
  • 2 approximation algorithm
  • Show More...

  • Refine by Type
  • 23 document
  • 1 volume

  • Refine by Publication Year
  • 5 2018
  • 4 2019
  • 4 2021
  • 2 2020
  • 2 2022
  • Show More...

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail