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Volume

LIPIcs, Volume 293

40th International Symposium on Computational Geometry (SoCG 2024)

SoCG 2024, June 11-14, 2024, Athens, Greece

Editors: Wolfgang Mulzer and Jeff M. Phillips

Document

DOI: 10.4230/LIPIcs.ITCS.2026.1

Triangle Detection in H-Free Graphs

Authors: Amir Abboud, Ron Safier, and Nathan Wallheimer

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We initiate the study of combinatorial algorithms for Triangle Detection in H-free graphs. The goal is to decide if a graph that forbids a fixed pattern H as a subgraph contains a triangle, using only "combinatorial" methods that notably exclude fast matrix multiplication. Our work aims to classify which patterns admit a subcubic speedup, working towards a dichotomy theorem. On the lower bound side, we show that if H is not 3-colorable or contains more than one triangle, the complexity of the problem remains unchanged, and no combinatorial speedup is likely possible. On the upper bound side, we develop an embedding approach that results in a strongly subcubic, combinatorial algorithm for a rich class of "embeddable" patterns. Specifically, for an embeddable pattern of size k, our algorithm runs in Õ(n^{3-1/(2^{k-3)}}) time, where Õ(⋅) hides poly-logarithmic factors. This algorithm also extends to listing all the triangles within the same time bound. We supplement this main result with two generalizations: - A generalization to patterns that are embeddable up to a single obstacle that arises from a triangle in the pattern. This completes our classification for small patterns, yielding a dichotomy theorem for all patterns of size up to eight. - An H-sensitive algorithm for embeddable patterns, which runs faster when the number of copies of H is significantly smaller than the maximum possible Ω(n^{k}). Finally, we focus on the special case of odd cycles. We present specialized Triangle Detection algorithms that are very efficient: - A combinatorial algorithm for C_{2k+1}-free graphs that runs in Õ(m+n^{1+2/k}) time for every k ≥ 2, where m is the number of edges in the graph. - A combinatorial C₅-sensitive algorithm that runs in Õ(n² + n^{4/3} t^{1/3}) time, where t is the number of 5-cycles in the graph.

Cite as

Amir Abboud, Ron Safier, and Nathan Wallheimer. Triangle Detection in H-Free Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{abboud_et_al:LIPIcs.ITCS.2026.1,
  author =	{Abboud, Amir and Safier, Ron and Wallheimer, Nathan},
  title =	{{Triangle Detection in H-Free Graphs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{1:1--1:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.1},
  URN =		{urn:nbn:de:0030-drops-252885},
  doi =		{10.4230/LIPIcs.ITCS.2026.1},
  annote =	{Keywords: fine-grained complexity, triangle detection, H-free graphs}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2026.31

Delaunay Triangulations with Predictions

Authors: Sergio Cabello, Timothy M. Chan, and Panos Giannopoulos

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

We investigate algorithms with predictions in computational geometry, specifically focusing on the basic problem of computing 2D Delaunay triangulations. Given a set P of n points in the plane and a triangulation G that serves as a "prediction" of the Delaunay triangulation, we would like to use G to compute the correct Delaunay triangulation DT(P) more quickly when G is "close" to DT(P). We obtain a variety of results of this type, under different deterministic and probabilistic settings, including the following: 1) Define D to be the number of edges in G that are not in DT(P). We present a deterministic algorithm to compute DT(P) from G in O(n + Dlog³ n) time, and a randomized algorithm in O(n+Dlog n) expected time, the latter of which is optimal in terms of D. 2) Let R be a random subset of the edges of DT(P), where each edge is chosen independently with probability ρ. Suppose G is any triangulation of P that contains R. We present an algorithm to compute DT(P) from G in O(nlog log n + nlog(1/ρ)) time with high probability. 3) Define d_{vio} to be the maximum number of points of P strictly inside the circumcircle of a triangle in G (the number is 0 if G is equal to DT(P)). We present a deterministic algorithm to compute DT(P) from G in O(nlog^*n + nlog d_{vio}) time. We also obtain results in similar settings for related problems such as 2D Euclidean minimum spanning trees, and hope that our work will open up a fruitful line of future research.

Cite as

Sergio Cabello, Timothy M. Chan, and Panos Giannopoulos. Delaunay Triangulations with Predictions. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 31:1-31:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cabello_et_al:LIPIcs.ITCS.2026.31,
  author =	{Cabello, Sergio and Chan, Timothy M. and Giannopoulos, Panos},
  title =	{{Delaunay Triangulations with Predictions}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{31:1--31:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.31},
  URN =		{urn:nbn:de:0030-drops-253186},
  doi =		{10.4230/LIPIcs.ITCS.2026.31},
  annote =	{Keywords: Delaunay Triangulation, Minimum Spanning Tree, Algorithms with Predictions}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.6

A Dimension-Reducing Fréchet Simplification Oracle

Authors: Boris Aronov, Tsuri Farhana, Matthew J. Katz, and Indu Ramesh

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

Let P be a polygonal curve with n vertices in the plane. We construct a data structure of size O(n log n) suited for simplification queries of the following kind. Given a query line 𝓁 and an integer k ≥ 1, find a curve Q on 𝓁 with at most k vertices that minimizes the discrete Fréchet distance to P, among all such curves. Using our data structure, a query can be handled in O(k² log³ n + k log⁴n) time. More generally, a geometric tree T on n vertices in the plane can be preprocessed into a near-linear-size structure so that, given a pair u, v of its vertices, a line 𝓁, and an integer k ≥ 1, one can find a curve Q on 𝓁 with at most k vertices that minimizes the discrete Fréchet distance to the path from u to v in T, in time O(k² polylog n). For the general dimension-reduction problem, where P is a curve in ℝ^d (d ≥ 3), 0 < ε₀ < 1 is a real parameter, and a query specifies a g-flat h (1 ≤ g ≤ d-1) and an integer k ≥ 1, we construct a data structure of size O(nlog n + f(ε₀) n), where f(ε₀) = (1+1/ε₀)^{(d-1)/2}, that allows us to find a curve Q on h with at most k vertices, whose discrete Fréchet distance to P is at most 1+ε₀ times the distance of Q^* to P, where Q^* is such a curve that minimizes the distance to P. The query handling time is O(f(ε₀) k² log² n).

Cite as

Boris Aronov, Tsuri Farhana, Matthew J. Katz, and Indu Ramesh. A Dimension-Reducing Fréchet Simplification Oracle. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aronov_et_al:LIPIcs.ISAAC.2025.6,
  author =	{Aronov, Boris and Farhana, Tsuri and Katz, Matthew J. and Ramesh, Indu},
  title =	{{A Dimension-Reducing Fr\'{e}chet Simplification Oracle}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{6:1--6:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.6},
  URN =		{urn:nbn:de:0030-drops-249149},
  doi =		{10.4230/LIPIcs.ISAAC.2025.6},
  annote =	{Keywords: Computational geometry, discrete Fr\'{e}chet distance, curve simplification oracle, restricted minimum enclosing disk queries}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.46

Realizing Metric Spaces with Convex Obstacles

Authors: Sándor Kisfaludi-Bak and Leonidas Theocharous

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

The presence of obstacles has a significant impact on distance computation, motion-planning, and visibility. These problems have been studied extensively in the planar setting, while our understanding of these problems in 3- and higher-dimensional spaces is still rudimentary. In this paper, we study the impact of different types of obstacles on the induced geodesic metric in 3-dimensional Euclidean space. We say that a finite metric space (X, dist_X) is approximately realizable by a collection 𝒯 of obstacles in ℝ³ if for any ε > 0 it can be embedded into (ℝ³⧵⋃_{T∈𝒯} T, dist_𝒯) with worst-case multiplicative distortion 1+ε, where dist_𝒯 denotes the geodesic distance in the free space induced by 𝒯. We focus on three key geometric properties of obstacles -convexity, disjointness, and fatness- and examine how dropping each one of them affects the existence of such embeddings. Our main result concerns dropping the fatness property: we demonstrate that any finite metric space is realizable with 1+ε worst-case multiplicative distortion using a collection of convex and pairwise disjoint obstacles in ℝ³, even if the obstacles are congruent and equilateral triangles. Based on the same construction, we can also show that if we require fatness but drop any of the other two properties instead, then we can still approximately realize any finite metric space. Our results have important implications on the approximability of tsp with obstacles, a natural variant of tsp introduced recently by Alkema et al. (ESA 2022). Specifically, we use the recent results of Banerjee et al. on tsp in doubling spaces (FOCS 2024) and of Chew et al. on distances among obstacles (Inf. Process. Lett. 2002) to show that tsp with obstacles admits a PTAS if the obstacles are convex, fat, and pairwise disjoint. If any of these three properties is dropped, then our results, combined with the APX-hardness of Metric tsp, demonstrate that tsp with obstacles is APX-hard.

Cite as

Sándor Kisfaludi-Bak and Leonidas Theocharous. Realizing Metric Spaces with Convex Obstacles. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 46:1-46:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kisfaludibak_et_al:LIPIcs.ISAAC.2025.46,
  author =	{Kisfaludi-Bak, S\'{a}ndor and Theocharous, Leonidas},
  title =	{{Realizing Metric Spaces with Convex Obstacles}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{46:1--46:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.46},
  URN =		{urn:nbn:de:0030-drops-249545},
  doi =		{10.4230/LIPIcs.ISAAC.2025.46},
  annote =	{Keywords: traveling salesman, geodesic distance}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.45

BFS and Reverse Shortest Paths for Ball Intersection Graphs in Three and Higher Dimensions

Authors: Matthew J. Katz, Rachel Saban, and Micha Sharir

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

Let ℬ be a collection of n arbitrary balls in ℝ³, and let G₀(ℬ) be their intersection graph. We provide an algorithm for performing BFS on G₀(ℬ), which runs in O^*(n^{4/3}) time, where the O^*(⋅) notation hides subpolynomial factors. For r ≥ 0, let G_r(ℬ) be the intersection graph of the set ℬ_r = {B+r ∣ B ∈ ℬ}, where B+r is the ball concentric with B whose radius is larger by r than the radius of B. We provide an efficient algorithm for the reverse shortest path (RSP) problem, where we are given two designated balls B_s, B_t of ℬ and a parameter 0 < λ < n, and seek the smallest value r^* for which G_{r^*}(ℬ) contains a path from B_s to B_t of at most λ edges. For the special case of congruent balls (equivalently, for points in ℝ³), the algorithm runs in O^*(n^{29/21}) ≈ O^*(n^{1.381}) time. For the general case, the algorithm runs in O^*(n^{56/39}) ≈ O^*(n^{1.436}) time. We also extend the technique to handle other measures of expansion and higher dimensions.

Cite as

Matthew J. Katz, Rachel Saban, and Micha Sharir. BFS and Reverse Shortest Paths for Ball Intersection Graphs in Three and Higher Dimensions. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{katz_et_al:LIPIcs.ISAAC.2025.45,
  author =	{Katz, Matthew J. and Saban, Rachel and Sharir, Micha},
  title =	{{BFS and Reverse Shortest Paths for Ball Intersection Graphs in Three and Higher Dimensions}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{45:1--45:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.45},
  URN =		{urn:nbn:de:0030-drops-249535},
  doi =		{10.4230/LIPIcs.ISAAC.2025.45},
  annote =	{Keywords: Computational geometry, reverse shortest paths, breadth-first search, shrink-and-bifurcate, intersection graphs}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.21

On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers

Authors: Parinya Chalermsook, Ly Orgo, and Minoo Zarsav

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

This paper considers the Zarankiewicz problem in bipartite graphs with low-dimensional geometric representation (i.e., low Ferrers dimension). Let Z(n;k) be the maximum number of edges in a bipartite graph with n nodes and is free of a k-by-k biclique. Note that Z(n;k) ∈ Ω(nk) for all "natural" graph classes. Our first result reveals a separation between bipartite graphs of Ferrers dimension three and four: while we show that Z(n;k) ≤ 9n(k-1) for graphs of Ferrers dimension three, Z(n;k) ∈ Ω(n k ⋅ (log n)/(log log n)) for Ferrers dimension four graphs (Chan & Har-Peled, 2023) (Chazelle, 1990). To complement this, we derive a tight upper bound of 2n(k-1) for chordal bipartite graphs and 54n(k-1) for grid intersection graphs (GIG), a prominent graph class residing in four Ferrers dimensions and capturing planar bipartite graphs as well as bipartite intersection graphs of rectangles. Previously, the best-known bound for GIG was Z(n;k) ∈ O(2^{O(k)} n), implied by the results of Fox & Pach (2006) and Mustafa & Pach (2016). Our results advance and offer new insights into the interplay between Ferrers dimensions and extremal combinatorics.

Cite as

Parinya Chalermsook, Ly Orgo, and Minoo Zarsav. On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chalermsook_et_al:LIPIcs.GD.2025.21,
  author =	{Chalermsook, Parinya and Orgo, Ly and Zarsav, Minoo},
  title =	{{On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{21:1--21:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.21},
  URN =		{urn:nbn:de:0030-drops-250074},
  doi =		{10.4230/LIPIcs.GD.2025.21},
  annote =	{Keywords: Bipartite graph classes, extremal graph theory, geometric intersection graphs, Zarankiewicz problem, bicliques}
}

Document

DOI: 10.4230/LIPIcs.GD.2025.5

Constrained Flips in Plane Spanning Trees

Authors: Oswin Aichholzer, Joseph Dorfer, and Birgit Vogtenhuber

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

A flip in a plane spanning tree T is the operation of removing one edge from T and adding another edge such that the resulting structure is again a plane spanning tree. For trees on a set of points in convex position we study two classic types of constrained flips: (1) Compatible flips are flips in which the removed and inserted edge do not cross each other. We relevantly improve the previous upper bound of 2n-O(√n) on the diameter of the compatible flip graph to (5n/3)-O(1), by this matching the upper bound for unrestricted flips by Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber [SODA 2025] up to an additive constant of 1. We further show that no shortest compatible flip sequence removes an edge that is already in its target position. Using this so-called happy edge property, we derive a fixed-parameter tractable algorithm to compute the shortest compatible flip sequence between two given trees. (2) Rotations are flips in which the removed and inserted edge share a common vertex. Besides showing that the happy edge property does not hold for rotations, we improve the previous upper bound of 2n-O(1) for the diameter of the rotation graph to (7n/4)-O(1).

Cite as

Oswin Aichholzer, Joseph Dorfer, and Birgit Vogtenhuber. Constrained Flips in Plane Spanning Trees. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aichholzer_et_al:LIPIcs.GD.2025.5,
  author =	{Aichholzer, Oswin and Dorfer, Joseph and Vogtenhuber, Birgit},
  title =	{{Constrained Flips in Plane Spanning Trees}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.5},
  URN =		{urn:nbn:de:0030-drops-249913},
  doi =		{10.4230/LIPIcs.GD.2025.5},
  annote =	{Keywords: Non-crossing spanning trees, Flip Graphs, Diameter, Complexity, Happy edges}
}

Document

Poster Abstract

DOI: 10.4230/LIPIcs.GD.2025.47

Reconfigurations of Plane Caterpillars and Paths (Poster Abstract)

Authors: Todor Antić, Guillermo Gamboa Quintero, and Jelena Glišić

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract

Let S be a point set in the plane, and let 𝒫(S) and 𝒞(S) be the sets of all plane spanning paths and caterpillars on S. We study reconfiguration operations on 𝒫(S) and 𝒞(S). In particular, we prove that all of the commonly studied reconfigurations on plane spanning trees still yield connected reconfiguration graphs for caterpillars when S is in convex position. If S is in general position, we show that the rotation, compatible flip and flip graphs of 𝒞(S) are connected while the slide graph is sometimes disconnected, but always has a component of size 1/4(3ⁿ-1). We then study sizes of connected components in reconfiguration graphs of plane spanning paths. In this direction, we show that no component of size at most 7 can exist in the flip graph on 𝒫(S).

Cite as

Todor Antić, Guillermo Gamboa Quintero, and Jelena Glišić. Reconfigurations of Plane Caterpillars and Paths (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 47:1-47:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{antic_et_al:LIPIcs.GD.2025.47,
  author =	{Anti\'{c}, Todor and Gamboa Quintero, Guillermo and Gli\v{s}i\'{c}, Jelena},
  title =	{{Reconfigurations of Plane Caterpillars and Paths}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{47:1--47:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.47},
  URN =		{urn:nbn:de:0030-drops-250337},
  doi =		{10.4230/LIPIcs.GD.2025.47},
  annote =	{Keywords: reconfiguration graph, caterpillar, path, geometric graph}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.25

Instance-Optimal Imprecise Convex Hull

Authors: Sarita de Berg, Ivor van der Hoog, Eva Rotenberg, Daniel Rutschmann, and Sampson Wong

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

Imprecise measurements of a point set P = (p₁, …, p_n) can be modelled by a family of regions F = (R₁, …, R_n), where each imprecise region R_i ∈ F contains a unique point p_i ∈ P. A retrieval models an accurate measurement by replacing an imprecise region R_i with its corresponding point p_i. We construct the convex hull of an imprecise point set in the plane, by determining the cyclic ordering of the convex hull vertices of P as efficiently as possible. Efficiency is interpreted in two ways: (i) minimising the number of retrievals, and (ii) the computation time to determine the set of regions that must be retrieved. Previous works focused on only one of these two aspects: either minimising retrievals or optimising algorithmic runtime. Our contribution is the first to simultaneously achieve both. Let r(F, P) denote the minimal number of retrievals required by any algorithm to determine the convex hull of P for a given instance (F, P). For a family F of n constant-complexity polygons, our main result is a reconstruction algorithm that performs Θ(r(F, P)) retrievals in O(r(F, P) log³ n) time. Compared to previous approaches that achieve optimal retrieval counts, we improve the runtime per retrieval from polynomial to polylogarithmic. We extend the generality of previous results to simple k-gons, to pairwise disjoint disks with radii in [1,k], and to unit disks where at most k disks overlap in a single point. Our runtime scales linearly with k.

Cite as

Sarita de Berg, Ivor van der Hoog, Eva Rotenberg, Daniel Rutschmann, and Sampson Wong. Instance-Optimal Imprecise Convex Hull. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{deberg_et_al:LIPIcs.ESA.2025.25,
  author =	{de Berg, Sarita and van der Hoog, Ivor and Rotenberg, Eva and Rutschmann, Daniel and Wong, Sampson},
  title =	{{Instance-Optimal Imprecise Convex Hull}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{25:1--25:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.25},
  URN =		{urn:nbn:de:0030-drops-244932},
  doi =		{10.4230/LIPIcs.ESA.2025.25},
  annote =	{Keywords: convex hull, imprecise geometry preprocessing model, partial information}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.24

Fréchet Distance in Unweighted Planar Graphs

Authors: Ivor van der Hoog, Thijs van der Horst, Eva Rotenberg, and Lasse Wulf

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

The Fréchet distance is a distance measure between trajectories in ℝ^d or walks in a graph G. Given constant-time shortest path queries, the Discrete Fréchet distance D_G(P, Q) between two walks P and Q can be computed in O(|P|⋅|Q|) time using a dynamic program. Driemel, van der Hoog, and Rotenberg [SoCG'22] show that for weighted planar graphs this approach is likely tight, as there can be no strongly-subquadratic algorithm to compute a 1.01-approximation of D_G(P, Q) unless the Orthogonal Vector Hypothesis (OVH) fails. Such quadratic-time conditional lower bounds are common to many Fréchet distance variants. However, they can be circumvented by assuming that the input comes from some well-behaved class: There exist (1+ε)-approximations, both in weighted graphs and in ℝ^d, that take near-linear time for c-packed or κ-straight walks in the graph. In ℝ^d there also exists a near-linear time algorithm to compute the Fréchet distance whenever all input edges are long compared to the distance. We consider computing the Fréchet distance in unweighted planar graphs. We show that there exist no strongly-subquadratic 1.25-approximations of the discrete Fréchet distance between two disjoint simple paths in an unweighted planar graph in strongly subquadratic time, unless OVH fails. This improves the previous lower bound, both in terms of generality and approximation factor. We subsequently show that adding graph structure circumvents this lower bound: If the graph is a regular tiling with unit-weighted edges, then there exists an Õ((|P|+|Q|)^{1.5})-time algorithm to compute D_G(P, Q). Our result has natural implications in the plane, as it allows us to define a new class of well-behaved curves that facilitate (1+ε)-approximations of their discrete Fréchet distance in subquadratic time.

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Ivor van der Hoog, Thijs van der Horst, Eva Rotenberg, and Lasse Wulf. Fréchet Distance in Unweighted Planar Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 24:1-24:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhoog_et_al:LIPIcs.ESA.2025.24,
  author =	{van der Hoog, Ivor and van der Horst, Thijs and Rotenberg, Eva and Wulf, Lasse},
  title =	{{Fr\'{e}chet Distance in Unweighted Planar Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{24:1--24:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.24},
  URN =		{urn:nbn:de:0030-drops-244924},
  doi =		{10.4230/LIPIcs.ESA.2025.24},
  annote =	{Keywords: Fr\'{e}chet distance, planar graphs, lower bounds, approximation algorithms}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.31

An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs

Authors: Bruce W. Brewer and Haitao Wang

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

Given in the plane a set S of n points and a set of disks centered at these points, the disk graph G(S) induced by these disks has vertex set S and an edge between two vertices if their disks intersect. Note that the disks may have different radii. We consider the problem of computing shortest paths from a source point s ∈ S to all vertices in G(S) where the length of a path in G(S) is defined as the number of edges in the path. The previously best algorithm solves the problem in O(nlog² n) time. A lower bound of Ω(nlog n) is also known for this problem under the algebraic decision tree model. In this paper, we present an O(nlog n) time algorithm, which matches the lower bound and thus is optimal. Another virtue of our algorithm is that it is quite simple.

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Bruce W. Brewer and Haitao Wang. An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 31:1-31:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{brewer_et_al:LIPIcs.ESA.2025.31,
  author =	{Brewer, Bruce W. and Wang, Haitao},
  title =	{{An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{31:1--31:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.31},
  URN =		{urn:nbn:de:0030-drops-244997},
  doi =		{10.4230/LIPIcs.ESA.2025.31},
  annote =	{Keywords: disk graphs, weighted Voronoi diagrams, shortest paths}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.67

Compact Representation of Semilinear and Terrain-Like Graphs

Authors: Jean Cardinal and Yelena Yuditsky

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

We consider the existence and construction of biclique covers of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The size of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size O(npolylog n). This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz’s problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (Forum Math. Sigma, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size O(nlog³ n). This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (Discrete Comput. Geom. 1994). Finally, we prove that there exists families of unit disk graphs on n vertices that do not admit biclique coverings of size o(n^{4/3}), showing that we are unlikely to improve on Szemerédi-Trotter type incidence bounds for higher-degree semialgebraic graphs.

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Jean Cardinal and Yelena Yuditsky. Compact Representation of Semilinear and Terrain-Like Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cardinal_et_al:LIPIcs.ESA.2025.67,
  author =	{Cardinal, Jean and Yuditsky, Yelena},
  title =	{{Compact Representation of Semilinear and Terrain-Like Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{67:1--67:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.67},
  URN =		{urn:nbn:de:0030-drops-245359},
  doi =		{10.4230/LIPIcs.ESA.2025.67},
  annote =	{Keywords: Biclique covers, intersection graphs, visibility graphs, Zarankiewicz’s problem}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.39

Going Beyond Surfaces in Diameter Approximation

Authors: Michał Włodarczyk

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

Calculating the diameter of an undirected graph requires quadratic running time under the Strong Exponential Time Hypothesis and this barrier works even against any approximation better than 3/2. For planar graphs with positive edge weights, there are known (1+ε)-approximation algorithms with running time poly(1/ε, log n)⋅ n. However, these algorithms rely on shortest path separators and this technique falls short to yield efficient algorithms beyond graphs of bounded genus. In this work we depart from embedding-based arguments and obtain diameter approximations relying on VC set systems and the local treewidth property. We present two orthogonal extensions of the planar case by giving (1+ε)-approximation algorithms with the following running times: - 𝒪_h((1/ε)^𝒪(h) ⋅ nlog² n)-time algorithm for graphs excluding an apex graph of size h as a minor, - 𝒪_d((1/ε)^𝒪(d) ⋅ nlog² n)-time algorithm for the class of d-apex graphs. As a stepping stone, we obtain efficient (1+ε)-approximate distance oracles for graphs excluding an apex graph of size h as a minor. Our oracle has preprocessing time 𝒪_h((1/ε)⁸⋅ nlog nlog W) and query time 𝒪_h((1/ε)²⋅log n log W), where W is the metric stretch. Such oracles have been so far only known for bounded genus graphs. All our algorithms are deterministic.

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Michał Włodarczyk. Going Beyond Surfaces in Diameter Approximation. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wlodarczyk:LIPIcs.ESA.2025.39,
  author =	{W{\l}odarczyk, Micha{\l}},
  title =	{{Going Beyond Surfaces in Diameter Approximation}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{39:1--39:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.39},
  URN =		{urn:nbn:de:0030-drops-245076},
  doi =		{10.4230/LIPIcs.ESA.2025.39},
  annote =	{Keywords: diameter, approximation, distance oracles, graph minors, treewidth}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.35

The Geodesic Fréchet Distance Between Two Curves Bounding a Simple Polygon

Authors: Thijs van der Horst, Marc van Kreveld, Tim Ophelders, and Bettina Speckmann

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

The Fréchet distance is a popular similarity measure that is well-understood for polygonal curves in ℝ^d: near-quadratic time algorithms exist, and conditional lower bounds suggest that these results cannot be improved significantly, even in one dimension and when approximating with a factor less than three. We consider the special case where the curves bound a simple polygon and distances are measured via geodesics inside this simple polygon. Here the conditional lower bounds do not apply; Efrat et al. (2002) were able to give a near-linear time 2-approximation algorithm. In this paper, we significantly improve upon their result: we present a (1+ε)-approximation algorithm, for any ε > 0, that runs in 𝒪(1/(ε) (n+m log n) log nm log 1/(ε)) time for a simple polygon bounded by two curves with n and m vertices, respectively. To do so, we show how to compute the reachability of specific groups of points in the free space at once, by interpreting the free space as one between separated one-dimensional curves. We solve this one-dimensional problem in near-linear time, generalizing a result by Bringmann and Künnemann (2015). Finally, we give a linear time exact algorithm if the two curves bound a convex polygon.

Cite as

Thijs van der Horst, Marc van Kreveld, Tim Ophelders, and Bettina Speckmann. The Geodesic Fréchet Distance Between Two Curves Bounding a Simple Polygon. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhorst_et_al:LIPIcs.ESA.2025.35,
  author =	{van der Horst, Thijs and van Kreveld, Marc and Ophelders, Tim and Speckmann, Bettina},
  title =	{{The Geodesic Fr\'{e}chet Distance Between Two Curves Bounding a Simple Polygon}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{35:1--35:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.35},
  URN =		{urn:nbn:de:0030-drops-245038},
  doi =		{10.4230/LIPIcs.ESA.2025.35},
  annote =	{Keywords: Fr\'{e}chet distance, approximation, geodesic, simple polygon}
}

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