11 Search Results for "Klost, Katharina"

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.IPEC.2025.30
A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles II: Vertex and Edge Deletion Numbers

Authors: Jesse Beisegel, Katharina Klost, Kristin Knorr, Fabienne Ratajczak, and Robert Scheffler

Published in: LIPIcs, Volume 358, 20th International Symposium on Parameterized and Exact Computation (IPEC 2025)

Abstract
We consider the problem of finding a Hamiltonian path or cycle with precedence constraints in the form of a partial order on the vertex set. We study the complexity for graph width parameters for which the ordinary problems Hamiltonian Path and Hamiltonian Cycle are in FPT. In particular, we focus on parameters that describe how many vertices and edges have to be deleted to become a member of a certain graph class. We show that the problems are W[1]-hard for such restricted cases as vertex distance to path and vertex distance to clique. We complement these results by showing that the problems can be solved in XP time for vertex distance to outerplanar and vertex distance to block. Furthermore, we present some FPT algorithms, e.g., for edge distance to block. Additionally, we prove para-NP-hardness when considered with the edge clique cover number.
Cite as

Jesse Beisegel, Katharina Klost, Kristin Knorr, Fabienne Ratajczak, and Robert Scheffler. A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles II: Vertex and Edge Deletion Numbers. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{beisegel_et_al:LIPIcs.IPEC.2025.30,
  author =	{Beisegel, Jesse and Klost, Katharina and Knorr, Kristin and Ratajczak, Fabienne and Scheffler, Robert},
  title =	{{A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles II: Vertex and Edge Deletion Numbers}},
  booktitle =	{20th International Symposium on Parameterized and Exact Computation (IPEC 2025)},
  pages =	{30:1--30:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-407-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{358},
  editor =	{Agrawal, Akanksha and van Leeuwen, Erik Jan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2025.30},
  URN =		{urn:nbn:de:0030-drops-251623},
  doi =		{10.4230/LIPIcs.IPEC.2025.30},
  annote =	{Keywords: Hamiltonian path, Hamiltonian cycle, partial order, graph width parameter, parameterized complexity}
}
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.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.
Cite as

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.2025.81
An O(nlog n) Algorithm for Single-Source Shortest Paths in Disk Graphs

Authors: Mark de Berg and Sergio Cabello

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

Abstract
We prove that the single-source shortest-path problem on disk graphs can be solved in O(n log n) expected time, and that it can be solved on intersection graphs of fat triangles in O(n log³ n) time.
Cite as

Mark de Berg and Sergio Cabello. An O(nlog n) Algorithm for Single-Source Shortest Paths in Disk Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 81:1-81:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{deberg_et_al:LIPIcs.ESA.2025.81,
  author =	{de Berg, Mark and Cabello, Sergio},
  title =	{{An O(nlog n) Algorithm for Single-Source Shortest Paths in Disk Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{81:1--81:15},
  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.2025.81},
  URN =		{urn:nbn:de:0030-drops-245494},
  doi =		{10.4230/LIPIcs.ESA.2025.81},
  annote =	{Keywords: shortest path, geometric intersection graph, disk graph, fat triangles}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.7
Single-Source Shortest Path Problem in Weighted Disk Graphs

Authors: Shinwoo An, Eunjin Oh, and Jie Xue

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
In this paper, we present efficient algorithms for the single-source shortest path problem in weighted disk graphs. A disk graph is the intersection graph of a family of disks in the plane. Here, the weight of an edge is defined as the Euclidean distance between the centers of the disks corresponding to the endpoints of the edge. Given a family of n disks in the plane whose radii lie in [1,Ψ] and a source disk, we can compute a shortest path tree from a source vertex in the weighted disk graph in O(nlog² n log Ψ) time. Moreover, in the case that the radii of disks are arbitrarily large, we can compute a shortest path tree from a source vertex in the weighted disk graph in O(nlog⁴ n) time. This improves the best-known algorithm running in O(nlog⁶ n) time presented in ESA'23.
Cite as

Shinwoo An, Eunjin Oh, and Jie Xue. Single-Source Shortest Path Problem in Weighted Disk Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{an_et_al:LIPIcs.SoCG.2025.7,
  author =	{An, Shinwoo and Oh, Eunjin and Xue, Jie},
  title =	{{Single-Source Shortest Path Problem in Weighted Disk Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.7},
  URN =		{urn:nbn:de:0030-drops-231594},
  doi =		{10.4230/LIPIcs.SoCG.2025.7},
  annote =	{Keywords: Disk graphs, shortest path problem, compressed quadtrees}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.18
Finding a Shortest Curve That Separates Few Objects from Many

Authors: Therese Biedl, Éric Colin de Verdière, Fabrizio Frati, Anna Lubiw, and Günter Rote

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We present a fixed-parameter tractable (FPT) algorithm to find a shortest curve that encloses a set of k required objects in the plane while paying a penalty for enclosing unwanted objects. The input is a set of interior-disjoint simple polygons in the plane, where k of the polygons are required to be enclosed and the remaining optional polygons have non-negative penalties. The goal is to find a closed curve that is disjoint from the polygon interiors and encloses the k required polygons, while minimizing the length of the curve plus the penalties of the enclosed optional polygons. If the penalties are high, the output is a shortest curve that separates the required polygons from the others. The problem is NP-hard if k is not fixed, even in very special cases. The runtime of our algorithm is O(3^k n³), where n is the number of vertices of the input polygons. We extend the result to a graph version of the problem where the input is a connected plane graph with positive edge weights. There are k required faces; the remaining faces are optional and have non-negative penalties. The goal is to find a closed walk in the graph that encloses the k required faces, while minimizing the weight of the walk plus the penalties of the enclosed optional faces. We also consider an inverted version of the problem where the required objects must lie outside the curve. Our algorithms solve some other well-studied problems, such as geometric knapsack.
Cite as

Therese Biedl, Éric Colin de Verdière, Fabrizio Frati, Anna Lubiw, and Günter Rote. Finding a Shortest Curve That Separates Few Objects from Many. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{biedl_et_al:LIPIcs.SoCG.2025.18,
  author =	{Biedl, Therese and Colin de Verdi\`{e}re, \'{E}ric and Frati, Fabrizio and Lubiw, Anna and Rote, G\"{u}nter},
  title =	{{Finding a Shortest Curve That Separates Few Objects from Many}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{18:1--18:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.18},
  URN =		{urn:nbn:de:0030-drops-231701},
  doi =		{10.4230/LIPIcs.SoCG.2025.18},
  annote =	{Keywords: Enclosure, curve, separation, weakly simple polygon, Euler tour}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.73
Dominating Set, Independent Set, Discrete k-Center, Dispersion, and Related Problems for Planar Points in Convex Position

Authors: Anastasiia Tkachenko and Haitao Wang

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
Given a set P of n points in the plane, its unit-disk graph G(P) is a graph with P as its vertex set such that two points of P are connected by an edge if their (Euclidean) distance is at most 1. We consider several classical problems on G(P) in a special setting when points of P are in convex position. These problems are all NP-hard in the general case. We present efficient algorithms for these problems under the convex position assumption. ● For the problem of finding the smallest dominating set of G(P), we present an O(knlog n) time algorithm, where k is the smallest dominating set size. We also consider the weighted case in which each point of P has a weight and the goal is to find a dominating set in G(P) with minimum total weight; our algorithm runs in O(n³log² n) time. In particular, for a given k, our algorithm can compute in O(kn²log² n) time a minimum weight dominating set of size at most k (if it exists). ● For the discrete k-center problem, which is to find a subset of k points in P (called centers) for a given k, such that the maximum distance between any point in P and its nearest center is minimized. We present an algorithm that solves the problem in O(min{n^{4/3}log n+knlog² n,k² nlog²n}) time, which is O(n²log² n) in the worst case when k = Θ(n). For comparison, the runtime of the current best algorithm for the continuous version of the problem where centers can be anywhere in the plane is O(n³ log n). ● For the problem of finding a maximum independent set in G(P), we give an algorithm of O(n^{7/2}) time and another randomized algorithm of O(n^{37/11}) expected time, which improve the previous best result of O(n⁶log n) time. Our algorithms can be extended to compute a maximum-weight independent set in G(P) with the same time complexities when points of P have weights. - If we are looking for an (unweighted) independent set of size 3, we derive an algorithm of O(nlog n) time; the previous best algorithm runs in O(n^{4/3}log² n) time (which works for the general case where points of P are not necessarily in convex position). - If points of P have weights and are not necessarily in convex position, we present an algorithm that can find a maximum-weight independent set of size 3 in O(n^{5/3+δ}) time for an arbitrarily small constant δ > 0. By slightly modifying the algorithm, a maximum-weight clique of size 3 can also be found within the same time complexity. ● For the dispersion problem, which is to find a subset of k points from P for a given k, such that the minimum pairwise distance of the points in the subset is maximized. We present an algorithm of O(n^{7/2}log n) time and another randomized algorithm of O(n^{37/11}log n) expected time, which improve the previous best result of O(n⁶) time. - If k = 3, we present an algorithm of O(nlog² n) time and another randomized algorithm of O(nlog n) expected time; the previous best algorithm runs in O(n^{4/3}log² n) time (which works for the general case where points of P are not necessarily in convex position).
Cite as

Anastasiia Tkachenko and Haitao Wang. Dominating Set, Independent Set, Discrete k-Center, Dispersion, and Related Problems for Planar Points in Convex Position. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{tkachenko_et_al:LIPIcs.STACS.2025.73,
  author =	{Tkachenko, Anastasiia and Wang, Haitao},
  title =	{{Dominating Set, Independent Set, Discrete k-Center, Dispersion, and Related Problems for Planar Points in Convex Position}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{73:1--73:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.73},
  URN =		{urn:nbn:de:0030-drops-228982},
  doi =		{10.4230/LIPIcs.STACS.2025.73},
  annote =	{Keywords: Dominating set, k-center, geometric set cover, independent set, clique, vertex cover, unit-disk graphs, convex position, dispersion, maximally separated sets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.23
Long Plane Trees

Authors: Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec

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

Abstract
In the longest plane spanning tree problem, we are given a finite planar point set 𝒫, and our task is to find a plane (i.e., noncrossing) spanning tree T_OPT for 𝒫 with maximum total Euclidean edge length |T_OPT|. Despite more than two decades of research, it remains open if this problem is NP-hard. Thus, previous efforts have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates |T_OPT|. The approximate trees in these algorithms all have small unweighted diameter, typically three or four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree T_ALG with diameter at most four and |T_ALG| ≥ 0.546 ⋅ |T_OPT|. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter d ≥ 3, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most d (compared to a longest plane tree without constraints).
Cite as

Sergio Cabello, Michael Hoffmann, Katharina Klost, Wolfgang Mulzer, and Josef Tkadlec. Long Plane Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cabello_et_al:LIPIcs.SoCG.2022.23,
  author =	{Cabello, Sergio and Hoffmann, Michael and Klost, Katharina and Mulzer, Wolfgang and Tkadlec, Josef},
  title =	{{Long Plane Trees}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{23:1--23:17},
  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.2022.23},
  URN =		{urn:nbn:de:0030-drops-160311},
  doi =		{10.4230/LIPIcs.SoCG.2022.23},
  annote =	{Keywords: geometric network design, spanning trees, plane straight-line graphs, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.49
Dynamic Connectivity in Disk Graphs

Authors: Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth

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

Abstract
Let S ⊆ ℝ² be a set of n planar sites, such that each s ∈ S has an associated radius r_s > 0. Let 𝒟(S) be the disk intersection graph for S. It has vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii r_s, r_t intersect. Our goal is to design data structures that maintain the connectivity structure of 𝒟(S) as sites are inserted and/or deleted. First, we consider unit disk graphs, i.e., r_s = 1, for all s ∈ S. We describe a data structure that has O(log² n) amortized update and O(log n/log log n) amortized query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ r_s ≤ Ψ, for a Ψ ≥ 1 known in advance. In the fully dynamic case, we achieve amortized update time O(Ψ λ₆(log n) log⁷ n) and query time O(log n/log log n), where λ_s(n) is the maximum length of a Davenport-Schinzel sequence of order s on n symbols. In the incremental case, where only insertions are allowed, we get logarithmic dependency on Ψ, with O(α(n)) query time and O(logΨ λ₆(log n) log⁷ n) update time. For the decremental setting, where only deletions are allowed, we first develop an efficient disk revealing structure: given two sets R and B of disks, we can delete disks from R, and upon each deletion, we receive a list of all disks in B that no longer intersect the union of R. Using this, we get decremental data structures with amortized query time O(log n/log log n) that support m deletions in O((nlog⁵ n + m log⁷ n) λ₆(log n) + nlog Ψ log⁴n) overall time for bounded radius ratio Ψ and O((nlog⁶ n + m log⁸n) λ₆(log n)) for arbitrary radii.
Cite as

Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth. Dynamic Connectivity in Disk Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 49:1-49:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kaplan_et_al:LIPIcs.SoCG.2022.49,
  author =	{Kaplan, Haim and Kauer, Alexander and Klost, Katharina and Knorr, Kristin and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul},
  title =	{{Dynamic Connectivity in Disk Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{49:1--49:17},
  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.2022.49},
  URN =		{urn:nbn:de:0030-drops-160572},
  doi =		{10.4230/LIPIcs.SoCG.2022.49},
  annote =	{Keywords: Disk Graphs, Connectivity, Lower Envelopes}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.64
Triangles and Girth in Disk Graphs and Transmission Graphs

Authors: Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
Let S subset R^2 be a set of n sites, where each s in S has an associated radius r_s > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if |st| <= r_s + r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| <= r_s, i.e., if t lies in the disk with center s and radius r_s. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.
Cite as

Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Triangles and Girth in Disk Graphs and Transmission Graphs. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kaplan_et_al:LIPIcs.ESA.2019.64,
  author =	{Kaplan, Haim and Klost, Katharina and Mulzer, Wolfgang and Roditty, Liam and Seiferth, Paul and Sharir, Micha},
  title =	{{Triangles and Girth in Disk Graphs and Transmission Graphs}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{64:1--64:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.64},
  URN =		{urn:nbn:de:0030-drops-111859},
  doi =		{10.4230/LIPIcs.ESA.2019.64},
  annote =	{Keywords: disk graph, transmission graph, triangle, girth}
}
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