24 Search Results for "Seiferth, Paul"

Document
DOI: 10.4230/LIPIcs.SWAT.2026.36
Exact Subquadratic Algorithm for Many-To-Many Matching on Planar Point Sets with Integer Coordinates

Authors: Seongbin Park and Eunjin Oh

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
In this paper, we study the many-to-many matching problem on planar point sets with integer coordinates: Given two disjoint sets R,B ⊂ [Δ]² with |R|+|B| = n, the goal is to select a set of edges between R and B so that every point is incident to at least one edge and the total Euclidean length is minimized. In the general case that R and B are point sets in the plane, the best-known algorithm for the many-to-many matching problem takes Õ(n²) time. We present an exact Õ(n^{1.5} log Δ) time algorithm for point sets in [Δ]². To the best of our knowledge, this is the first subquadratic exact algorithm for planar many-to-many matching under bounded integer coordinates.
Cite as

Seongbin Park and Eunjin Oh. Exact Subquadratic Algorithm for Many-To-Many Matching on Planar Point Sets with Integer Coordinates. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{park_et_al:LIPIcs.SWAT.2026.36,
  author =	{Park, Seongbin and Oh, Eunjin},
  title =	{{Exact Subquadratic Algorithm for Many-To-Many Matching on Planar Point Sets with Integer Coordinates}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{36:1--36:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.36},
  URN =		{urn:nbn:de:0030-drops-260728},
  doi =		{10.4230/LIPIcs.SWAT.2026.36},
  annote =	{Keywords: Edge cover, many-to-many matching, similarity, geometric matching}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2026.20
Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs

Authors: Jie Gao, Paweł Gawrychowski, Panos Giannopoulos, Wolfgang Mulzer, Satyam Singh, Frank Staals, and Meirav Zehavi

Published in: LIPIcs, Volume 370, 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)

Abstract
A disk graph is the intersection graph of (closed) disks in the plane. We consider the classic problem of finding a maximum clique in a disk graph. For general disk graphs, the complexity of this problem is still open, but for unit disk graphs, it is well known to be in P. The currently fastest algorithm runs in time O(n^{7/3+ o(1)}), where n denotes the number of disks [Jared Espenant et al., 2023; J. Mark Keil and Debajyoti Mondal, 2025]. Moreover, for the case of disk graphs with t distinct radii, the problem has also recently been shown to be in XP. More specifically, it is solvable in time O^*(n^{2t}) [J. Mark Keil and Debajyoti Mondal, 2025]. In this paper, we present algorithms with improved running times by allowing for approximate solutions and by using randomization: [(i)] 1) for unit disk graphs, we give an algorithm that, with constant success probability, computes a (1-ε)-approximate maximum clique in expected time Õ(n/ε²); and 2) for disk graphs with t distinct radii, we give a parameterized approximation scheme that, with a constant success probability, computes a (1-ε)-approximate maximum clique in expected time Õ(f(t)⋅ (1/ε)^{O(t)} ⋅ n), for some (exponential) function f(t).
Cite as

Jie Gao, Paweł Gawrychowski, Panos Giannopoulos, Wolfgang Mulzer, Satyam Singh, Frank Staals, and Meirav Zehavi. Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{gao_et_al:LIPIcs.SWAT.2026.20,
  author =	{Gao, Jie and Gawrychowski, Pawe{\l} and Giannopoulos, Panos and Mulzer, Wolfgang and Singh, Satyam and Staals, Frank and Zehavi, Meirav},
  title =	{{Near-Linear and Parameterized Approximations for Maximum Cliques in Disk Graphs}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-421-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{370},
  editor =	{Fraigniaud, Pierre},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2026.20},
  URN =		{urn:nbn:de:0030-drops-260563},
  doi =		{10.4230/LIPIcs.SWAT.2026.20},
  annote =	{Keywords: Maximum Clique, Disk Graphs, Unit Disk Graphs, FPT Approximation}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.23
Shortest Paths in Geodesic Unit-Disk Graphs

Authors: Bruce W. Brewer and Haitao Wang

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Let S be a set of n points in a polygon P with m vertices. The geodesic unit-disk graph G(S) induced by S has vertex set S and contains an edge between two vertices whenever their geodesic distance in P is at most one. In the weighted version, each edge is assigned weight equal to the geodesic distance between its endpoints; in the unweighted version, every edge has weight 1. Given a source point s ∈ S, we study the problem of computing shortest paths from s to all vertices of G(S). To the best of our knowledge, this problem has not been investigated previously. A naive approach constructs G(S) explicitly and then applies a standard shortest path algorithm for general graphs, but this requires quadratic time in the worst case, since G(S) may contain Ω(n²) edges. In this paper, we give the first subquadratic-time algorithms for this problem. For the weighted case, when P is a simple polygon, we obtain an O(m + n log³ n log² m)-time algorithm. For the unweighted case, we provide an O(m + n log n log² m)-time algorithm for simple polygons, and an O(√n (n+m)log(n+m))-time algorithm for polygons with holes.
Cite as

Bruce W. Brewer and Haitao Wang. Shortest Paths in Geodesic Unit-Disk Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brewer_et_al:LIPIcs.SoCG.2026.23,
  author =	{Brewer, Bruce W. and Wang, Haitao},
  title =	{{Shortest Paths in Geodesic Unit-Disk Graphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{23:1--23:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.23},
  URN =		{urn:nbn:de:0030-drops-258297},
  doi =		{10.4230/LIPIcs.SoCG.2026.23},
  annote =	{Keywords: unit-disk graph, geodesic distance, shortest paths, geodesic Voronoi diagrams, range emptiness queries, dynamic data structures}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.30
Computing the Girth of a Segment Intersection Graph

Authors: Timothy M. Chan and Yuancheng Yu

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We present an algorithm that computes the girth of the intersection graph of n given line segments in the plane in O(n^1.483) expected time. This is the first such algorithm with O(n^{3/2-ε}) running time for a positive constant ε, and makes progress towards an open question posed by Chan (SODA 2023). The main techniques include (i) the usage of recent subcubic algorithms for bounded-difference min-plus matrix multiplication, and (ii) an interesting variant of the planar graph separator theorem. The result extends to intersection graphs of connected algebraic curves or semialgebraic sets of constant description complexity.
Cite as

Timothy M. Chan and Yuancheng Yu. Computing the Girth of a Segment Intersection Graph. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 30:1-30:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2026.30,
  author =	{Chan, Timothy M. and Yu, Yuancheng},
  title =	{{Computing the Girth of a Segment Intersection Graph}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{30:1--30:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.30},
  URN =		{urn:nbn:de:0030-drops-258364},
  doi =		{10.4230/LIPIcs.SoCG.2026.30},
  annote =	{Keywords: Geometric intersection graphs, girth, shortest paths, graph separators, matrix multiplication}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.69
Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain

Authors: Joost van der Laan, Frank Staals, and Lorenzo Theunissen

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We present efficient data structures for approximate nearest neighbor searching and approximate 2-point shortest path queries in a two-dimensional polygonal domain P with n vertices. Our goal is to store a dynamic set of m point sites S in P so that we can efficiently find a site s ∈ S closest to an arbitrary query point q. We will allow both insertions and deletions in the set of sites S. However, as even just computing the distance between an arbitrary pair of points q,s ∈ P requires a substantial amount of space, we allow for approximating the distances. Given a parameter ε > 0, we build an O(n/(ε)log n) space data structure that can compute a 1+ε-approximation of the distance between q and s in O((1/ε²)log n) time. Building on this, we then obtain an O((n+m)/ε log n + m/ε log m) space data structure that allows us to report a site s ∈ S so that the distance between query point q and s is at most (1+ε)-times the distance between q and its true nearest neighbor in O((1/ε²)log n + 1/(ε)log n log m + (1/ε)log² m) time. Our data structure supports updates in O((1/ε²)log n + (1/ε)log n log m + (1/ε)log² m) amortized time.
Cite as

Joost van der Laan, Frank Staals, and Lorenzo Theunissen. Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 69:1-69:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{vanderlaan_et_al:LIPIcs.SoCG.2026.69,
  author =	{van der Laan, Joost and Staals, Frank and Theunissen, Lorenzo},
  title =	{{Approximate Dynamic Nearest Neighbor Searching in a Polygonal Domain}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{69:1--69:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.69},
  URN =		{urn:nbn:de:0030-drops-258769},
  doi =		{10.4230/LIPIcs.SoCG.2026.69},
  annote =	{Keywords: dynamic data structure, nearest neighbor search, polygonal domain}
}
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.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.WADS.2025.27
A QPTAS for Facility Location on Unit Disk Graphs

Authors: Zachary Friggstad, Mohsen Rezapour, Mohammad R. Salavatipour, and Hao Sun

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
We study the classic (Uncapacitated) Facility Location problem on Unit Disk Graphs (UDGs). For a given point set P in the plane, the unit disk graph UDG(P) on P has vertex set P and an edge between two distinct points p, q ∈ P if and only if their Euclidean distance |pq| is at most 1. The weight of the edge pq is equal to their distance |pq|. An instance of {Facility Location} on UDG(P) consists of a set C ⊆ P of clients and a set F ⊆ P of facilities, each having an opening cost f_i. The goal is to pick a subset F' ⊆ F to open while minimizing ∑_{i ∈ F'} f_i + ∑_{v ∈ C} d(v,F'), where d(v,F') is the distance of v to nearest facility in F' through UDG(P). In this paper, we present the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem. While approximation schemes are well-established for facility location problems on sparse geometric graphs (such as planar graphs), there is a lack of such results for dense graphs. Specifically, prior to this study, to the best of our knowledge, there was no approximation scheme for any facility location problem on UDGs in the general setting.
Cite as

Zachary Friggstad, Mohsen Rezapour, Mohammad R. Salavatipour, and Hao Sun. A QPTAS for Facility Location on Unit Disk Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{friggstad_et_al:LIPIcs.WADS.2025.27,
  author =	{Friggstad, Zachary and Rezapour, Mohsen and Salavatipour, Mohammad R. and Sun, Hao},
  title =	{{A QPTAS for Facility Location on Unit Disk Graphs}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.27},
  URN =		{urn:nbn:de:0030-drops-242586},
  doi =		{10.4230/LIPIcs.WADS.2025.27},
  annote =	{Keywords: Facility Location, Unit Disk Graphs, Approximation Algorithms}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.12
Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework

Authors: Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou

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

Abstract
Given a set S of n colored sites, each s ∈ S associated with a distance-to-site function δ_s : ℝ² → ℝ, we consider two distance-to-color functions for each color: one takes the minimum of δ_s for sites s ∈ S in that color and the other takes the maximum. These two sets of distance functions induce two families of higher-order Voronoi diagrams for colors in the plane, namely, the minimal and maximal order-k color Voronoi diagrams, which include various well-studied Voronoi diagrams as special cases. In this paper, we derive an exact upper bound 4k(n-k)-2n on the total number of vertices in both the minimal and maximal order-k color diagrams for a wide class of distance functions δ_s that satisfy certain conditions, including the case of point sites S under convex distance functions and the L_p metric for any 1 ≤ p ≤ ∞. For the L_1 (or, L_∞) metric, and other convex polygonal metrics, we show that the order-k minimal diagram of point sites has O(min{k(n-k), (n-k)²}) complexity, while its maximal counterpart has O(min{k(n-k), k²}) complexity. To obtain these combinatorial results, we extend the Clarkson-Shor framework to colored objects, and demonstrate its application to several fundamental geometric structures, including higher-order color Voronoi diagrams, colored j-facets, and levels in the arrangements of piecewise linear/algebraic curves/surfaces. We also present iterative algorithms to compute higher-order color Voronoi diagrams.
Cite as

Sang Won Bae, Nicolau Oliver, and Evanthia Papadopoulou. Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bae_et_al:LIPIcs.SoCG.2025.12,
  author =	{Bae, Sang Won and Oliver, Nicolau and Papadopoulou, Evanthia},
  title =	{{Higher-Order Color Voronoi Diagrams and the Colorful Clarkson-Shor Framework}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{12:1--12:19},
  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.2025.12},
  URN =		{urn:nbn:de:0030-drops-231647},
  doi =		{10.4230/LIPIcs.SoCG.2025.12},
  annote =	{Keywords: higher-order Voronoi diagrams, color Voronoi diagrams, Hausdorff Voronoi diagrams, colored j-facets, arrangements, Clarkson-Shor technique}
}
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.2025.72
Geometric Bipartite Matching Based Exact Algorithms for Server Problems

Authors: Sharath Raghvendra, Pouyan Shirzadian, and Rachita Sowle

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

Abstract
For any given metric space, obtaining an offline optimal solution to the classical k-server problem can be reduced to solving a minimum-cost partial bipartite matching between two point sets A and B within that metric space. For d-dimensional 𝓁_p metric space, we present an Õ(min{nk, n^{2-1/(2d+1)}log Δ}⋅ Φ(n)) time algorithm for solving this instance of minimum-cost partial bipartite matching; here, Δ represents the spread of the point set, and Φ(n) is the query/update time of a d-dimensional dynamic weighted nearest neighbor data structure. Our algorithm improves upon prior algorithms that require at least Ω(nkΦ(n)) time. The design of minimum-cost (partial) bipartite matching algorithms that make sub-quadratic queries to a weighted nearest-neighbor data structure, even for bounded spread instances, is a major open problem in computational geometry. We resolve this problem at least for the instances that are generated by the offline version of the k-server problem. Our algorithm employs a hierarchical partitioning approach, dividing the points of A∪ B into rectangles. It maintains a partial minimum-cost matching where any point b ∈ B is either matched to another point a ∈ A or to the boundary of the rectangle it is located in. The algorithm involves iteratively merging pairs of rectangles by erasing the shared boundary between them and recomputing the minimum-cost partial matching. This continues until all boundaries are erased and we obtain the desired minimum-cost partial matching of A and B. We exploit geometry in our analysis to show that each point participates in only Õ(n^{1-1/(2d+1)}log Δ) number of augmenting paths, leading to a total execution time of Õ(n^{2-1/(2d+1)}Φ(n)log Δ). We also show that, for the 𝓁₁ norm and d dimensions, any algorithm that can solve instances of the offline n-server problem with an exponential spread in T(n) time can be used to compute minimum-cost bipartite matching in a complete graph defined on two (d-1)-dimensional point sets under the 𝓁₁ norm within T(n) time. This suggests that removing spread from the execution time of our algorithm may be difficult as it immediately results in a sub-quadratic algorithm for bipartite matching under the 𝓁₁ norm.
Cite as

Sharath Raghvendra, Pouyan Shirzadian, and Rachita Sowle. Geometric Bipartite Matching Based Exact Algorithms for Server Problems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 72:1-72:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{raghvendra_et_al:LIPIcs.SoCG.2025.72,
  author =	{Raghvendra, Sharath and Shirzadian, Pouyan and Sowle, Rachita},
  title =	{{Geometric Bipartite Matching Based Exact Algorithms for Server Problems}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{72:1--72:15},
  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.2025.72},
  URN =		{urn:nbn:de:0030-drops-232240},
  doi =		{10.4230/LIPIcs.SoCG.2025.72},
  annote =	{Keywords: Minimum-Cost Bipartite Matching, Server Problems, Primal-Dual Approach}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.32
Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-And-Bifurcate Technique

Authors: Timothy M. Chan and Zhengcheng Huang

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

Abstract
In a series of papers, Avraham, Filtser, Kaplan, Katz, and Sharir (SoCG'14), Kaplan, Katz, Saban, and Sharir (ESA'23), and Katz, Saban, and Sharir (ESA'24) studied a class of geometric optimization problems - including reverse shortest path in unweighted and weighted unit-disk graphs, discrete Fréchet distance with one-sided shortcuts, and reverse shortest path in visibility graphs on 1.5-dimensional terrains - for which standard parametric search does not work well due to a lack of efficient parallel algorithms for the corresponding decision problems. The best currently known algorithms for all the above problems run in O^*(n^{6/5}) = O^*(n^{1.2}) time (ignoring subpolynomial factors), and they were obtained using a technique called shrink-and-bifurcate. We improve the running time to Õ(n^{8/7}) ≈ O(n^{1.143}) for these problems. Furthermore, specifically for reverse shortest path in unweighted unit-disk graphs, we improve the running time further to Õ(n^{9/8}) = Õ(n^{1.125}).
Cite as

Timothy M. Chan and Zhengcheng Huang. Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-And-Bifurcate Technique. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2025.32,
  author =	{Chan, Timothy M. and Huang, Zhengcheng},
  title =	{{Faster Algorithms for Reverse Shortest Path in Unit-Disk Graphs and Related Geometric Optimization Problems: Improving the Shrink-And-Bifurcate Technique}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{32:1--32:14},
  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.2025.32},
  URN =		{urn:nbn:de:0030-drops-231845},
  doi =		{10.4230/LIPIcs.SoCG.2025.32},
  annote =	{Keywords: Geometric optimization problems, parametric search, shortest path, disk graphs, Fr\'{e}chet distance, visibility, distance selection, randomized algorithms}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.59
Incremental Planar Nearest Neighbor Queries with Optimal Query Time

Authors: John Iacono and Yakov Nekrich

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

Abstract
In this paper we show that two-dimensional nearest neighbor queries can be answered in optimal O(log n) time while supporting insertions in O(log^{1+ε} n) time. No previous data structure was known that supports O(log n)-time queries and polylog-time insertions. In order to achieve logarithmic queries our data structure uses a new technique related to fractional cascading that leverages the inherent geometry of this problem. Our method can be also used in other semi-dynamic scenarios.
Cite as

John Iacono and Yakov Nekrich. Incremental Planar Nearest Neighbor Queries with Optimal Query Time. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 59:1-59:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{iacono_et_al:LIPIcs.SoCG.2025.59,
  author =	{Iacono, John and Nekrich, Yakov},
  title =	{{Incremental Planar Nearest Neighbor Queries with Optimal Query Time}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{59:1--59:15},
  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.2025.59},
  URN =		{urn:nbn:de:0030-drops-232117},
  doi =		{10.4230/LIPIcs.SoCG.2025.59},
  annote =	{Keywords: Data Structures, Dynamic Data Structures, Nearest Neighbor Queries}
}
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