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Volume

LIPIcs, Volume 349

19th International Symposium on Algorithms and Data Structures (WADS 2025)

WADS 2025, August 11-15, 2025, York University, Toronto, Canada

Editors: Pat Morin and Eunjin Oh

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2026.41

Connected Dominating Sets in Triangulations

Authors: Prosenjit Bose, Vida Dujmović, Hussein Houdrouge, Pat Morin, and Saeed Odak

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

Abstract

A dominating set of a graph G is connected if it induces a connected graph in G. For planar triangulations, it has been known since 1990 that every n-vertex triangulation admits a connected dominating set of size at most n/2 - 1, and no improvement to this bound was known for over three decades. We break this longstanding barrier by showing that every n-vertex triangulation has a connected dominating set of size at most 10n/21. Equivalently, every triangulation admits a spanning tree with at least 11n/21 leaves. Moreover, we present an algorithm that computes such a set in optimal linear time. Our result narrows the gap to the best known lower bound and has graph drawing applications, establishing a bound for one-bend free sets and improving the known bound for simultaneous planar embeddings.

Cite as

Prosenjit Bose, Vida Dujmović, Hussein Houdrouge, Pat Morin, and Saeed Odak. Connected Dominating Sets in Triangulations. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 41:1-41:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bose_et_al:LIPIcs.ICALP.2026.41,
  author =	{Bose, Prosenjit and Dujmovi\'{c}, Vida and Houdrouge, Hussein and Morin, Pat and Odak, Saeed},
  title =	{{Connected Dominating Sets in Triangulations}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{41:1--41:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.41},
  URN =		{urn:nbn:de:0030-drops-264300},
  doi =		{10.4230/LIPIcs.ICALP.2026.41},
  annote =	{Keywords: connected domination, triangulations, planar graphs, graph drawing, collinear sets}
}

Document

DOI: 10.4230/LIPIcs.CPM.2026.9

Indexing and Encoding Arrays for Element Distinctness Queries

Authors: Johannes Fischer and Filippo Lari

Published in: LIPIcs, Volume 369, 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)

Abstract

We introduce the data structure variant of the well-known element distinctness problem. Given an array of n elements, the goal is to preprocess the array into a data structure that supports queries asking whether all elements within a given query range are distinct. This has applications in text indexing and possibly also in other algorithmic domains. In the indexing model (where access to the input array is allowed), we design a data structure using O((n log b)/b) bits and answering queries in the time needed to solve an online element distinctness instance of size O(b), for any b ≥ 1. As a concrete instantiation of this, there exists an index that answers queries in O(log log log n) time using O({n log²(log log log n)}/{log log log n}) bits of additional space. Moving to the encoding model (where access to the input array is not allowed), we begin by proving an information-theoretic lower bound for the space usage of 2n-O(log n) bits, and then design a matching encoding with O(1) time queries. We then consider the case in which the alphabet size σ is constant. In this setting, the lower bound can be refined to n log(r_σ) - 3 log(σ+2) + O(1) bits, where r_σ = 4cos²(π/(σ+2)). This lower bound is matched by an encoding with O(1) time queries.

Cite as

Johannes Fischer and Filippo Lari. Indexing and Encoding Arrays for Element Distinctness Queries. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fischer_et_al:LIPIcs.CPM.2026.9,
  author =	{Fischer, Johannes and Lari, Filippo},
  title =	{{Indexing and Encoding Arrays for Element Distinctness Queries}},
  booktitle =	{37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-420-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{369},
  editor =	{Bille, Philip and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2026.9},
  URN =		{urn:nbn:de:0030-drops-259350},
  doi =		{10.4230/LIPIcs.CPM.2026.9},
  annote =	{Keywords: element distinctness, range queries, lower bounds, succinct data structures}
}

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.SWAT.2026.2

On the Fragile Complexity of Geometric Algorithms

Authors: Boris Aronov, Mayank Goswami, John Iacono, and Indu Ramesh

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

Abstract

Surprisingly, the question of bounding the maximum number of operations undergone by each individual element in an algorithm - known as the fragile complexity of the algorithm - has not received much attention. In a foundational paper, Afshani et al. (2019) developed the concept of fragility and explored classic problems such as sorting and selection from this perspective. Motivated by a suggestion for future research by Afshani et al., we initiate a study of fragile complexity in computational geometry. We obtain bounds on several time-honored questions in 2D such as computing the maxima, closest pair, convex hull, triangulation, and approximate Euclidean Minimum Spanning Tree (apx-EMST). Our algorithms for the maxima, convex hull, and triangulation problems are competitive with the classical algorithms in terms of worst-case runtime and guarantee polylogarithmic fragility. We present an O(nlog²n) time algorithm that returns a 1.0125-apx-EMST and achieves O(log² n) fragility, thus matching the best known performance up to polylogarithmic factors.

Cite as

Boris Aronov, Mayank Goswami, John Iacono, and Indu Ramesh. On the Fragile Complexity of Geometric Algorithms. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aronov_et_al:LIPIcs.SWAT.2026.2,
  author =	{Aronov, Boris and Goswami, Mayank and Iacono, John and Ramesh, Indu},
  title =	{{On the Fragile Complexity of Geometric Algorithms}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{2:1--2:18},
  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.2026.2},
  URN =		{urn:nbn:de:0030-drops-260386},
  doi =		{10.4230/LIPIcs.SWAT.2026.2},
  annote =	{Keywords: Fragile complexity, convex hull, maxima, closest pair, algorithmic complexity}
}

Document

DOI: 10.4230/LIPIcs.SWAT.2026.37

Faster Linear-Space Data Structures for Path Frequency Queries

Authors: Ovidiu Rața

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

Abstract

We present linear-space data structures for several frequency queries on trees, namely: path mode, path least frequent element, and path α-minority queries. We present the first linear-space data structures, requiring O(n √{nw}) preprocessing time, that can answer path mode and path least frequent element queries in O(√{n/w}) time. This improves upon the best previously known bound of O(log log n √{n/w}) achieved by Durocher et al. [Durocher et al., 2016] in 2016. For the path α-minority problem, where α is specified at query time, we reduce the query time of the linear-space data structure of Durocher et al. [Durocher et al., 2016] from O(α^{-1}log log n) down to O(α^{-1}) by employing a simple randomized algorithm with a success probability ≥ 1/2. We also present the first linear-space data structure supporting "Path Maximum g-value Color" queries in O(√{n/w}) time, requiring O(n √{nw}) preprocessing time. This general framework encapsulates both path mode and path least frequent element queries. For our data structures, we consider the word-RAM model with w ∈ Ω(log n), where w is the word size in bits.

Cite as

Ovidiu Rața. Faster Linear-Space Data Structures for Path Frequency Queries. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{rata:LIPIcs.SWAT.2026.37,
  author =	{Rața, Ovidiu},
  title =	{{Faster Linear-Space Data Structures for Path Frequency Queries}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{37:1--37:17},
  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.2026.37},
  URN =		{urn:nbn:de:0030-drops-260732},
  doi =		{10.4230/LIPIcs.SWAT.2026.37},
  annote =	{Keywords: Data structure, Range query, Mode, Minority, Least frequent element, Trees, Linear-space, Path query}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.26

Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice

Authors: Guangya Cai

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

Abstract

We study the problem of finding a fair linear scoring function over (numerical) attributes for top-k selection, ensuring fairness through a proportional representation constraint on the protected group. Existing algorithms do not scale efficiently, particularly in higher dimensions. Our hardness analysis shows that in more than two dimensions, no algorithm is likely to scale efficiently with respect to dataset size, and the computational complexity is likely to grow rapidly with dimensionality. However, the hardness results also provide key insights guiding algorithm design, leading to our two-pronged solution: (1) For small k, our analysis reveals a gap in the hardness barrier. By addressing various engineering challenges, including achieving efficient parallelism, we turn this potential of efficiency into an optimized geometry-based algorithm delivering substantial performance gains. (2) For large k, where the hardness is robust, we employ a practically efficient optimization-based algorithm which, despite being theoretically worse, achieves superior real-world performance. Experimental evaluations on real-world datasets then explore scenarios where worst-case behavior does not manifest, identifying areas critical to practical performance. Our solution achieves speedups of up to several orders of magnitude compared to the state of the art, an efficiency made possible through a tight integration of hardness analysis, algorithm design, practical engineering, and empirical evaluation.

Cite as

Guangya Cai. Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cai:LIPIcs.SoCG.2026.26,
  author =	{Cai, Guangya},
  title =	{{Finding a Fair Scoring Function for Top-k Selection: From Hardness to Practice}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{26:1--26:17},
  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.2026.26},
  URN =		{urn:nbn:de:0030-drops-258320},
  doi =		{10.4230/LIPIcs.SoCG.2026.26},
  annote =	{Keywords: Fairness, Top-k, Integration}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.29

Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above

Authors: Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, Hung Le, and Da Wei Zheng

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

Abstract

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case [Duraj et al., 2024; Hsien-Chih Chang et al., 2024; Chan et al., 2025] where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [Karl Bringmann et al., 2022] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter Δ is at least Ω(log n), hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: 1) A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. 2) A truly subquadratic time lower bound for Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be quadratic hard when the diameter is Ω(log n) [Karl Bringmann et al., 2022]. 3) A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D [Karl Bringmann et al., 2022]. 4) A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

Cite as

Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, Hung Le, and Da Wei Zheng. Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{chan_et_al:LIPIcs.SoCG.2026.29,
  author =	{Chan, Timothy M. and Chang, Hsien-Chih and Gao, Jie and Kisfaludi-Bak, S\'{a}ndor and Le, Hung and Zheng, Da Wei},
  title =	{{Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{29:1--29:15},
  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.2026.29},
  URN =		{urn:nbn:de:0030-drops-258357},
  doi =		{10.4230/LIPIcs.SoCG.2026.29},
  annote =	{Keywords: Graph Diameter, Geometric Intersection Graphs, Unit Ball Graphs}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.43

Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs

Authors: Tsuri Farhana, Omrit Filtser, and Shalev Goldshtein

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

Abstract

We study unlabeled multi-robot motion planning for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Solovey et al. (RSS'15) provide a near-optimal solution assuming that start/target positions must have pairwise distance at least 4, and at least √5≈2.236 from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different trade-offs on the robots-separation and obstacles-separation bounds, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) the robots-separation is 2 2/3 and the obstacles-separation is 1 2/3, or (ii) the robots-separation is ≈3.291 and the obstacles-separation ≈1.354. Additionally, we introduce a different strategy yielding a polynomial-time solution when the robots-separation is only 2, and the obstacles-separation is 3. Finally, we show that without any robots-separation assumption, obstacles-separation of at least 1.5 may be necessary for a solution to exist.

Cite as

Tsuri Farhana, Omrit Filtser, and Shalev Goldshtein. Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{farhana_et_al:LIPIcs.SoCG.2026.43,
  author =	{Farhana, Tsuri and Filtser, Omrit and Goldshtein, Shalev},
  title =	{{Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-Offs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{43:1--43:16},
  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.2026.43},
  URN =		{urn:nbn:de:0030-drops-258495},
  doi =		{10.4230/LIPIcs.SoCG.2026.43},
  annote =	{Keywords: multi-robot motion planning}
}

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.SoCG.2026.18

Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs

Authors: Thomas Bläsius, Emil Dohse, Deborah Haun, and Laura Merker

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

Abstract

Hyperbolic uniform disk graphs (HUDGs) are intersection graphs of disks with some radius r in the hyperbolic plane, where r may be constant or depend on the number of vertices in a family of HUDGs. We show that HUDGs with constant clique number do not admit product structure, i.e., that there is no constant c such that every such graph is a subgraph of H ⊠ P for some graph H of treewidth at most c. This justifies that HUDGs are described as not having a grid-like structure in the literature, and is in contrast to unit disk graphs in the Euclidean plane, whose grid-like structure is evident from the fact that they are subgraphs of the strong product of two paths and a clique of constant size [Dvořák et al., '21, MATRIX Annals]. By allowing H to be any graph of constant treewidth instead of a path-like graph, we reject the possibility of a grid-like structure not merely by the maximum degree (which is unbounded for HUDGs) but due to their global structure. We complement this by showing that for every (sub-)constant r, HUDGs admit product structure, whereas the typical hyperbolic behavior is observed if r grows with the number of vertices. Our proof involves a family of n-vertex HUDGs with radius log n that has bounded clique number but unbounded treewidth, and one for which the ratio of treewidth and clique number is log n / log log n. Up to a log log n factor, this negatively answers a question raised by Bläsius et al. [SoCG '25] asking whether balanced separators of HUDGs with radius log n can be covered by less than log n cliques. Our results also imply that the local and layered tree-independence number of HUDGs are both unbounded, answering an open question of Dallard et al. [arXiv '25].

Cite as

Thomas Bläsius, Emil Dohse, Deborah Haun, and Laura Merker. Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{blasius_et_al:LIPIcs.SoCG.2026.18,
  author =	{Bl\"{a}sius, Thomas and Dohse, Emil and Haun, Deborah and Merker, Laura},
  title =	{{Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{18:1--18:17},
  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.2026.18},
  URN =		{urn:nbn:de:0030-drops-258249},
  doi =		{10.4230/LIPIcs.SoCG.2026.18},
  annote =	{Keywords: hyperbolic uniform disk graphs, product structure, treewidth}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.20

The Spanning Ratio of the Directed Θ₆-Graph Is 5

Authors: Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill

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

Abstract

Given a finite set P ⊂ ℝ², the directed Theta-6 graph, denoted Θ₆(P), is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The Θ₆(P)-graph is defined as follows: the plane around each point u ∈ P is partitioned into 6 equiangular cones with apex u, and in each cone, u is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the Θ₆(P)-graph contains an edge from u to v exactly when the interior of ∇_u^v is disjoint from P, where ∇_u^v is the unique equilateral triangle containing u on a corner, v on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the Θ₆(P)-graph is between 4 and 7 in the worst case (Akitaya, Biniaz, and Bose Comput. Geom., 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any Θ_k(P)-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.

Cite as

Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill. The Spanning Ratio of the Directed Θ₆-Graph Is 5. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bose_et_al:LIPIcs.SoCG.2026.20,
  author =	{Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John and Hill, Darryl},
  title =	{{The Spanning Ratio of the Directed \Theta₆-Graph Is 5}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{20:1--20:18},
  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.2026.20},
  URN =		{urn:nbn:de:0030-drops-258268},
  doi =		{10.4230/LIPIcs.SoCG.2026.20},
  annote =	{Keywords: Geometric Spanners, Theta Graphs, Directed Theta Graphs, Spanning Ratio, Computational Geometry}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.15

Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

Authors: Sujoy Bhore, Sándor Kisfaludi‑Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong

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

Abstract

A Euclidean noncrossing Steiner (1+ε)-spanner for a point set P ⊂ ℝ² is a planar straight-line graph that, for any two points a, b ∈ P, contains a path whose length is at most 1+ε times the Euclidean distance between a and b. We construct a Euclidean noncrossing Steiner (1+ε)-spanner with O(n/ε^{3/2}) edges for any set of n points in the plane. This result improves upon the previous best upper bound of O(n/ε⁴) obtained nearly three decades ago. We also establish an almost matching lower bound: There exist n points in the plane for which any Euclidean noncrossing Steiner (1+ε)-spanner has Ω_μ(n/ε^{3/2-μ}) edges for any μ > 0. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.

Cite as

Sujoy Bhore, Sándor Kisfaludi‑Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong. Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.15,
  author =	{Bhore, Sujoy and Kisfaludi‑Bak, S\'{a}ndor and Milenkovi\'{c}, Lazar and T\'{o}th, Csaba D. and W\k{e}grzycki, Karol and Wong, Sampson},
  title =	{{Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{15:1--15:18},
  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.2026.15},
  URN =		{urn:nbn:de:0030-drops-258210},
  doi =		{10.4230/LIPIcs.SoCG.2026.15},
  annote =	{Keywords: geometric network design, spanners, crossing number, incidences}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2026.13

Dynamic Light Spanners in Doubling Metrics

Authors: Sujoy Bhore, Jonathan Conroy, and Arnold Filtser

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

Abstract

A t-spanner of a point set X in a metric space (𝒳, δ) is a graph G with vertex set P such that, for any pair of points u,v ∈ X, the distance between u and v in G is at most t times δ(u,v). We study the problem of maintaining a spanner for a dynamic point set X - that is, when X undergoes a sequence of insertions and deletions - in a metric space of constant doubling dimension. For any constant ε > 0, we maintain a (1+ε)-spanner of P whose total weight remains within a constant factor of the weight of the minimum spanning tree of X. Each update (insertion or deletion) can be performed in poly(log Φ) time, where Φ denotes the aspect ratio of X. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.

Cite as

Sujoy Bhore, Jonathan Conroy, and Arnold Filtser. Dynamic Light Spanners in Doubling Metrics. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.13,
  author =	{Bhore, Sujoy and Conroy, Jonathan and Filtser, Arnold},
  title =	{{Dynamic Light Spanners in Doubling Metrics}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{13:1--13:16},
  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.2026.13},
  URN =		{urn:nbn:de:0030-drops-258193},
  doi =		{10.4230/LIPIcs.SoCG.2026.13},
  annote =	{Keywords: Dynamic data structures, spanners, light-weight, Euclidean metrics, doubling metrics}
}

Document

DOI: 10.4230/OASIcs.NINeS.2026.10

Scalable Routing in a City-Scale Wi-Fi Network for Disaster Recovery

Authors: Ziqian Liu, Om Chabra, James Lynch, Aaron Martin, Chenning Li, and Hari Balakrishnan

Published in: OASIcs, Volume 139, 1st New Ideas in Networked Systems (NINeS 2026)

Abstract

This paper presents CityMesh, a city-scale decentralized mesh network designed for disaster recovery and emergency scenarios. When wide-area Internet connectivity is unavailable or severely degraded, CityMesh leverages both static access points and mobile devices equipped with Wi-Fi to provide intra-city connectivity and reach opportunistic gateways to the Internet (e.g., via satellite links). The main contribution of this paper is a scalable routing protocol that supports millions of devices, addressing a long-standing limitation of wireless mesh and mobile ad hoc networks. Unlike prior approaches, CityMesh exploits rich building-location and building-geometry data from widely available city maps to guide route computation, improving packet delivery while significantly reducing transmission overhead. Simulation results from 70 cities show that CityMesh improves packet delivery rates by 88% over WEAVE (a state-of-the-art geographic routing protocol). A campus-scale deployment of 300 Wi-Fi devices across 31 buildings shows the practical deployability of CityMesh. These results demonstrate the promise of map-aware routing as a foundation for scalable, resilient city-wide Wi-Fi networks.

Cite as

Ziqian Liu, Om Chabra, James Lynch, Aaron Martin, Chenning Li, and Hari Balakrishnan. Scalable Routing in a City-Scale Wi-Fi Network for Disaster Recovery. In 1st New Ideas in Networked Systems (NINeS 2026). Open Access Series in Informatics (OASIcs), Volume 139, pp. 10:1-10:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{liu_et_al:OASIcs.NINeS.2026.10,
  author =	{Liu, Ziqian and Chabra, Om and Lynch, James and Martin, Aaron and Li, Chenning and Balakrishnan, Hari},
  title =	{{Scalable Routing in a City-Scale Wi-Fi Network for Disaster Recovery}},
  booktitle =	{1st New Ideas in Networked Systems (NINeS 2026)},
  pages =	{10:1--10:31},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-414-7},
  ISSN =	{2190-6807},
  year =	{2026},
  volume =	{139},
  editor =	{Argyraki, Katerina and Panda, Aurojit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.NINeS.2026.10},
  URN =		{urn:nbn:de:0030-drops-255954},
  doi =		{10.4230/OASIcs.NINeS.2026.10},
  annote =	{Keywords: mesh networking, disaster recovery, geographic routing, scalability, Wi-Fi}
}

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