53 Search Results for "Staals, Frank"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.33
Visibility Queries in Simple Polygons

Authors: Sujoy Bhore, Chih-Hung Liu, Anurag Murty Naredla, Yakov Nekrich, Eunjin Oh, André van Renssen, Frank Staals, Haitao Wang, and Jie Xue

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

Abstract
Given a simple polygon P with n vertices, we consider the problem of constructing a data structure for visibility queries: for any query point q ∈ P, compute the visibility polygon of q in P. To obtain O(log n + k) query time, where k is the size of the visibility polygon of q, the previous best result requires O(n³) space. In this paper, we propose a new data structure that uses O(n^{2+ε}) space, for any ε > 0, while achieving the same query time. If only O(n²) space is available, the best known result provides O(log² n + k) query time. We improve this to O(log n log log n + k) time. When restricted to o(n²) space, the only previously known approach, aside from the O(n)-time algorithm that computes the visibility polygon without preprocessing, is an O(n)-space data structure that supports O(k log n)-time queries. We construct a data structure using O(n log n) space that answers visibility queries in O(n^{1/2+ε} + k) time. In addition, for the special case in which q lies on the boundary of P, we build a data structure of O(n log n) space supporting O(log² n + k) query time; alternatively, we achieve O(log n + k) query time using O(n^{1+ε}) space. To achieve our results, we propose a new method for decomposing simple polygons, which may be of independent interest.
Cite as

Sujoy Bhore, Chih-Hung Liu, Anurag Murty Naredla, Yakov Nekrich, Eunjin Oh, André van Renssen, Frank Staals, Haitao Wang, and Jie Xue. Visibility Queries in Simple Polygons. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 33:1-33:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.ICALP.2026.33,
  author =	{Bhore, Sujoy and Liu, Chih-Hung and Naredla, Anurag Murty and Nekrich, Yakov and Oh, Eunjin and van Renssen, Andr\'{e} and Staals, Frank and Wang, Haitao and Xue, Jie},
  title =	{{Visibility Queries in Simple Polygons}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{33:1--33:22},
  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.2026.33},
  URN =		{urn:nbn:de:0030-drops-264222},
  doi =		{10.4230/LIPIcs.ICALP.2026.33},
  annote =	{Keywords: simple polygons, visibility polygons, visibility queries, polygon decompositions}
}
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.2026.29
Bichromatic Classifications of Points Using Strips

Authors: Jaegun Lee, Chaeyoon Chung, and Hee-Kap Ahn

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

Abstract
Given a set of n points in the plane, each colored either blue or red, we study the problem of finding a strip that separates the blue points from the red points. Specifically, we consider the following two variants: (1) locating a strip that contains no red points while maximizing the number of blue points within the strip, and (2) locating a strip that contains all blue points while minimizing the number of red points within the strip. For variant (1), we present an O(n²)-time algorithm, improving upon the previously best O(n²log n)-time result. We also show that this running time is optimal under the standard 3SUM conjecture. We also give an output-sensitive algorithm with running time O(k_{opt} n log n) that returns a strip, where k_{opt} is the number of blue points not contained within the strip in an optimal solution. We extend our results to the case of up to t parallel strips, obtaining an O(n²log n)-time algorithm. For variant (2), an optimal Θ(nlog n)-time algorithm is known for t = 1. We show 3SUM-hardness for t = 2 and give an O(n²)-time algorithm. For any t ≥ 3, we present an O(n²log n)-time algorithm.
Cite as

Jaegun Lee, Chaeyoon Chung, and Hee-Kap Ahn. Bichromatic Classifications of Points Using Strips. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{lee_et_al:LIPIcs.SWAT.2026.29,
  author =	{Lee, Jaegun and Chung, Chaeyoon and Ahn, Hee-Kap},
  title =	{{Bichromatic Classifications of Points Using Strips}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{29:1--29:17},
  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.2026.29},
  URN =		{urn:nbn:de:0030-drops-260659},
  doi =		{10.4230/LIPIcs.SWAT.2026.29},
  annote =	{Keywords: Bichromatic Classification, Separation, Strip, Duality}
}
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.2026.35
On Computing the (Exact) Fréchet Distance with a Frog

Authors: Jacobus Conradi, Ivor van der Hoog, and Eva Rotenberg

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

Abstract
The continuous Fréchet distance 𝒟_F(π,σ) between two polygonal curves π and σ is classically computed by exploring the free space diagram over the two curves. [SoCG'25] recently proposed a radically different approach: they approximate 𝒟_F(π,σ) by computing paths in a discrete graph that models a joint traversal of π and σ, recursively bisecting edges until the discrete distance converges to the continuous one. They implement their "frog-based" technique, and claim that it yields substantial practical speedups compared to the state-of-the-art implementations. In this paper, we revisit this technique. We observe that, in its current form, it has three limitations: (i) it does not use exact arithmetic, (ii) its recursive bisection introduces the required monotonicity events to realise the Fréchet distance only in the limit, and (iii) it applies a heuristic simplification technique which is overly conservative. Motivated by theoretical interest, we develop new techniques that guarantee exactness, polynomial-time convergence and near-optimal lossless simplifications. We provide an open-source C++ implementation of our variant. Our primary contribution is an extensive empirical evaluation on a broad, publically available, suite of real-world and synthetic data sets. Among the frog-based variants, exact computation indeed introduces overhead and increases median runtime. Yet, our new approach is often faster in the worst case, worst ten percent, or even the average runtime due to its worst-case convergence guarantees. More surprisingly, the implementation of [SoCG'19] dominates all frog-based implementations in performance - this finding contrasts previously published claims. These results provide a much-needed nuanced perspective on the capabilities and limitations of frog-based techniques: we showcase its theoretical appeal, but highlight its limited practical feasibility.
Cite as

Jacobus Conradi, Ivor van der Hoog, and Eva Rotenberg. On Computing the (Exact) Fréchet Distance with a Frog. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 35:1-35:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{conradi_et_al:LIPIcs.SoCG.2026.35,
  author =	{Conradi, Jacobus and van der Hoog, Ivor and Rotenberg, Eva},
  title =	{{On Computing the (Exact) Fr\'{e}chet Distance with a Frog}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{35:1--35:20},
  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.2026.35},
  URN =		{urn:nbn:de:0030-drops-258414},
  doi =		{10.4230/LIPIcs.SoCG.2026.35},
  annote =	{Keywords: Algorithms engineering, Fr\'{e}chet distance}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.40
Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface

Authors: Loïc Dubois

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

Abstract
Every surface that is intrinsically polyhedral can be represented by a portalgon: a collection of polygons in the Euclidean plane with some pairs of equally long edges abstractly identified. While this representation is arguably simpler than meshes (flat polygons in ℝ³ forming a surface), it has unbounded happiness: a shortest path in the surface may visit the same polygon arbitrarily many times. This pathological behavior is an obstacle towards efficient algorithms. On the other hand, Löffler, Ophelders, Staals, and Silveira [SoCG 2023] recently proved that the (intrinsic) Delaunay triangulations have bounded happiness. In this paper, given a closed polyhedral surface S, represented by a triangular portalgon T, we provide an algorithm to compute the Delaunay triangulation of S whose vertices are the singularities of S (the points whose surrounding angle is distinct from 2π). The time complexity of our algorithm is polynomial in the number of triangles and in the logarithm of the aspect ratio r of T. Within our model of computation, we show that the dependency in log r is unavoidable. Our algorithm can be used to pre-process a triangular portalgon before computing shortest paths on its surface, and to determine whether the surfaces of two triangular portalgons are isometric.
Cite as

Loïc Dubois. Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{dubois:LIPIcs.SoCG.2026.40,
  author =	{Dubois, Lo\"{i}c},
  title =	{{Computing the Intrinsic Delaunay Triangulation of a Closed Polyhedral Surface}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{40:1--40:15},
  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.2026.40},
  URN =		{urn:nbn:de:0030-drops-258460},
  doi =		{10.4230/LIPIcs.SoCG.2026.40},
  annote =	{Keywords: Polyhedral surface, intrinsic Delaunay triangulation, algorithmic complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.45
Line Segment Visibility in Simple Polygons: Exact, Robust, Scalable Computation and Applications

Authors: Sándor P. Fekete, Prahlad Narasimhan Kasthurirangan, Phillip Keldenich, Fabian Kollhoff, Chek-Manh Loi, and Michael Perk

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

Abstract
The weak visibility polygon of a line segment s inside a simple polygon P, denoted by V_P(s), is the region of the polygon that is visible from at least one point on s. Given its fundamental nature in computational geometry, several algorithms have been proposed to compute weak visibility polygons efficiently, each with different trade-offs in terms of preprocessing time, query time, and space complexity. Although there are many applications that require computing these polygons such as computer graphics, robot motion planning, and network communication systems, there is a lack of any implementations of these algorithms in the literature - not to mention one that is exact, robust, and scalable. Furthermore, weak segment visibility polygons are used as basic building blocks in several other algorithms, such as in minimum-link path computation. In this work, we present an implementation of an optimal linear-time algorithm for computing the weak visibility polygon of a segment inside a triangulated simple polygon. Our implementation provides exact, robust geometric primitives and optimizations to handle large inputs with more than 18,000,000 vertices. We demonstrate two concrete applications: (1) construction of window partitions, a standard data structure in visibility algorithms, and (2) support for optimal minimum-link path queries between two points in a simple polygon, the latter serving as a direct use case of the former. Experimental results on a variety of polygon families confirm that the end-to-end running time scales linearly with the size of the polygon and is dominated by the cost of computing the triangulation, validating the practicality and scalability of the approach. The implementation is released as open source in the format of a CGAL package to support reproducibility and further research.
Cite as

Sándor P. Fekete, Prahlad Narasimhan Kasthurirangan, Phillip Keldenich, Fabian Kollhoff, Chek-Manh Loi, and Michael Perk. Line Segment Visibility in Simple Polygons: Exact, Robust, Scalable Computation and Applications. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fekete_et_al:LIPIcs.SoCG.2026.45,
  author =	{Fekete, S\'{a}ndor P. and Kasthurirangan, Prahlad Narasimhan and Keldenich, Phillip and Kollhoff, Fabian and Loi, Chek-Manh and Perk, Michael},
  title =	{{Line Segment Visibility in Simple Polygons: Exact, Robust, Scalable Computation and Applications}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{45:1--45:19},
  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.2026.45},
  URN =		{urn:nbn:de:0030-drops-258516},
  doi =		{10.4230/LIPIcs.SoCG.2026.45},
  annote =	{Keywords: Visibility, line segments, link distance, window partition, computation, implementation, robustness, scalability, exactness, CGAL}
}
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.2026.89
Approximating Convex Hulls via Range Queries

Authors: Thomas Schibler, Jie Xue, and Jiumu Zhu

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

Abstract
Recently, motivated by the rapid increase of the data size in various applications, Monemizadeh [APPROX'23] and Driemel, Monemizadeh, Oh, Staals, and Woodruff [SoCG'25] studied geometric problems in the setting where the only access to the input point set is via querying a range-search oracle. Algorithms in this setting are evaluated on two criteria: (i) the number of queries to the oracle and (ii) the error of the output. In this paper, we continue this line of research and investigate one of the most fundamental geometric problems in the oracle setting, i.e., the convex hull problem. Let P be an unknown set of points in [0,1]^d equipped with a range-emptiness oracle. Via querying the oracle, the algorithm is supposed to output a convex polygon C ⊆ [0,1]^d as an estimation of the convex hull CH(P) of P. The error of the output is defined as the volume of the symmetric difference C ⊕ CH(P) = (C∖CH(P)) ∪ (CH(P)∖C). We prove tight and near-tight tradeoffs between the number of queries and the error of the output for different variants of the problem, depending on the type of the range-emptiness queries and whether the queries are non-adaptive or adaptive. - Orthogonal emptiness queries in d-dimensional space: We show that the minimum error a deterministic algorithm can achieve with q queries is Θ(q^{-1/d}) if the queries are non-adaptive, and Θ(q^{-1/(d-1)}) if the queries are adaptive. In particular, in 2D, the bounds are Θ(1/√q) and Θ(1/q) for non-adaptive and adaptive queries, respectively. - Halfplane emptiness queries in 2D: We show that the minimum error a deterministic algorithm can achieve with q queries is Θ(1/√q) if the queries are non-adaptive, and Θ̃(1/q²) if the queries are adaptive. Here Θ̃(⋅) hides logarithmic factors.
Cite as

Thomas Schibler, Jie Xue, and Jiumu Zhu. Approximating Convex Hulls via Range Queries. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 89:1-89:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{schibler_et_al:LIPIcs.SoCG.2026.89,
  author =	{Schibler, Thomas and Xue, Jie and Zhu, Jiumu},
  title =	{{Approximating Convex Hulls via Range Queries}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{89:1--89:15},
  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.2026.89},
  URN =		{urn:nbn:de:0030-drops-258956},
  doi =		{10.4230/LIPIcs.SoCG.2026.89},
  annote =	{Keywords: convex hull, range searching}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.12
Dynamic and Streaming Algorithms for Union Volume Estimation

Authors: Sujoy Bhore, Karl Bringmann, Timothy M. Chan, and Yanheng Wang

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

Abstract
The union volume estimation problem asks to (1±ε)-approximate the volume of the union of n given objects X₁,…,X_n ⊂ ℝ^d. In their seminal work in 1989, Karp, Luby, and Madras solved this problem in time O(n/ε²) in an oracle model where each object X_i can be accessed via three types of queries: obtain the volume of X_i, sample a random point from X_i, and test whether X_i contains a given point x. This running time was recently shown to be optimal [Bringmann, Larsen, Nusser, Rotenberg, and Wang, SoCG'25]. In another line of work, Meel, Vinodchandran, and Chakraborty [PODS'21] designed algorithms that read the objects in one pass using polylogarithmic time per object and polylogarithmic space; this can be phrased as a dynamic algorithm supporting insertions of objects for union volume estimation in the oracle model. In this paper, we study algorithms for union volume estimation in the oracle model that support both insertions and deletions of objects. We obtain the following results: 1) an algorithm supporting insertions and deletions in polylogarithmic update and query time and linear space (this is the first such dynamic algorithm, even for 2D triangles); 2) an algorithm supporting insertions and suffix queries (which generalizes the sliding window setting) in polylogarithmic update and query time and space; 3) an algorithm supporting insertions and deletions of convex bodies of constant dimension in polylogarithmic update and query time and space.
Cite as

Sujoy Bhore, Karl Bringmann, Timothy M. Chan, and Yanheng Wang. Dynamic and Streaming Algorithms for Union Volume Estimation. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2026.12,
  author =	{Bhore, Sujoy and Bringmann, Karl and Chan, Timothy M. and Wang, Yanheng},
  title =	{{Dynamic and Streaming Algorithms for Union Volume Estimation}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{12:1--12:15},
  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.2026.12},
  URN =		{urn:nbn:de:0030-drops-258180},
  doi =		{10.4230/LIPIcs.SoCG.2026.12},
  annote =	{Keywords: union volume estimation, dynamic algorithms, streaming algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.3
Circle-Segment Intersection Queries in Connected Geometric Graphs

Authors: Peyman Afshani, Yannick Bosch, and Sabine Storandt

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

Abstract
In this paper, we study the problem of efficiently reporting all intersections between a given set of line segments in the plane and a query circle, focusing on the case where the segments form the edges of a connected geometric graph. While previous data structures for circle-segment intersection queries on general segment sets incur high space or query time costs, we exploit the connectivity of the input to obtain significantly improved performance. In fact, we propose a new circle-segment intersection data structure that can be constructed in 𝒪((n + C) log³ n) time and space on connected graphs with n edges and C edge crossings. It answers intersection queries in 𝒪(k log³ n) time, where k denotes the output size. Our method relies on the construction of efficient circle-graph intersection oracles as well as a novel linear-time algorithm to partition the edges of the graph into balanced, connected components, which might be of independent interest. In a proof-of-concept experimental study on real-world road networks, we show that our novel data structure also performs well in practice. Even on networks with millions of edges, the construction time is within minutes and queries are answered in a few milliseconds.
Cite as

Peyman Afshani, Yannick Bosch, and Sabine Storandt. Circle-Segment Intersection Queries in Connected Geometric Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{afshani_et_al:LIPIcs.ISAAC.2025.3,
  author =	{Afshani, Peyman and Bosch, Yannick and Storandt, Sabine},
  title =	{{Circle-Segment Intersection Queries in Connected Geometric Graphs}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{3:1--3:16},
  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.2025.3},
  URN =		{urn:nbn:de:0030-drops-249114},
  doi =		{10.4230/LIPIcs.ISAAC.2025.3},
  annote =	{Keywords: Intersection data structure, Graph partitioning, Dobkin-Kirkpatrick hierarchy}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2025.46
Realizing Metric Spaces with Convex Obstacles

Authors: Sándor Kisfaludi-Bak and Leonidas Theocharous

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

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

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

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@InProceedings{kisfaludibak_et_al:LIPIcs.ISAAC.2025.46,
  author =	{Kisfaludi-Bak, S\'{a}ndor and Theocharous, Leonidas},
  title =	{{Realizing Metric Spaces with Convex Obstacles}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{46:1--46:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.46},
  URN =		{urn:nbn:de:0030-drops-249545},
  doi =		{10.4230/LIPIcs.ISAAC.2025.46},
  annote =	{Keywords: traveling salesman, geodesic distance}
}
Document
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2025.44
Recovering Graphs from Their Witness Unit Square Representation (Poster Abstract)

Authors: Maarten Löffler, Frank Staals, and Soeren Terziadis

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

Abstract
A wUSR of a graph G is a set of unit squares in the plane, one per vertex, if two vertices have an edge in G if their squares overlap and the overlap contains no witness. We present an output sensitive algorithm to compute a graph G based on its given witness unit square representation.
Cite as

Maarten Löffler, Frank Staals, and Soeren Terziadis. Recovering Graphs from Their Witness Unit Square Representation (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 44:1-44:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{loffler_et_al:LIPIcs.GD.2025.44,
  author =	{L\"{o}ffler, Maarten and Staals, Frank and Terziadis, Soeren},
  title =	{{Recovering Graphs from Their Witness Unit Square Representation}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{44:1--44:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.44},
  URN =		{urn:nbn:de:0030-drops-250306},
  doi =		{10.4230/LIPIcs.GD.2025.44},
  annote =	{Keywords: proximity graphs, geometric intersection graphs, witness representation, unit square intersection graph, output sensitive algorithm, range searching}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.24
Fréchet Distance in Unweighted Planar Graphs

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

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

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

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

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@InProceedings{vanderhoog_et_al:LIPIcs.ESA.2025.24,
  author =	{van der Hoog, Ivor and van der Horst, Thijs and Rotenberg, Eva and Wulf, Lasse},
  title =	{{Fr\'{e}chet Distance in Unweighted Planar Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{24:1--24:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.24},
  URN =		{urn:nbn:de:0030-drops-244924},
  doi =		{10.4230/LIPIcs.ESA.2025.24},
  annote =	{Keywords: Fr\'{e}chet distance, planar graphs, lower bounds, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.23
Computing Largest Subsets of Points Whose Convex Hulls Have Bounded Area and Diameter

Authors: Gianmarco Picarella, Marc van Kreveld, Frank Staals, and Sjoerd de Vries

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

Abstract
We study the problem of computing a convex region with bounded area and diameter that contains the maximum number of points from a given point set P. We show that this problem can be solved in O(n⁶k) time and O(n³k) space, where n is the size of P and k is the maximum number of points in the found region. We experimentally compare this new algorithm with an existing algorithm that does the same but without the diameter constraint, which runs in O(n³k) time. For the new algorithm, we use different diameters. We use both synthetic data and data from an application in cancer detection, which motivated our research.
Cite as

Gianmarco Picarella, Marc van Kreveld, Frank Staals, and Sjoerd de Vries. Computing Largest Subsets of Points Whose Convex Hulls Have Bounded Area and Diameter. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{picarella_et_al:LIPIcs.ESA.2025.23,
  author =	{Picarella, Gianmarco and van Kreveld, Marc and Staals, Frank and de Vries, Sjoerd},
  title =	{{Computing Largest Subsets of Points Whose Convex Hulls Have Bounded Area and Diameter}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.23},
  URN =		{urn:nbn:de:0030-drops-244919},
  doi =		{10.4230/LIPIcs.ESA.2025.23},
  annote =	{Keywords: convex polygon, dynamic programming, implementation}
}
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