55 Search Results for "O'Rourke, Joseph"

Document
DOI: 10.4230/LIPIcs.SWAT.2026.16
Online Hitting Set for Axis-Aligned Squares

Authors: Minati De, Satyam Singh, and Csaba D. Tóth

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

Abstract
Given a set P of n points in the plane and a sequence of axis-aligned squares that arrive in an online fashion, the online hitting set problem consists of maintaining, by adding new points from P if necessary, a hitting set H ⊆ P, which contains at least one point in every input square that has already arrived. We present an O(log n)-competitive deterministic algorithm for this problem. The competitive ratio is the best possible, apart from constant factors. In fact, this is the first O(log n)-competitive algorithm for the online hitting set problem that works for geometric objects of arbitrary sizes (i.e., unbounded scaling factors) in the plane. We further generalize this result to positive homothets of a polygon with k ≥ 3 vertices in the plane and provide an O(k²log n)-competitive algorithm.
Cite as

Minati De, Satyam Singh, and Csaba D. Tóth. Online Hitting Set for Axis-Aligned Squares. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{de_et_al:LIPIcs.SWAT.2026.16,
  author =	{De, Minati and Singh, Satyam and T\'{o}th, Csaba D.},
  title =	{{Online Hitting Set for Axis-Aligned Squares}},
  booktitle =	{20th Scandinavian Symposium on Algorithm Theory (SWAT 2026)},
  pages =	{16:1--16:18},
  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.2026.16},
  URN =		{urn:nbn:de:0030-drops-260528},
  doi =		{10.4230/LIPIcs.SWAT.2026.16},
  annote =	{Keywords: axis-aligned squares, hitting set, homothets of a polygon, online algorithm}
}
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.2026.1
Covering and Partitioning Complex Objects with Small Pieces

Authors: Anders Aamand, Mikkel Abrahamsen, Reilly Browne, Mayank Goswami, Prahlad Narasimhan Kasthurirangan, Linda Kleist, Joseph S. B. Mitchell, Valentin Polishchuk, and Jack Stade

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

Abstract
We study the problems of covering or partitioning a polygon P (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to write P as a union of small pieces, and in partitioning, we furthermore require the pieces to be pairwise interior-disjoint. We show that these problems are in fact equivalent: Optimum covers and partitions have the same number of pieces. For covering, a natural local search algorithm repeatedly attempts to replace k pieces from a candidate cover with k-1 pieces. In two dimensions and for sufficiently large k, we show that when no such swap is possible, the cover is a 1+ O(1/√k) approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a 13-approximation that only works for polygons without holes [Abrahamsen and Rasmussen, SODA 2025]. In contrast, in the three dimensional version of the problem, for a polyhedron P of complexity n, we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in n, even if P is simple, i.e., has genus 0 and no holes.
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Anders Aamand, Mikkel Abrahamsen, Reilly Browne, Mayank Goswami, Prahlad Narasimhan Kasthurirangan, Linda Kleist, Joseph S. B. Mitchell, Valentin Polishchuk, and Jack Stade. Covering and Partitioning Complex Objects with Small Pieces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aamand_et_al:LIPIcs.SoCG.2026.1,
  author =	{Aamand, Anders and Abrahamsen, Mikkel and Browne, Reilly and Goswami, Mayank and Kasthurirangan, Prahlad Narasimhan and Kleist, Linda and Mitchell, Joseph S. B. and Polishchuk, Valentin and Stade, Jack},
  title =	{{Covering and Partitioning Complex Objects with Small Pieces}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{1:1--1:16},
  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.2026.1},
  URN =		{urn:nbn:de:0030-drops-258077},
  doi =		{10.4230/LIPIcs.SoCG.2026.1},
  annote =	{Keywords: Covering, partitioning, polygon, small piece, PTAS}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.38
On the Size of k-Irreducible Triangulations

Authors: Vincent Delecroix, Oscar Fontaine, and Arnaud de Mesmay

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

Abstract
A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of an orientable surface of genus g has O(k²g) triangles, which is optimal. This is an improvement over the previous best bound k^O(k) g² of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].
Cite as

Vincent Delecroix, Oscar Fontaine, and Arnaud de Mesmay. On the Size of k-Irreducible Triangulations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{delecroix_et_al:LIPIcs.SoCG.2026.38,
  author =	{Delecroix, Vincent and Fontaine, Oscar and de Mesmay, Arnaud},
  title =	{{On the Size of k-Irreducible Triangulations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{38:1--38:16},
  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.2026.38},
  URN =		{urn:nbn:de:0030-drops-258446},
  doi =		{10.4230/LIPIcs.SoCG.2026.38},
  annote =	{Keywords: surface, irreducible triangulation, system of curves, minimal position, systolic geometry}
}
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.
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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.2026.53
Constructing Doppelgängers of Greedy Geometric Spanners in Practice

Authors: Anirban Ghosh

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

Abstract
Greedy geometric spanners are considered to be the gold standard for their near-optimal guarantees in terms of sparsity and total weight. However, their inefficient construction poses significant challenges for large-scale geometric networks, especially for low values of stretch factors (< 2). We present Θ-Greedy, a simple and practical parallel algorithm engineered for constructing doppelgängers of greedy geometric spanners that empirically resemble the greedy spanners in key structural and performance metrics, including average degree, degree, and lightness. Unlike approximate greedy spanners, doppelgängers of greedy spanners are almost indistinguishable from the actual greedy spanners in practice. In our experiments, Θ-Greedy consistently produced greedy spanner doppelgängers across a broad range of synthetic and real-world datasets, offering the first practical alternative to the computationally intensive greedy spanners. Θ-Greedy can construct a 1.1-spanner on a 128K-element uniformly distributed point set in well under 5 minutes. In contrast, Bucketing, the most practical greedy spanner algorithm, takes around 3 hours. For million-sized point sets, Θ-Greedy can run to completion in a few hours, making it much faster than Bucketing, which takes days to finish. In extensive experiments on synthetic and real-world datasets, Θ-Greedy delivered speedups of up to 147x over Bucketing while preserving greedy-like sparsity and weight. For broader uses of the algorithm and reproducibility, we share our engineered C++ code.
Cite as

Anirban Ghosh. Constructing Doppelgängers of Greedy Geometric Spanners in Practice. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ghosh:LIPIcs.SoCG.2026.53,
  author =	{Ghosh, Anirban},
  title =	{{Constructing Doppelg\"{a}ngers of Greedy Geometric Spanners in Practice}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{53:1--53:21},
  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.2026.53},
  URN =		{urn:nbn:de:0030-drops-258599},
  doi =		{10.4230/LIPIcs.SoCG.2026.53},
  annote =	{Keywords: geometric graph, geometric spanners, greedy spanners, algorithm engineering}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.57
Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps

Authors: Petar Hristov, Ingrid Hotz, and Talha Bin Masood

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

Abstract
We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a shared domain, a situation that frequently arises in practical applications. While the Reeb graph has become a standard tool in computer graphics, shape analysis, and scientific visualization, the Reeb space is still in the early stages of adoption. Although several algorithms for computing the Reeb space have been proposed, none offer an implementation that is both exact and efficient, which has substantially limited its practical use. To address this gap, we introduce singular arrange and traverse, a new algorithm built upon the arrange and traverse framework [Hristov et al., 2025]. Our method exploits the fact that, in the bivariate case, only singular edges contribute to the structure of Reeb space, allowing us to ignore many regular edges [Tierny and Carr, 2017]. This observation results in substantial efficiency gains on datasets where most edges are regular, which is common in many numerical simulations of physical systems. We provide an implementation of our method and benchmark it against the original arrange and traverse algorithm, showing performance gains of up to four orders of magnitude on real-world datasets.
Cite as

Petar Hristov, Ingrid Hotz, and Talha Bin Masood. Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hristov_et_al:LIPIcs.SoCG.2026.57,
  author =	{Hristov, Petar and Hotz, Ingrid and Masood, Talha Bin},
  title =	{{Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{57:1--57:17},
  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.2026.57},
  URN =		{urn:nbn:de:0030-drops-258644},
  doi =		{10.4230/LIPIcs.SoCG.2026.57},
  annote =	{Keywords: Computational topology, Reeb graph, Reeb space, Multivariate data, Multifield, Geometric arrangement}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.80
Hardness of High-Dimensional Linear Classification

Authors: Alexander Munteanu, Simon Omlor, and Jeff M. Phillips

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

Abstract
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only O(n^d) and respectively Õ(1/ε^d) upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and k-Sum problems. Our reductions yield matching lower bounds of Ω̃(n^d) and respectively Ω̃(1/ε^d) based on Affine Degeneracy testing, and Ω̃(n^{d/2}) and respectively Ω̃(1/ε^{d/2}) conditioned on k-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.
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Alexander Munteanu, Simon Omlor, and Jeff M. Phillips. Hardness of High-Dimensional Linear Classification. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 80:1-80:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{munteanu_et_al:LIPIcs.SoCG.2026.80,
  author =	{Munteanu, Alexander and Omlor, Simon and Phillips, Jeff M.},
  title =	{{Hardness of High-Dimensional Linear Classification}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{80:1--80:16},
  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.2026.80},
  URN =		{urn:nbn:de:0030-drops-258871},
  doi =		{10.4230/LIPIcs.SoCG.2026.80},
  annote =	{Keywords: Conditional Hardness, k-Sum, Affine Degeneracy, Halfspace Discrepancy, Classification}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.84
The Voronoi Diagram of Four Lines in ℝ³

Authors: Evanthia Papadopoulou and Zeyu Wang

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

Abstract
We consider the Voronoi diagram of lines in ℝ³ under the Euclidean metric, and give a full classification of its structure in the base case of four lines in general position. We first show that the number of vertices in the Voronoi diagram of four lines in general position is always even, between 0 and 8, and all such numbers can be realized. We identify a key structure for the diagram formation, called a twist, which is a pair of consecutive intersections among trisector branches; only two types of twists are possible, so-called full and partial twists. A full twist is a purely local structure, which can be inserted or removed without affecting the rest of the diagram. Assuming no full twists, the nearest and the farthest Voronoi diagrams of four lines, each have 15 distinct topologies, which are in one-to-one correspondence; the two-dimensional faces are all unbounded, and the total number of vertices is at most six. The unbounded features of the farthest diagram, encoded in a two-dimensional spherical map, are also in one-to-one correspondence. The identified topologies are all realizable. Any Voronoi diagram of four lines in general position in ℝ³ can be obtained from one of these topologies by inserting full twists; each twist induces a bounded face of exactly two vertices in both the nearest and farthest diagrams. We obtain the classification by an exhaustive search algorithm using some new structural and combinatorial observations of line Voronoi diagrams.
Cite as

Evanthia Papadopoulou and Zeyu Wang. The Voronoi Diagram of Four Lines in ℝ³. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 84:1-84:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{papadopoulou_et_al:LIPIcs.SoCG.2026.84,
  author =	{Papadopoulou, Evanthia and Wang, Zeyu},
  title =	{{The Voronoi Diagram of Four Lines in \mathbb{R}³}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{84:1--84:17},
  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.2026.84},
  URN =		{urn:nbn:de:0030-drops-258916},
  doi =		{10.4230/LIPIcs.SoCG.2026.84},
  annote =	{Keywords: Voronoi diagram, lines, three dimensions, structural properties}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2026.98
Interactive Uniform Floodlight Illumination and Rotating Rays Voronoi Diagrams (Media Exposition)

Authors: Carlos Alegría, Ioannis Mantas, Marko Savić, and Martin Suderland

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

Abstract
Floodlight illumination problems are art-gallery variants, where a target domain needs to be illuminated by guards, each associated with a field of view. The rotating rays Voronoi diagram is a Voronoi diagram with rays as sites under the angular distance. There is a natural connection of this Voronoi structure with the problem of finding the minimum aperture such that a given set of uniform aperture floodlights illuminates a target domain. In this work we present an interactive visualization software for such problems, supporting different angular distances, namely, oriented and unoriented versions, and for different domains, namely, the plane and simple polygons.
Cite as

Carlos Alegría, Ioannis Mantas, Marko Savić, and Martin Suderland. Interactive Uniform Floodlight Illumination and Rotating Rays Voronoi Diagrams (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 98:1-98:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{alegria_et_al:LIPIcs.SoCG.2026.98,
  author =	{Alegr{\'\i}a, Carlos and Mantas, Ioannis and Savi\'{c}, Marko and Suderland, Martin},
  title =	{{Interactive Uniform Floodlight Illumination and Rotating Rays Voronoi Diagrams}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{98:1--98:7},
  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.2026.98},
  URN =		{urn:nbn:de:0030-drops-259048},
  doi =		{10.4230/LIPIcs.SoCG.2026.98},
  annote =	{Keywords: rotating rays Voronoi diagram, oriented angular distance, unoriented angular distance, Brocard angle, floodlight illumination, coverage problems, visualization software}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2026.101
From Chaos to Continents: Voronoi-Based Procedural Terrain Generation with Hydrology and 3D Visualization (Media Exposition)

Authors: Batsambuu Batbold and Lori Ziegelmeier

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

Abstract
Procedural content generation often employs grid-based methods to create virtual environments. We present a pipeline that utilizes Voronoi diagrams and Lloyd’s Relaxation to construct an irregular mesh for terrain generation. We implement a customizable "Land Anchor" system combined with Perlin noise to determine landmass shapes, distinct from standard radial distribution methods. Furthermore, we simulate hydrology using priority-flood routing on the Voronoi edges and assign biomes via a Gaussian-smoothed Whittaker classification. The full pipeline is exposed through an interactive application that enables real-time parameter tuning and terrain export, and resulting geometric data is extruded in Blender to produce a 3D terrain model.
Cite as

Batsambuu Batbold and Lori Ziegelmeier. From Chaos to Continents: Voronoi-Based Procedural Terrain Generation with Hydrology and 3D Visualization (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 101:1-101:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{batbold_et_al:LIPIcs.SoCG.2026.101,
  author =	{Batbold, Batsambuu and Ziegelmeier, Lori},
  title =	{{From Chaos to Continents: Voronoi-Based Procedural Terrain Generation with Hydrology and 3D Visualization}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{101:1--101:7},
  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.2026.101},
  URN =		{urn:nbn:de:0030-drops-259077},
  doi =		{10.4230/LIPIcs.SoCG.2026.101},
  annote =	{Keywords: Procedural Content Generation, Voronoi Diagrams, Lloyd’s Relaxation, Perlin Noise, Blender}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2026.104
Interactive Visualization and Verification Tools for Tesseract Path Unfoldings (Media Exposition)

Authors: Soham Samanta, Hugo A. Akitaya, Erik Demaine, and Martin Demaine

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

Abstract
This paper introduces interactive software tools for studying 2-face path unfoldings of the tesseract (4D hypercube). We present: (1) an algorithm to verify whether a given 24-omino is a valid path unfolding of the tesseract, (2) a web-based visualization tool for exploring and animating unfolding sequences with smooth 3D interpolation, and (3) a design interface integrated with SVG Painter, with a similar design to Demaine’s SVG Painter, to create custom unfoldings. We demonstrate these tools by designing a geometric font of 36 path unfoldings resembling Latin letters and digits, illustrating the rich diversity and accessibility of tesseract geometry.
Cite as

Soham Samanta, Hugo A. Akitaya, Erik Demaine, and Martin Demaine. Interactive Visualization and Verification Tools for Tesseract Path Unfoldings (Media Exposition). In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 104:1-104:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{samanta_et_al:LIPIcs.SoCG.2026.104,
  author =	{Samanta, Soham and A. Akitaya, Hugo and Demaine, Erik and Demaine, Martin},
  title =	{{Interactive Visualization and Verification Tools for Tesseract Path Unfoldings}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{104:1--104:5},
  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.2026.104},
  URN =		{urn:nbn:de:0030-drops-259104},
  doi =		{10.4230/LIPIcs.SoCG.2026.104},
  annote =	{Keywords: unfolding, polyominoes, tesseract, path unfolding, visualization}
}
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.ITCS.2026.31
Delaunay Triangulations with Predictions

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

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

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

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

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@InProceedings{cabello_et_al:LIPIcs.ITCS.2026.31,
  author =	{Cabello, Sergio and Chan, Timothy M. and Giannopoulos, Panos},
  title =	{{Delaunay Triangulations with Predictions}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{31:1--31:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.31},
  URN =		{urn:nbn:de:0030-drops-253186},
  doi =		{10.4230/LIPIcs.ITCS.2026.31},
  annote =	{Keywords: Delaunay Triangulation, Minimum Spanning Tree, Algorithms with Predictions}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.53
Average Sensitivity of Geometric Algorithms

Authors: Matthijs Ebbens and Yuichi Yoshida

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

Abstract
In modern applications of geometric algorithms, it is often unrealistic to assume that the input representation fully captures all relevant aspects of the problem, because the input data is often large and dynamic. To address this challenge, we consider the notion of average sensitivity, which is defined as the average earth mover’s distance between the output distributions of the algorithm when run on an input and the same input with one point removed, where the average is over removed points and the distance between two outputs is measured using the symmetric difference size. We start by showing that a number of classical problems from computational geometry, in particular the convex hull, Delaunay triangulation, and Voronoi diagram problems, are "simple" from the viewpoint of average sensitivity by proving tight bounds for the average sensitivity of any algorithm for these problems. Then, we continue by constructing an algorithm with low average sensitivity that computes, for any ε > 0, a set of (1/3+ε)n guards for the art gallery problem. This is the main technical contribution of this work, which combines algorithms from computational geometry with results from the theory of local computation algorithms (LCAs) and property testing.
Cite as

Matthijs Ebbens and Yuichi Yoshida. Average Sensitivity of Geometric Algorithms. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{ebbens_et_al:LIPIcs.ITCS.2026.53,
  author =	{Ebbens, Matthijs and Yoshida, Yuichi},
  title =	{{Average Sensitivity of Geometric Algorithms}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{53:1--53:16},
  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.2026.53},
  URN =		{urn:nbn:de:0030-drops-253409},
  doi =		{10.4230/LIPIcs.ITCS.2026.53},
  annote =	{Keywords: Average Sensitivity, Convex Hull, Delaunay Triangulation, Voronoi Diagram, Art Gallery}
}
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