3 Search Results for "Karavelas, Menelaos I."

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.SoCG.2025.46
Non-Euclidean Erdős-Anning Theorems

Authors: David Eppstein

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

Abstract
The Erdős-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter δ, at most O(δ²) points can have integer distances from all three triangle vertices. We prove the same results for any strictly convex distance function on the plane, and analogous results for every two-dimensional complete Riemannian manifold of bounded genus and for geodesic distance on the boundary of every three-dimensional Euclidean convex set. As a consequence, we resolve a 1983 question of Richard Guy on the equilateral dimension of Riemannian manifolds. Our proofs are based on the properties of additively weighted Voronoi diagrams of these distances.
Cite as

David Eppstein. Non-Euclidean Erdős-Anning Theorems. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eppstein:LIPIcs.SoCG.2025.46,
  author =	{Eppstein, David},
  title =	{{Non-Euclidean Erd\H{o}s-Anning Theorems}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{46:1--46:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.46},
  URN =		{urn:nbn:de:0030-drops-231983},
  doi =		{10.4230/LIPIcs.SoCG.2025.46},
  annote =	{Keywords: integer distances, additively weighted Voronoi diagrams, convex distance functions, Riemannian manifolds, geodesic distance}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.81
A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes

Authors: Menelaos I. Karavelas and Eleni Tzanaki

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
We derive tight expressions for the maximum number of k-faces, k=0,...,d-1, of the Minkowski sum, P_1+...+P_r, of r convex d-polytopes P_1,...,P_r in R^d, where d >= 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal [1]. In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f- and h-vector calculus, stellar subdivisions and shellings, and generalizes the methodology used in [10] and [9] for proving upper bounds on the f-vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P_1+...+P_r as a section of the Cayley polytope C of the summands; bounding the k-faces of P_1+...+P_r reduces to bounding the subset of the (k+r-1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds.
Cite as

Menelaos I. Karavelas and Eleni Tzanaki. A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 81-95, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{karavelas_et_al:LIPIcs.SOCG.2015.81,
  author =	{Karavelas, Menelaos I. and Tzanaki, Eleni},
  title =	{{A Geometric Approach for the Upper Bound Theorem for Minkowski Sums of Convex Polytopes}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{81--95},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.81},
  URN =		{urn:nbn:de:0030-drops-51428},
  doi =		{10.4230/LIPIcs.SOCG.2015.81},
  annote =	{Keywords: Convex polytopes, Minkowski sum, upper bound}
}
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