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Documents authored by van Wordragen, Geert

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DOI: 10.4230/LIPIcs.SoCG.2025.21
Structure and Independence in Hyperbolic Uniform Disk Graphs

Authors: Thomas Bläsius, Jean-Pierre von der Heydt, Sándor Kisfaludi-Bak, Marcus Wilhelm, and Geert van Wordragen

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

Abstract
We consider intersection graphs of disks of radius r in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of r, where very small r corresponds to an almost-Euclidean setting and r ∈ Ω(log n) corresponds to a firmly hyperbolic setting. We observe that larger values of r create simpler graph classes, at least in terms of separators and the computational complexity of the Independent Set problem. First, we show that intersection graphs of disks of radius r in the hyperbolic plane can be separated with 𝒪((1+1/r)log n) cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set S of n points with pairwise distance at least 2r in the hyperbolic plane, the corresponding Delaunay complex has outerplanarity 1+𝒪((log n)/r), which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes. Using this outerplanarity (and treewidth) bound we prove that Independent Set can be solved in n^𝒪(1+(log n)/r) time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, Independent Set is polynomial-time solvable in the firmly hyperbolic setting of r ∈ Ω(log n). Finally, in the case when the disks have ply (depth) at most 𝓁, we give a PTAS for Maximum Independent Set that has only quasi-polynomial dependence on 1/ε and 𝓁. Our PTAS is a further generalization of our exact algorithm.
Cite as

Thomas Bläsius, Jean-Pierre von der Heydt, Sándor Kisfaludi-Bak, Marcus Wilhelm, and Geert van Wordragen. Structure and Independence in Hyperbolic Uniform Disk Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blasius_et_al:LIPIcs.SoCG.2025.21,
  author =	{Bl\"{a}sius, Thomas and von der Heydt, Jean-Pierre and Kisfaludi-Bak, S\'{a}ndor and Wilhelm, Marcus and van Wordragen, Geert},
  title =	{{Structure and Independence in Hyperbolic Uniform Disk Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{21:1--21:16},
  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.2025.21},
  URN =		{urn:nbn:de:0030-drops-231731},
  doi =		{10.4230/LIPIcs.SoCG.2025.21},
  annote =	{Keywords: hyperbolic geometry, unit disk graphs, independent set, treewidth}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.25
Fine-Grained Complexity of Earth Mover’s Distance Under Translation

Authors: Karl Bringmann, Frank Staals, Karol Węgrzycki, and Geert van Wordragen

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
The Earth Mover’s Distance is a popular similarity measure in several branches of computer science. It measures the minimum total edge length of a perfect matching between two point sets. The Earth Mover’s Distance under Translation (EMDuT) is a translation-invariant version thereof. It minimizes the Earth Mover’s Distance over all translations of one point set. For EMDuT in ℝ¹, we present an 𝒪̃(n²)-time algorithm. We also show that this algorithm is nearly optimal by presenting a matching conditional lower bound based on the Orthogonal Vectors Hypothesis. For EMDuT in ℝ^d, we present an 𝒪̃(n^{2d+2})-time algorithm for the L₁ and L_∞ metric. We show that this dependence on d is asymptotically tight, as an n^o(d)-time algorithm for L_1 or L_∞ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.
Cite as

Karl Bringmann, Frank Staals, Karol Węgrzycki, and Geert van Wordragen. Fine-Grained Complexity of Earth Mover’s Distance Under Translation. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2024.25,
  author =	{Bringmann, Karl and Staals, Frank and W\k{e}grzycki, Karol and van Wordragen, Geert},
  title =	{{Fine-Grained Complexity of Earth Mover’s Distance Under Translation}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.25},
  URN =		{urn:nbn:de:0030-drops-199706},
  doi =		{10.4230/LIPIcs.SoCG.2024.25},
  annote =	{Keywords: Earth Mover’s Distance, Earth Mover’s Distance under Translation, Fine-Grained Complexity, Maximum Weight Bipartite Matching}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.68
A Quadtree, a Steiner Spanner, and Approximate Nearest Neighbours in Hyperbolic Space

Authors: Sándor Kisfaludi-Bak and Geert van Wordragen

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
We propose a data structure in d-dimensional hyperbolic space that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces. Using these quadtrees and the L-order we build geometric spanners. Near-linear size (1+ε)-spanners do not exist in hyperbolic spaces, but we create a Steiner spanner that achieves a spanning ratio of 1+ε with O_{d,ε}(n) edges, using a simple construction that can be maintained dynamically. As a corollary we also get a (2+ε)-spanner (in the classical sense) of the same size, where the spanning ratio 2+ε is almost optimal among spanners of subquadratic size. Finally, we show that our Steiner spanner directly provides an elegant solution to the approximate nearest neighbour problem: given a point set P in d-dimensional hyperbolic space we build the data structure in O_{d,ε}(nlog n) time, using O_{d,ε}(n) space. Then for any query point q we can find a point p ∈ P that is at most 1+ε times farther from q than its nearest neighbour in P in O_{d,ε}(log n) time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time O_{d,ε}(log n). This is the first dynamic nearest neighbour data structure in hyperbolic space with proven efficiency guarantees.
Cite as

Sándor Kisfaludi-Bak and Geert van Wordragen. A Quadtree, a Steiner Spanner, and Approximate Nearest Neighbours in Hyperbolic Space. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kisfaludibak_et_al:LIPIcs.SoCG.2024.68,
  author =	{Kisfaludi-Bak, S\'{a}ndor and van Wordragen, Geert},
  title =	{{A Quadtree, a Steiner Spanner, and Approximate Nearest Neighbours in Hyperbolic Space}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{68:1--68:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.68},
  URN =		{urn:nbn:de:0030-drops-200133},
  doi =		{10.4230/LIPIcs.SoCG.2024.68},
  annote =	{Keywords: hyperbolic geometry, Steiner spanner, dynamic approximate nearest neighbours}
}
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