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Documents authored by van Wordragen, Geert

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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.
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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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