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DOI: 10.4230/LIPIcs.SoCG.2020.45

A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread

Authors: Kyle Fox and Jiashuai Lu

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract

The geometric transportation problem takes as input a set of points P in d-dimensional Euclidean space and a supply function μ : P → ℝ. The goal is to find a transportation map, a non-negative assignment τ : P × P → ℝ_{≥ 0} to pairs of points, so the total assignment leaving each point is equal to its supply, i.e., ∑_{r ∈ P} τ(q, r) - ∑_{p ∈ P} τ(p, q) = μ(q) for all points q ∈ P. The goal is to minimize the weighted sum of Euclidean distances for the pairs, ∑_{(p, q) ∈ P × P} τ(p, q) ⋅ ||q - p||₂. We describe the first algorithm for this problem that returns, with high probability, a (1 + ε)-approximation to the optimal transportation map in O(n poly(1 / ε) polylog n) time. In contrast to the previous best algorithms for this problem, our near-linear running time bound is independent of the spread of P and the magnitude of its real-valued supplies.

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Kyle Fox and Jiashuai Lu. A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 45:1-45:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2020.45,
  author =	{Fox, Kyle and Lu, Jiashuai},
  title =	{{A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{45:1--45:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.45},
  URN =		{urn:nbn:de:0030-drops-122034},
  doi =		{10.4230/LIPIcs.SoCG.2020.45},
  annote =	{Keywords: Transportation map, earth mover’s distance, shape matching, approximation algorithms}
}

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A Near-Linear Time Approximation Scheme for Geometric Transportation with Arbitrary Supplies and Spread

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