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

Authors Kyle Fox, Jiashuai Lu

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Kyle Fox
  • Department of Computer Science, The University of Texas at Dallas, TX, USA
Jiashuai Lu
  • Department of Computer Science, The University of Texas at Dallas, TX, USA

Acknowledgements

The authors would like to thank Hsien-Chih Chang for some helpful discussions that took place with the first author at Dagstuhl seminar 19181 "Computational Geometry". We would also like to thank the anonymous reviewers for many helpful comments and suggestions.

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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) https://doi.org/10.4230/LIPIcs.SoCG.2020.45

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Network flows
Keywords
  • Transportation map
  • earth mover’s distance
  • shape matching
  • approximation algorithms

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References

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