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Fine-Grained Complexity of Earth Mover’s Distance Under Translation

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

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  • Theory of computation → Computational geometry
Keywords
  • Earth Mover’s Distance
  • Earth Mover’s Distance under Translation
  • Fine-Grained Complexity
  • Maximum Weight Bipartite Matching

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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) https://doi.org/10.4230/LIPIcs.SoCG.2024.25

Author Details

Karl Bringmann
  • Saarland University and Max-Planck-Institute for Informatics, Saarbrücken, Germany
Frank Staals
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Karol Węgrzycki
  • Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany
Geert van Wordragen
  • Department of Computer Science, Aalto University, Espoo, Finland

Funding

Karl Bringmann and Karol Węgrzycki: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 850979).

Acknowledgements

This work was initiated at the Workshop on New Directions in Geometric Algorithms, May 14-19 2023, Utrecht, The Netherlands.

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