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 (\(\mathrm{EMDuT}\)) is a translation-invariant version thereof. It minimizes the Earth Mover’s Distance over all translations of one point set.

For \(\mathrm{EMDuT}\) in \(\mathbb{R}^1\), we present an \(\widetilde{\mathcal{O}}(n^2)\)-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 \(\mathrm{EMDuT}\) in \(\mathbb{R}^d\), we present an \(\widetilde{\mathcal{O}}(n^{2d+2})\)-time algorithm for the \(L_1\) and \(L_\infty\) metric. We show that this dependence on \(d\) is asymptotically tight, as an \(n^{o(d)}\)-time algorithm for \(L_1\) or \(L_\infty\) would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.

Corresponding Publications

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

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

Proc. 40th Annual Symposium on Computational Geometry, 2024

To appear.

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@inproceedings{emdut2024,
  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 = {Proc. 40th Annual Symposium on Computational Geometry},
  year = {2024},
  location = {Athens, Greece},
  series = {Leibniz International Proceedings in Informatics (LIPIcs)},
  publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  category = {other},
  note = {To appear.},
}