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

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openAccess
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Conference article in proceedings
Date
2024-06
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Mcode
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Language
en
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Leibniz International Proceedings in Informatics, LIPIcs ; Volume 293
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 Oe(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 Rd, we present an Oe(n2d+2)-time algorithm for the L1 and L∞ metric. We show that this dependence on d is asymptotically tight, as an no(d)-time algorithm for L1 or L∞ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.
Description
Publisher Copyright: © Karl Bringmann, Frank Staals, Karol Węgrzycki, and Geert van Wordragen.
Keywords
Earth Mover’s Distance, Earth Mover’s Distance under Translation, Fine-Grained Complexity, Maximum Weight Bipartite Matching
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Citation
Bringmann, K, Staals, F, Węgrzycki, K & van Wordragen, G 2024, Fine-Grained Complexity of Earth Mover’s Distance Under Translation . in W Mulzer & J M Phillips (eds), 40th International Symposium on Computational Geometry, SoCG 2024 ., 25, Leibniz International Proceedings in Informatics, LIPIcs, vol. 293, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, International Symposium on Computational Geometry, Athens, Greece, 11/06/2024 . https://doi.org/10.4230/LIPIcs.SoCG.2024.25