# Fine-Grained Complexity of Earth Mover's Distance under Translation

CoRR（2024）

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 ℝ^1, we present an
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 EMDuT in ℝ^d, we
present an 𝒪(n^2d+2)-time algorithm for the L_1 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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