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

arXiv: https://arxiv.org/abs/2403.04356 · HTML: https://mlsanity.com/paper/2403.04356/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Mar 2024.

## Built on by (most cited first, 1 of 1)

- [APML: Adaptive Probabilistic Matching Loss for Robust 3D Point Cloud Reconstruction](https://mlsanity.com/paper/2509.08104/) (2025)

## Abstract (arXiv)

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 tildeO(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 tildeO(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.