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.

翻译：推土机距离是计算机科学多个分支中广泛使用的相似性度量，它衡量两个点集之间完美匹配的最小总边长。平移下推土机距离（$\mathrm{EMDuT}$）是其平移不变版本，通过优化一个点集所有平移变换下的推土机距离来实现最小化。针对$\mathbb{R}^1$中的$\mathrm{EMDuT}$问题，我们提出一种$\widetilde{\mathcal{O}}(n^2)$时间复杂度的算法，并通过基于正交向量假设的条件下界证明该算法近乎最优。针对$\mathbb{R}^d$中$L_1$和$L_\infty$度量下的$\mathrm{EMDuT}$问题，我们提出一种$\widetilde{\mathcal{O}}(n^{2d+2})$时间复杂度的算法，并证明其对维度$d$的依赖具有渐近紧致性——若存在$L_1$或$L_\infty$度量下$n^{o(d)}$时间复杂度的算法，将违背指数时间假设（ETH）。在本研究之前，这些问题仅存在近似算法。