Given two distributions $P$ and $S$ of equal total mass, the Earth Mover's Distance measures the cost of transforming one distribution into the other, where the cost of moving a unit of mass is equal to the distance over which it is moved. We give approximation algorithms for the Earth Mover's Distance between various sets of geometric objects. We give a $(1 + \varepsilon)$-approximation when $P$ is a set of weighted points and $S$ is a set of line segments, triangles or $d$-dimensional simplices. When $P$ and $S$ are both sets of line segments, sets of triangles or sets of simplices, we give a $(1 + \varepsilon)$-approximation with a small additive term. All algorithms run in time polynomial in the size of $P$ and $S$, and actually calculate the transport plan (that is, a specification of how to move the mass), rather than just the cost. To our knowledge, these are the first combinatorial algorithms with a provable approximation ratio for the Earth Mover's Distance when the objects are continuous rather than discrete points.

中文翻译：

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估算两组几何对象之间地球移动器的距离

给定两个分布$ P $和$ S $相等的总质量，地行者的距离衡量将一个分布转换为另一分布的成本，其中移动一个质量单位的成本等于其移动的距离。我们给出了各种几何对象集之间的地球移动器距离的近似算法。当$ P $是一组加权点而$ S $是一组线段，三角形或$ d $维单形时，我们给出$（1 + \ varepsilon）$近似值。当$ P $和$ S $都是线段集，三角形集或单纯形集时，我们给出一个小加法项的$（1 + \ varepsilon）$近似值。所有算法都以$ P $和$ S $的大小按时间多项式运行，并实际计算出运输计划（即，如何移动质量的规范），而不仅仅是成本。