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.

翻译：给定两个总质量相等的分布 $P$ 和 $S$，推土机距离用于衡量将一个分布转化为另一个分布所需的代价，其中移动单位质量的质量的代价等于其移动距离。我们给出了各种几何对象集之间推土机距离的近似算法。当 $P$ 为加权点集且 $S$ 为线段集、三角形集或 $d$ 维单纯形集时，我们给出了一个 $(1 + \varepsilon)$-近似算法。当 $P$ 和 $S$ 均为线段集、三角形集或单纯形集时，我们给出了一个带有小附加项的 $(1 + \varepsilon)$-近似算法。所有算法的时间复杂度均为 $P$ 和 $S$ 规模的多项式级，并且实际计算了运输方案（即质量移动的详细说明），而不仅仅是代价。据我们所知，这是首个针对连续对象（而非离散点）的推土机距离、具有可证近似比的组合算法。