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Given two distributions $P$ and $S$ of equal total mass, the Earth Mover'sDistance 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 itis moved.  We give approximation algorithms for the Earth Mover's Distance betweenvarious sets of geometric objects. We give a $(1 + \varepsilon)$-approximationwhen $P$ 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 linesegments, sets of triangles or sets of simplices, we give a $(1 +\varepsilon)$-approximation with a small additive term. All algorithms run intime polynomial in the size of $P$ and $S$, and actually calculate thetransport plan (that is, a specification of how to move the mass), rather thanjust the cost. To our knowledge, these are the first combinatorial algorithmswith a provable approximation ratio for the Earth Mover's Distance when theobjects are continuous rather than discrete points.

Approximating the Earth Mover's Distance between sets of geometric objects