Computing the Monge–Kantorovich distance

Abstract

The Monge–Kantorovich distance gives a metric between probability distributions on a metric space ${\mathbb X}$ and the MK distance is tied to the underlying metric on ${\mathbb X}$. The MK distance (or a closely related metric) has been used in many areas of image comparison and retrieval and thus it is of significant interest to compute it efficiently. In the context of finite discrete measures on a graph, we give a linear time algorithm for trees and reduce the case of a general graph to that of a tree. Next, we give a linear time algorithm which computes an approximation to the MK distance between two finite discrete distributions on any graph. Finally, we extend our results to continuous distributions on graphs and give some very general theoretical results in this context.