Comp. Appl. Math. (2017) 36:1389–1402
Computing the Monge–Kantorovich distance
Received: 13 July 2015 / Revised: 25 November 2015 / Accepted: 19 December 2015 /
Published online: 7 January 2016
© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract The Monge–Kantorovich distance gives a metric between probability distributions
on a metric space
and the MK distance is tied to the underlying metric on
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.
Keywords Monge-Kantorovich distance · Quotient norm · Graph quotient · Matrix
pseudo-inverse · Digital image comparison · Earth-mover’s distance
Mathematics Subject Classiﬁcation Primary 60B05; Secondary 60A10 · 90C08
The Monge–Kantorovich Distance is a metric between two probability measures on a metric
. The MK distance is linked to the underlying metric on
and is related to a mass
transportation problem deﬁned by the two distributions. In this paper, we consider this dis-
tance when the metric space is the underlying space of a ﬁnite connected graph, in which
case the MK distance yields the weak-* topology.
Communicated by Jinyun Yuan.
Department of Mathematics and Statistics, Acadia University, Wolfville, Canada