# Computing the Monge–Kantorovich distance

Computing the Monge–Kantorovich distance The Monge–Kantorovich distance gives a metric between probability distributions on a metric space \$\${\mathbb X}\$\$ X and the MK distance is tied to the underlying metric on \$\${\mathbb X}\$\$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational and Applied Mathematics Springer Journals

# Computing the Monge–Kantorovich distance

, Volume 36 (3) – Jan 7, 2016
14 pages

/lp/springer_journal/computing-the-monge-kantorovich-distance-MS69i0zibN
Publisher
Springer International Publishing
Subject
Mathematics; Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematical Applications in the Physical Sciences; Mathematical Applications in Computer Science
ISSN
0101-8205
eISSN
1807-0302
D.O.I.
10.1007/s40314-015-0303-7
Publisher site
See Article on Publisher Site

### Abstract

The Monge–Kantorovich distance gives a metric between probability distributions on a metric space \$\${\mathbb X}\$\$ X and the MK distance is tied to the underlying metric on \$\${\mathbb X}\$\$ 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.

### Journal

Computational and Applied MathematicsSpringer Journals

Published: Jan 7, 2016

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