Improving the primal-dual algorithm for the transportation problem in the plane

TL;DR

This paper introduces improvements to the primal-dual algorithm for the planar transportation problem, significantly reducing computation time for Euclidean and squared Euclidean distance-based costs, with applications in image analysis.

Contribution

The paper presents novel enhancements to the primal-dual algorithm specifically tailored for the transportation problem in the plane, optimizing performance for Euclidean and squared Euclidean costs.

Findings

Reduced computation time for the transportation problem in the plane.

Effective handling of Euclidean and squared Euclidean distance functions.

Potential applications in digital image analysis.

Abstract

The transportation problem in the plane - how to move a set of objects from one set of points to another set of points in the cheapest way - is a very old problem going back several hundreds of years. In recent years the solution of the problem has found applications in the analysis of digital images when searching for similarities and discrepancies between images. The main drawback, however, is the long computation time for finding the solution. In this paper we present some new results by which the time for solving the transportation problem in the plane can be reduced substantially. As cost-function we choose a distance-function between points in the plane. We consider both the case when the distance-function is equal to the ordinary Euclidean distance, as well as the case when the distance-function is equal to the square of the Euclidean distance. This latter distance-function has the advantage that it is integer-valued if the coordinates of the points in the plane are integers.