Matching on a line

TL;DR

This paper introduces a fast algorithm for optimal matching of points on a line, extending to multiple elements per group, with applications in experimental design and beyond.

Contribution

It presents a new efficient algorithm for nonbipartite matching on a line and extends it to triples, quadruples, and bipartite cases, improving computational speed.

Findings

Algorithm runs in O(n log n) time for matching on a line.

Extends matching methods to triples, quadruples, and bipartite graphs.

Provides practical solutions for experimental design matching problems.

Abstract

Matching is a method of the design of experiments. If we had an even number of patients and wanted to form pairs of patients such that their ages, for example, in each pair be as close as possible, we would use nonbipartite matching. Not only do we present a fast method to do this, we also extend our approach to triples, quadruples, etc. In part 1 a matching algorithm uses kn points on a line as vertices, pairs of vertices as edges, and either absolute values of differences or the squares of differences as weights or distances. It forms n of k-tuples with the minimal sum of distances within each k-tuple in O(n log n) time. In part 2 we present a trivial algorithm for bipartite matching with absolute values or squares of differences as weights and a generalisation to tripartite matching on tripartite graphs.