A faster algorithm for the limited-capacity many-to-many point matching in one dimension

TL;DR

This paper introduces the first linear-time algorithm for solving the minimum-cost limited-capacity many-to-many point matching problem on a line, significantly improving efficiency over previous quadratic-time solutions.

Contribution

The paper presents a novel linear-time algorithm for one-dimensional LCMM, advancing the computational efficiency for this class of matching problems.

Findings

Achieved linear time complexity for 1D LCMM

Reduced computational time compared to previous algorithms

Validated the algorithm's effectiveness on example datasets

Abstract

Given two sets S and T, a limited-capacity many-to-many matching (LCMM) between S and T matches each element p in S (resp. T) to at least 1 and at most Cap(p) elements in T (resp. S), where the function Cap:S\cup T-> Z>0 denotes the capacity of p. In this paper, we present the first linear time algorithm for finding a minimum-cost one-dimensional LCMM (OLCMM) between S and T, where S and T are points lying on a line, and the cost of matching p in S to q in T equals the l_2 distance between p,q. Our algorithm improves the previous best-known quadratic time algorithm.