Randomized algorithms for the on-line minimum matching problem on euclidean space

Abstract

Suppose we are given two sets R and B, each of n points in the plane. Define the cost of a matching to be the toal distance of the edges in the matching. The minimum matching problem on Euclidean space is to find a complete bipartite matching which has the minimum cost on Euclidean space. In this paper, we are interested in the on-line minimum matching problem on Euclidean space. We assume the set R is known, but the points of set B are revealed one by one. When a point of set B arrives, we must decide what match to make involving the point. None of the matches can be changed after they are made. The on-line minimum matching problem on Euclidean space tries to minimize the cost of the complete bipartite matching that we find. This paper proposes a family of randomized algorithms, Algorithm RM(m) (1  m  n), for solving this problem. When a point in the set B arrives, Algorithm RM(m) randomly chooses at most m unmatched points in the set R, and adds the minimum edge between the arriving point and the chosen points to the matching. In each decision step, the algorithm only runs in O(m) time, which is superior to the known non-randomized algorithms for this problem. In this paper, we show that Algorithm RM(m) is not a competitive on-line algorithm for 1  m  n − 1. However, we further show that if 2n is large, the average cost incurred by Algorithm RM(m) (1  m  n) is bounded by 0(√n) times the average cost of the optimal Euclidean minimum matching.