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

Given two point sets S and T, a limited-capacity many-to-many matching (LCMM) between S and T matches each point p in S (resp. T) to at least 1 and at most Cap(p) points 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 lie on the 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.