In the minimum-weight many-to-many point matching problem, we are given a set R of red points and a set B of blue points in the plane, of total size N, and we want to pair up each point in R to one or more points in B and vice versa so that the sum of distances between the paired points is minimized. This problem can be solved in \(O(N^3)\) time by using a reduction to the minimum-weight perfect matching problem, and thus, it is not fast enough to be used for on-line systems where a large number of tunes need to be compared. Motivated by similarity problems in music theory, in this paper we study several constrained minimum-weight many-to-many point matching problems in which the allowed pairings are given by geometric restrictions, i.e., a bichromatic pair can be matched if and only if the corresponding points satisfy a specific condition of closeness. We provide algorithms to solve these constrained versions in O(N) time when the sets R and B are given ordered by abscissa.

在最小权重多对多点匹配问题中，我们得到平面上的一组红点R和一组蓝点B ，总大小为N ，并且我们希望将R中的每个点配对为B中的一个或多个点，反之亦然，以使配对点之间的距离之和最小化。这个问题可以通过使用归约到最小权重完美匹配问题来在\(O(N^3)\)时间内解决，因此，它不够快，无法用于有大量数据的在线系统需要比较的曲调。受音乐理论中相似性问题的启发，本文研究了几个约束最小权重多对多点匹配问题，其中允许的配对由几何限制给出，即双色对可以匹配当且仅当对应点满足特定的接近条件。当集合R和B按横坐标排序时，我们提供了在O ( N ) 时间内解决这些约束版本的算法。