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An O(M log M) Algorithm for Bipartite Matching with Roadmap Distances

An algorithm is presented which produces the minimum cost bipartite matching between two sets of M points each, where the cost of matching two points is proportional to the minimum distance by which a particle could reach one point from the other while constrained to travel on a connected set of curves, or roads. Given any such roadmap, the algorithm obtains O(M log M) total runtime in terms of M, which is the best possible bound in the sense that any algorithm for minimal matching has runtime Omega(M log M). The algorithm is strongly polynomial and is based on a capacity-scaling approach to the [minimum] convex cost flow problem. The result generalizes the known Theta(M log M) complexity of computing optimal matchings between two sets of points on (i) a line segment, and (ii) a circle.