The Shortest Network under a Given Topology

Abstract Given a set of fixed points, a set of moving points in the Euclidean plane, and a set of edges connecting these points, the problem we consider is that of locating the moving points so as to minimize the total length of edges, where zero-length edges are allowed. We study the special case where each point has degree at most three and show that it is related to the well-studied Steiner minimal tree problem. We prove that if a Steiner tree (a tree such that no two edges meet at an angle less than 120°) exists with the given topology, then it is the shortest network. We also propose an O(n2) time algorithm, where n is the number of fixed points, for obtaining the Steiner tree. Our algorithm can also be used to substantially improve existing algorithms for the Steiner minimal tree problem.