An O(n log n) Average Time Algorithm for Computing the Shortest Network under a Given Topology

Abstract. In 1992 F. K. Hwang and J. F. Weng published an O(n2) time algorithm for computing the shortest network under a given full Steiner topology interconnecting n fixed points in the Euclidean plane. The Hwang—Weng algorithm can be used to improve substantially existing algorithms for the Steiner minimum tree problem because it reduces the number of different Steiner topologies to be considered dramatically. In this paper we present an improved Hwang—Weng algorithm. While the worst-case time complexity of our algorithm is still O(n2) , its average time complexity over all the full Steiner topologies interconnecting n fixed points is O (n log n ).

[1]  Denton E. Hewgill,et al.  Exact Computation of Steiner Minimal Trees in the Plane , 1986, Inf. Process. Lett..

[2]  Q. Feng,et al.  On better heuristic for Euclidean Steiner minimum trees , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[3]  H. Pollak,et al.  Steiner Minimal Trees , 1968 .

[4]  Z. A. Melzak.  On the Problem of Steiner , 1961, Canadian Mathematical Bulletin.

[5]  Frank K. Hwang,et al.  The Shortest Network under a Given Topology , 1992, J. Algorithms.

[6]  F. Hwang,et al.  An improved algorithm for steiner trees , 1990 .

[7]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean TSP and other geometric problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[8]  Pawel Winter,et al.  An algorithm for the steiner problem in the euclidean plane , 1985, Networks.

[9]  David S. Johnson,et al.  The Complexity of Computing Steiner Minimal Trees , 1977 .

[10]  F. Hwang,et al.  A linear time algorithm for full steiner trees , 1986 .