# Minimum s-t cut of a planar undirected network in o(n log^{2}(n)) time

## Abstract

Let N be a planar undirected network with distinguished vertices s, t, a total of n vertices, and each edge labeled with a positive real (the edge's cost) from a set L. This paper presents an algorithm for computing a minimum (cost) s-t cut of N. For general L, this algorithm runs in time O(n log^{2}(n)) time on a (uniform cost criteria) RAM. For the case L contains only integers ≤n^{0(1)}, the algorithm runs in time O(n log(n)loglog(n)). Our algorithm also constructs a minimum s-t cut of a planar graph (i.e., for the case L= {1}) in time O(n log(n)).

The fastest previous algorithm for computing a minimum s-t cut of a planar undirected network [Gomory and Hu, 1961] and [Itai and Shiloach, 1979] has time O(n^{2} log(n)) and the best previous time bound for minimum s-t cut of a planar graph (Cheston, Probert, and Saxton, 1977] was O(n^{2}).

## Keywords

Planar Graph Multiple Edge Planar Network Recursive Call Dual Graph## References

- A. Aho, J. Hopcroft, J. Ullman,
*The Design and Analysis of Computer Algorithms*, Addison Wesley, Reading, Mass. (1974).Google Scholar - C. Berge and A. Ghouila-Honri,
*Programming, Games, and Transportation Networks*, Methuen, Agincourt, Ontario, 1965.Google Scholar - P. van Emde Boas, R. Kaas, E. Zijlstra, "Design and implementation of an efficient priority queue,"
*Mathematical Systems Theory*, 10, pp. 99–127 (1977).Google Scholar - G. Cheston, R. Probert, C. Saxton, "Fast algorithms for determination of connectivity sets for Planar graphs," Univ. Saskatchewant, Dept. Comp. Science, Dec. 1977.Google Scholar
- E. Dijkstra, "A note on two problems in connections with graphs,"
*Numerische Mathematik*, 1, pp. 269–271 (1959).Google Scholar - S. Even and R. Tarjan, "Network flow and testing graph connectivity,"
*SIAM J. Computing*, Vol. 4, No. 4, pp. 507–518 (Dec. 1975).Google Scholar - C. Ford and D. Fulkerson, "Maximal flow through a network,"
*Canadian J. Math.*, 8, pp. 399–404 (1956).Google Scholar - C. Ford and D. Fulkerson,
*Flows in Networks*, Princeton University Press, Princeton, N.J., 1962.Google Scholar - Z. Galil and A. Naamad, "Network flow and generalized path compression,"
*Proceedings of Symposium of Theory of Computing*, Atlanta, Georgia, 1979.Google Scholar - R. Gomory and T. Hu, "Multi-terminal network flows,"
*SIAM J. Appl. Math.*, pp. 551–570 (1961).Google Scholar - A. Itai and Y. Shiloach, "Maximum flow in planar networks,"
*SIAM J. of Computing*, Vol. 8, No. 2, pp. 135–150 (May 1979).Google Scholar - Y. Shiloach, "An O(n I·log
^{2}I) maximum-flow algorithm," Comp. Science Dept., Stanford Univ., Stanford, Cal. (Dec. 1978).Google Scholar - Y. Shiloach, "A multi-terminal minimum cut algorithm for planar graphs,"
*SIAM J. Computing*, Vol. 9, No. 2, pp. 214–219 (May 1980).Google Scholar