Minimum s-t cut of a planar undirected network in o(n log2(n)) time

  • John H. Reif
Session 2: F.P. Preparata, Chairman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 115)


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 log2(n)) time on a (uniform cost criteria) RAM. For the case L contains only integers ≤n0(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(n2 log(n)) and the best previous time bound for minimum s-t cut of a planar graph (Cheston, Probert, and Saxton, 1977] was O(n2).


Planar Graph Multiple Edge Planar Network Recursive Call Dual Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. A. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Computer Algorithms, Addison Wesley, Reading, Mass. (1974).Google Scholar
  2. C. Berge and A. Ghouila-Honri, Programming, Games, and Transportation Networks, Methuen, Agincourt, Ontario, 1965.Google Scholar
  3. 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
  4. 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
  5. E. Dijkstra, "A note on two problems in connections with graphs," Numerische Mathematik, 1, pp. 269–271 (1959).Google Scholar
  6. 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
  7. C. Ford and D. Fulkerson, "Maximal flow through a network," Canadian J. Math., 8, pp. 399–404 (1956).Google Scholar
  8. C. Ford and D. Fulkerson, Flows in Networks, Princeton University Press, Princeton, N.J., 1962.Google Scholar
  9. Z. Galil and A. Naamad, "Network flow and generalized path compression," Proceedings of Symposium of Theory of Computing, Atlanta, Georgia, 1979.Google Scholar
  10. R. Gomory and T. Hu, "Multi-terminal network flows," SIAM J. Appl. Math., pp. 551–570 (1961).Google Scholar
  11. 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
  12. Y. Shiloach, "An O(n I·log2I) maximum-flow algorithm," Comp. Science Dept., Stanford Univ., Stanford, Cal. (Dec. 1978).Google Scholar
  13. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • John H. Reif
    • 1
  1. 1.Aiken Computation LaboratoryHarvard UniversityCambridgeUSA

Personalised recommendations