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 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).
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This work was supported in part by the National Science Foundation Grant NSF-MCS79-21024 and the Office of Naval Research Contract N00014-80-C-0647.
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References
A. Aho, J. Hopcroft, J. Ullman, The Design and Analysis of Computer Algorithms, Addison Wesley, Reading, Mass. (1974).
C. Berge and A. Ghouila-Honri, Programming, Games, and Transportation Networks, Methuen, Agincourt, Ontario, 1965.
P. van Emde Boas, R. Kaas, E. Zijlstra, "Design and implementation of an efficient priority queue," Mathematical Systems Theory, 10, pp. 99–127 (1977).
G. Cheston, R. Probert, C. Saxton, "Fast algorithms for determination of connectivity sets for Planar graphs," Univ. Saskatchewant, Dept. Comp. Science, Dec. 1977.
E. Dijkstra, "A note on two problems in connections with graphs," Numerische Mathematik, 1, pp. 269–271 (1959).
S. Even and R. Tarjan, "Network flow and testing graph connectivity," SIAM J. Computing, Vol. 4, No. 4, pp. 507–518 (Dec. 1975).
C. Ford and D. Fulkerson, "Maximal flow through a network," Canadian J. Math., 8, pp. 399–404 (1956).
C. Ford and D. Fulkerson, Flows in Networks, Princeton University Press, Princeton, N.J., 1962.
Z. Galil and A. Naamad, "Network flow and generalized path compression," Proceedings of Symposium of Theory of Computing, Atlanta, Georgia, 1979.
R. Gomory and T. Hu, "Multi-terminal network flows," SIAM J. Appl. Math., pp. 551–570 (1961).
A. Itai and Y. Shiloach, "Maximum flow in planar networks," SIAM J. of Computing, Vol. 8, No. 2, pp. 135–150 (May 1979).
Y. Shiloach, "An O(n I·log2I) maximum-flow algorithm," Comp. Science Dept., Stanford Univ., Stanford, Cal. (Dec. 1978).
Y. Shiloach, "A multi-terminal minimum cut algorithm for planar graphs," SIAM J. Computing, Vol. 9, No. 2, pp. 214–219 (May 1980).
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© 1981 Springer-Verlag Berlin Heidelberg
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Reif, J.H. (1981). Minimum s-t cut of a planar undirected network in o(n log2(n)) time. In: Even, S., Kariv, O. (eds) Automata, Languages and Programming. ICALP 1981. Lecture Notes in Computer Science, vol 115. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-10843-2_5
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DOI: https://doi.org/10.1007/3-540-10843-2_5
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