Abstract
Computing flows in a network is a fundamental graph theory problem with numerous applications. In this paper, we present two algorithms for simplifying a flow network G=(V,E), i.e., detecting and removing from G all edges (and vertices) that have no impact on any source-to-sink flow in G. Such network simplification can reduce the size of the network and hence the amount of computation performed by maximum flow algorithms. For the undirected network case, we present the first linear time algorithm. For the directed network case, we present an O(|E|*(|V|+|E|)) time algorithm, an improvement over the previous best O(|V|+|E|2 log |V|) time solution. Both of our algorithms are quite simple.
This research was supported in part by the National Science Foundation under Grant CCR-9988468 and by a Summer Graduate Research Fellowship of the Center for Applied Mathematics of the University of Notre Dame.
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References
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)
Ahuja, R., Magnanti, T., Orlin, J.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)
Alstrup, S., Harel, D., Lauridsen, P., Thorup, M.: Dominators in linear time. SIAM Journal on Computing 28(6), 2117–2132 (1999)
Biedl, T.C., Brejová, B., Vinař, T.: Simplifying flow networks. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 202–211. Springer, Heidelberg (2000)
Brejová, B., Vinař, T.: Weihe’s algorithm for maximum flow in planar graphs (project report), University of Waterloo, Course CS760K (1999)
Cooper, K., Harvey, T., Kennedy, K.: A simple, fast dominance algorithm. Softw. Pract. Exper. 4, 1–10 (2001)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, McGraw-Hill, 2nd edn. McGraw-Hill, New York (2001)
Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)
Goldberg, A.: Recent developments in maximum flow algorithms, Technical Report #98-045, NEC Research Institute (1998)
Harary, F., Prins, G.: The block-cutpoint-tree of a graph. Publ. Math. Debrecen 13, 103–107 (1966)
Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13, 338–355 (1984)
Lawler, E.L.: Combinatorial Optimization: Networks and Matroids, Holt, Rinehart, and Winston (1976)
Manber, U.: Introduction to Algorithms, A Creative Approach. Addison-Wesley, Reading (1989)
Schieber, B., Vishkin, U.: On finding lowest common ancestors: Simplification and parallelization. SIAM Journal on Computing 17(6), 1253–1262 (1988)
Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. Journal of Computer and System Sciences 26(3), 362–391 (1983)
Weihe, K.: Maximum (s,t)-flows in planar networks in O(|V | log |V |) time. Journal of Computer and System Sciences 55(3), 454–475 (1997)
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Misiołek, E., Chen, D.Z. (2005). Efficient Algorithms for Simplifying Flow Networks. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_75
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DOI: https://doi.org/10.1007/11533719_75
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28061-3
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