Algorithmica

, Volume 61, Issue 1, pp 174–189 | Cite as

Maximum Flow in Directed Planar Graphs with Vertex Capacities

Article

Abstract

In this paper we present an O(nlog n) time algorithm for finding a maximum flow in a directed planar graph, where the vertices are subject to capacity constraints, in addition to the arcs. If the source and the sink are on the same face, then our algorithm can be implemented in O(n) time.

For general (not planar) graphs, vertex capacities do not make the maximum flow problem more difficult, as there is a simple reduction that eliminates vertex capacities. However, this reduction does not preserve the planarity of the graph. The essence of our algorithm is a different reduction that does preserve the planarity and can be implemented in linear time. For the special case of undirected planar graphs, an algorithm with the same time complexity was recently claimed, but we show that it has a flaw.

We also apply our technique to obtain a linear-time algorithm to convert a flow to an acyclic flow, and a linear-time algorithm to find a largest set of vertex-disjoint st paths, in a directed planar graph.

Keywords

Maximum flow Planar graph Vertex capacities Acyclic flow Vertex-disjoint paths 

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References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice Hall, Englewood Cliffs (1993) MATHGoogle Scholar
  2. 2.
    Borradaile, G., Klein, P.: An O(nlog n) algorithm for maximum st-flow in a directed planar graph. J. ACM 56, 1–30 (2009) MathSciNetGoogle Scholar
  3. 3.
    Brandes, U., Wagner, D.: A linear time algorithm for the arc disjoint Menger problem in planar directed graphs. Algorithmica 28, 16–36 (2000) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962) MATHGoogle Scholar
  5. 5.
    Hassin, R.: Maximum flow in (s,t) planar networks. Inf. Process. Lett. 13, 107 (1981) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hassin, R., Johnson, D.B.: An O(nlog 2 n) algorithm for maximum flow in undirected planar networks. SIAM J. Comput. 14, 612–624 (1985) MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Henzinger, M.R., Klein, P., Rao, S., Subramania, S.: Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci. 55, 3–23 (1997) CrossRefMATHGoogle Scholar
  8. 8.
    Itai, A., Shiloach, Y.: Maximum flow in planar networks. SIAM J. Comput. 8, 135–150 (1979) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Johnson, D.B.: Parallel algorithms for minimum cuts and maximum flows in planar networks. J. ACM 34, 950–967 (1987) CrossRefGoogle Scholar
  10. 10.
    Khuller, S., Naor, J.: Flow in planar graphs with vertex capacities. Algorithmica 11, 200–225 (1994) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khuller, S., Naor, J., Klein, P.: The lattice structure of flow in planar graphs. SIAM J. Discrete Math. 63, 477–490 (1993) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Nishizwki, T., Chiba, N.: Planar Graphs: Theory and Algorithms. Ann. Discrete Math., vol. 32. North-Holland, Amsterdam (1988) Google Scholar
  13. 13.
    Ripphausen-Lipa, H., Wagner, D., Weihe, K.: The vertex-disjoint Menger problem in planar graphs. SIAM J. Comput. 26, 331–349 (1997) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26, 362–391 (1983) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Weihe, K.: Edge-disjoint (s,t)-paths in undirected planar graphs in linear time. J. Algorithms 23, 121–138 (1997) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Weihe, K.: Maximum (s,t)-flows in planar networks in O(|V|log |V|)-time. J. Comput. Syst. Sci. 55, 454–476 (1997) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zhang, X., Liang, W., Jiang, H.: Flow equivalent trees in node-edge-capacitated undirected planar graphs. Inf. Process. Lett. 100, 100–115 (2006) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhang, X., Liang, W., Chen, G.: Computing maximum flows in undirected planar networks with both edge and vertex capacities. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 577–586. Springer, Heidelberg (2008) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.The Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael

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