Boundary-to-Boundary Flows in Planar Graphs

  • Glencora Borradaile
  • Anna Harutyunyan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8288)


We give an iterative algorithm for finding the maximum flow between a set of sources and sinks that lie on the boundary of a planar graph. Our algorithm uses only O(n) queries to simple data structures, achieving an O(n logn) running time that we expect to be practical given the use of simple primitives. The only existing algorithm for this problem uses divide and conquer and, in order to achieve an O(n logn) running time, requires the use of the (complicated) linear-time shortest-paths algorithm for planar graphs.


maximum flow multiple terminal planar graphs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bender, M.A., Cole, R., Demaine, E.D., Farach-Colton, M., Zito, J.: Two Simplified Algorithms for Maintaining Order in a List. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 152–164. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Borradaile, G., Harutyunyan, A.: Maximum st-flow in directed planar graphs via shortest paths. In: Lecroq, T., Mouchard, L. (eds.) IWOCA 2013. LNCS, vol. 8288, pp. 423–427. Springer, Heidelberg (2013)Google Scholar
  3. 3.
    Borradaile, G., Klein, P.: An O(n logn) algorithm for maximum st-flow in a directed planar graph. J. of the ACM 56(2), 1–30 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borradaile, G., Klein, P., Mozes, S., Nussbaum, Y., Wulff-Nilsen, C.: Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in Near-Linear Time. In: Proc. FOCS, pp. 170–179 (2011)Google Scholar
  5. 5.
    Borradaile, G.: Exploiting Planarity for Network Flow and Connectivity Problems. PhD thesis, Brown University (2008)Google Scholar
  6. 6.
    Dietz, P., Sleator, D.: Two algorithms for maintaining order in a list. In: Proc. STOC, pp. 365–372 (1987)Google Scholar
  7. 7.
    Ford, C., Fulkerson, D.: Maximal flow through a network. Canadian J. Math. 8, 399–404 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Goldberg, A., Harrelson, C.: Computing the shortest path: A search meets graph theory. In: Proc. SODA, pp. 156–165 (2005)Google Scholar
  9. 9.
    Hassin, R.: Maximum flow in (s,t) planar networks. IPL 13, 107 (1981)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Henzinger, M., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. JCSS 55(1), 3–23 (1997)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Khuller, S., Naor, J., Klein, P.: The lattice structure of flow in planar graphs. SIAM J. on Disc. Math. 6(3), 477–490 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Miller, G., Naor, J.: Flow in planar graphs with multiple sources and sinks. SIAM J. on Comp. 24(5), 1002–1017 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Whitney, H.: Planar Graphs. Fundamenta Mathematicae 21, 73–84 (1933)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Glencora Borradaile
    • 1
  • Anna Harutyunyan
    • 2
  1. 1.Oregon State UniversityUSA
  2. 2.Vrije Universiteit BrusselBelgium

Personalised recommendations