A Dynamic Data Structure for Maintaining Disjoint Paths Information in Digraphs

  • Torsten Tholey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2906)

Abstract

In this paper we present the first dynamic data structure for testing – in constant time – the existence of two edge- or quasi-internally vertex-disjoint paths p1 from s to t1 and p2 from s to t2 for any three given vertices s,t1, and t2 of a digraph. By quasi-internally vertex-disjoint we mean that no inner vertex of p1 appears on p2 and vice versa. Moreover, for two vertices s and t, the data structure supports the output of all vertices and all edges whose removal would disconnect s and t in a time linear in the size of the output. The update operations consist of edge insertions and edge deletions, where the implementation of edge deletions will be given only in the full version of this paper. The update time after an edge deletion is competitive with the reconstruction of a static data structure for testing the existence of disjoint paths in constant time, whereas our data structure performs much better in the case of edge insertions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahuja, R.K., Goldberg, A.V., Orlin, J.B., Tarjan, R.E.: Finding minimum-cost flows by double scaling. Math. Programming, Series A 53, 243–266 (1992)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Demetrescu, C., Italiano, G.F.: Fully dynamic transitive closure: Breaking through the O(n 2) barrier. In: Proc. 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2000), pp. 381–389 (2000)Google Scholar
  4. 4.
    Demetrescu, C., Italiano, G.F.: Fully dynamic all pairs shortest paths with real edge weights. In: Proc. 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2001), pp. 260–267 (2001)Google Scholar
  5. 5.
    Demetrescu, C., Italiano, G.F.: Improved bounds and new trade-offs for dynamic all pairs shortest paths. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 633–643. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theoretical Computer Science 10, 111–121 (1980)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13, 338–355 (1984)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48, 723–760 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Lee, S.-W., Wu, C.-S.: A K-best paths algorithm for highly reliable communication networks. IEICE Trans. Commun. E82-B, 586–590 (1999)Google Scholar
  10. 10.
    King, V.: Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In: Proc. 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1999), pp. 81–89 (1999)Google Scholar
  11. 11.
    King, V., Thorup, M.: A space saving trick for directed dynamic transitive closure and shortest path algorithms. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 268–277. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. 12.
    Suurballe, J.W., Tarjan, R.E.: A quick method for finding shortest pairs of disjoint paths. Networks 14, 325–336 (1984)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: Proc. 32nd Annual ACM Symposium on Theory of Computing (STOC 2000), pp. 343–350 (2000)Google Scholar
  14. 14.
    Thorup, M.: Fully-dynamic min-cut. In: Proc. 33rd Annual ACM Symposium on Theory of Computing (STOC 2001), pp. 224–230 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Torsten Tholey
    • 1
  1. 1.Institut für InformatikJohann Wolfgang Goethe-Universität FrankfurtFrankfurt am MainGermany

Personalised recommendations