Fully-Dynamic All-Pairs Shortest Paths: Faster and Allowing Negative Cycles

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3111)


We present a solution to the fully-dynamic all pairs shortest path problem for a directed graph with arbitrary weights allowing negative cycles. We support each vertex update in \(O(n^2({\rm log} n + {\rm log^2}(\overline{m}/n)))\) amortized time. Here, n is the number vertices, m the number of edges and \(\overline{m} = n + m\). A vertex update inserts or deletes a vertex with all incident edges, and we update a complete distance matrix accordingly. The algorithm runs on a comparison-addition based pointer-machine.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Demetrescu, C., Italiano, G.: A new approach to dynamic all pairs shortest paths. In: Proc. 35th STOC, pp. 159–166 (2003)Google Scholar
  2. 2.
    Demetrescu, C., Italiano, G.: A new approach to dynamic all pairs shortest paths (2004) ,Full version of [1] available at
  3. 3.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Henzinger, M.R., King, V.: Maintaining minimum spanning forests in dynamic graphs. SIAM J. Computing 31(2), 364–374 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    King, V.: Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs. In: Proc. 40th FOCS, pp. 81–89 (1999)Google Scholar
  7. 7.
    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
  8. 8.
    Pettie, S.: A faster all-pairs shortest path algorithm for real-weighted sparse graphs. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 85–97. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Thorup, M.: Integer priority queues with decrease key in constant time and the single source shortest paths problem. In: Proc. 35th STOC, pp. 149–158 (2003)Google Scholar
  10. 10.
    Thorup, M.: Worst-case update times for fully-dynamic all-pairs shortest paths (2004) (submitted)Google Scholar
  11. 11.
    Williams, J.W.J.: Algorithm 232. Comm. ACM 7(6), 347–348 (1964)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.AT&T Labs–Research, Shannon LaboratoryFlorham ParkUSA

Personalised recommendations