Skip to main content
SpringerLink
Log in

A simplified algorithm for the all pairs shortest path problem with O(n 2logn) expected time

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

The best known expected time for the all pairs shortest path problem on a directed graph with non-negative edge costs is O(n 2logn) by Moffat and Takaoka. Let the solution set be the set of vertices to which the given algorithm has so far established shortest paths. The Moffat-Takaoka algorithm maintains complexities before and after the critical point in balance, which is the moment when the size of the solution set is n−n/logn. In this paper, we remove the concept of critical point, whereby we make the algorithm simpler and seamless, resulting in a simpler analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. We notice that the condition for while in line 19 can be simplified to T[w]≤n−n/logn, and line 18 can be removed. The above version clarifies the meaning of CP better.

References

  • Bloniarz P (1983) A shortest path algorithm with expected time O(n 2lognlog∗ n). SIAM J Comput 12:588–600

    Article  MathSciNet  MATH  Google Scholar 

  • Chan T (2007) More algorithms for all pairs shortest paths. In: STOC 2007, pp 590–598

    Google Scholar 

  • Dantzig G (1960) On the shortest route in a network. Manag Sci 6:269–271

    Google Scholar 

  • Feller WH (1968) An introduction to probability and its applications, vol 1, 3rd edn. John-Wiley, New York

    MATH  Google Scholar 

  • Fredman M, Tarjan R (1987) Fibonacci heaps and their uses in improved network optimization problems. J ACM 34:596–615

    Article  MathSciNet  Google Scholar 

  • Goldberg AV, Tarjan RE (1996) Expected performance of Dijkstra’s shortest path algorithm. Technical report 96-062, NEC Research Institute, Inc, June 1996. URL: http://www.avglab.com/andrew/pub/neci-tr-96-062.ps

  • Han Y, Takaoka T (2012) An O(n 3loglogn/log2 n) time algorithm for ALl pairs shortest paths. In: SWAT 2012. LNCS, vol 7357, pp 131–141

    Google Scholar 

  • Mehlhorn K, Priebe V (1997) On the all-pairs shortest path algorithm of Moffat and Takaoka. Random Struct Algorithms 10:205–220

    Article  MathSciNet  MATH  Google Scholar 

  • Moffat A, Takaoka T (1987) An all pairs shortest path algorithm with expected running time O(n 2logn). SIAM J Comput 16:1023–1031

    Article  MathSciNet  MATH  Google Scholar 

  • Pettie S (2004) A new approach to all pairs shortest paths on real weighted graphs. Theor Comput Sci 312:47–74

    Article  MathSciNet  MATH  Google Scholar 

  • Spira P (1973) A new algorithm for finding all shortest paths in a graph of positive arcs in average time O(n 2log2 n). SIAM J Comput 2:28–32

    Article  MathSciNet  MATH  Google Scholar 

  • Takaoka T, Moffat A (1980) An O(n 2lognloglogn) expected time algorithm for the all pairs shortest path problem. In: MFCS 1980. LNCS, vol 88, pp 643–655

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Canterbury, Christchurch, New Zealand

    Tadao Takaoka

Authors
  1. Tadao Takaoka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tadao Takaoka.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Takaoka, T. A simplified algorithm for the all pairs shortest path problem with O(n 2logn) expected time. J Comb Optim 25, 326–337 (2013). https://doi.org/10.1007/s10878-012-9550-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-012-9550-3

Keywords

Search

Navigation