Computing shortest paths with comparisons and additions

  • Authors:

  • Seth Pettie;Vijaya Ramachandran

  • Affiliations:

  • The Univ. of Texas at Austin, Austin TX;The Univ. of Texas at Austin, Austin TX

  • Venue:
  • SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2002

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Abstract

We present an undirected all-pairs shortest paths (APSP) algorithm which runs on a pointer machine in time O(mnα(m,n)) while making O(mnlog α(m, n)) comparisons and additions, where m and n are the number of edges and vertices, respectively, and α(m, n) is Tarjan's inverse-Ackermann function. This improves upon all previous comparison & addition-based APSP algorithms when the graph is sparse, i.e., when m = o(n log n).At the heart of our APSP algorithm is a new single-source shortest paths algorithm which runs in time O(mα(m, n) + n log log r) on a pointer machine, where r is the ratio of the maximum-to-minimum edge length. So long as r no(1) this algorithm is faster than any implementation of Dijkstra's classical algorithm in the comparison-addition model.For directed graphs we give an O(m + n log r)-time comparison & addition-based SSSP algorithm on a pointer machine. Similar algorithms assuming integer weights or the RAM model were given earlier.