Is there any efficient algorithm to find shortest distance between all pairs of vertices?

The graph is:

  • Cyclic
  • Sparse (each vertex has either 2 or 3 edge)
  • undirected(bidirectional)
  • weighted
  • non-negative

Since Floyd-Warshall algorithm and Johnson's algorithm are best for finding shortest path for every pairs but:

  • Johnson's algorithm works on directed
  • Floyd-Warshall algorithm is best for dense graphs

I couldn't find any algo that is efficient for above graph conditions.

  • 2
    $\begingroup$ If algorithm works on directed, it works on undirected as well (each non-directed edge can be converted to a pair of directed). $\endgroup$
    – rus9384
    Oct 14, 2017 at 15:14
  • $\begingroup$ @rus9384 "non-directed edge can be converted to a pair of directed" --how? $\endgroup$
    – wrangler
    Oct 14, 2017 at 15:33
  • 1
    $\begingroup$ If you have nondirected edge A---B, just replace it with A-->B and A<--B. $\endgroup$
    – rus9384
    Oct 14, 2017 at 16:29

1 Answer 1


Mikkel Thorup's paper Undirected Single-Source Shortest Paths with Positive Integer Weights in Linear Time shows an $O(m)$ time algorithm for single-source shortest paths on undirected weighted graphs. That immediately implies an $O(mn)$ time algorithm for all pair shortest paths, which is the optimal $O(n^2)$ in your scenario.

  • $\begingroup$ The article seems to be paid and costly. Isn't any free link present? $\endgroup$
    – wrangler
    Oct 14, 2017 at 15:49
  • $\begingroup$ @wrangler I edited it. Now it should be a free link. $\endgroup$
    – Wei Zhan
    Oct 14, 2017 at 15:52

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