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We present an all-pairs shortest path algorithm whose running time on a complete directed graph onn vertices whose edge weights are chosen independently and uniformly at random from [0,1] isO (n 2 ), in expectation and with high probability. This resolves a long-standing open problem. The algorithm is a variant of the dynamic all-pairs shortest paths algorithm of Demetrescu and Italiano [2006]. The analysis relies on a proof that the number oflocally shortest paths in such randomly weighted graphs isO (n 2 ), in expectation and with high probability. We also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update inO (log2 n ) expected time.

All-pairs shortest paths in O ( n 2 ) time with high probability

All-pairs shortest paths in O ( n ² ) time with high probability