Finding Real-Valued Single-Source Shortest Paths in o(n3) Expected Time

Given ann-vertex,m-edge directed networkGwith real costs on the edges and a designated source vertexs, we give a new algorithm to compute shortest paths froms. Our algorithm is a simple deterministic one withO(n2logn) expected running time over a large class of input distributions. This is the first strongly polynomial algorithm in over 35 years to improve upon some aspect of theO(nm) running time of the Bellman?Ford algorithm. The result extends to anO(n2logn) expected running time algorithm for finding the minimum mean cycle, an improvement over Karp'sO(nm) worst-case time bound when the underlying graph is dense. Both of our time bounds are shown to be achieved with high probability.

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