We describe a new forward-backward variant of Dijkstra's and Spira's Single-Source Shortest Paths (SSSP) algorithms. While essentially all SSSP algorithm only scan edges forward, the new algorithm scans some edges backward. The new algorithm assumes that edges in the out-going and incoming adjacency lists of the vertices appear in non-decreasing order of weight. (Spira's algorithm makes the same assumption about the out-going adjacency lists, but does not use incoming adjacency lists.) The running time of the algorithm on a complete directed graph on n vertices with independent exponential edge weights is Õ(n), with very high probability. This improves on the previously best result of Õ(n log n), which is best possible if only forward scans are allowed, exhibiting an interesting separation between forward-only and forward-backward SSSP algorithms. As a consequence, we also get a new all-pairs shortest paths algorithm. The expected running time of the algorithm on complete graphs with independent exponential edge weights is Õ(n2), matching a recent result of Peres et al. Furthermore, the probability that the new algorithm requires more than Õ(n2) time is exponentially small, improving on the polynomially small probability of Peres et al.
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