On the all-pairs shortest-path algorithm of Moffat and Takaoka

Abstract

We review how to solve the all‐pairs shortest‐path problem in a nonnegatively weighted digraph with n vertices in expected time O(n2 log n). This bound is shown to hold with high probability for a wide class of probability distributions on nonnegatively weighted digraphs. We also prove that, for a large class of probability distributions, Ω(n log n) time is necessary with high probability to compute shortest‐path distances with respect to a single source. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 205–220 (1997)

Average-case complexity of single-source shortest-paths algorithms: lower and upper bounds

“…Mehlhorn and Priebe [45] proved that for the complete graph with random edge weights every SSSP algorithm has to check at least Ω(n · log n) edges with high probability. Noshita [53] and Goldberg and Tarjan [35] analyzed the expected number of decrease_Key operations in Dijkstra's algorithm; the time bound, however, does not improve on the worstcase complexity of the algorithm.…”

A Forward-Backward Single-Source Shortest Paths Algorithm

“…The fastest algorithms above solve each SSSP problem in O(n log n) expected time. Mehlhorn and Priebe [MP97] showed that for the endpoint independent model, Ω(n log n) expected time is best possible for algorithms that can only access the (sorted) outgoing adjacency lists of the graph.…”

“…Bloniarz [Blo83] improved it to O(n 2 log n log * n). Finally, Moffat and Takaoka [MT87] and Mehlhorn and Priebe [MP97] (see also recent simplifications by Takaoka and Hashim [TH10,Tak13]) improved the running time to O(n 2 log n). All these algorithms, like Dijkstra's and Spira's algorithm use only the outgoing adjacency lists of the graph.…”

“…Moffat and Takaoka [MT87] and Mehlhorn and Priebe [MP97] (see also Takaoka and Hashim [TH10,Tak13]) obtained further refinements of Bloniarz's algorithm. Their algorithms perform, on average, only O(n) extract-min operations, while still scanning only O(n log n) edges, yielding comparison-based algorithms with an expected running time of O(n log n).…”

“…Adapting the argument of Mehlhorn and Priebe [MP97] we show that any SSSP algorithm that can only access the (sorted) outgoing adjacency lists of the input graph, which is assumed to be drawn from the directed graph K n (Exp(1)), must examine Ω(n log n) edges, with high probability. As it is not clear how incoming adjacency lists could be used to speed up SSSP algorithms, this seems to give a negative answer to the question.…”