Finding the hidden path: time bounds for all-pairs shortest paths

The all-pairs shortest paths problem in weighted graphs is investigated. An algorithm called the hidden paths algorithm, which finds these paths in time O(m*+n n² log n), where m* is the number of edges participating in shortest paths, is presented. It is argued that m* is likely to be small in practice, since m*=O(n log n) with high probability for many probability distributions on edge weights. An Ω(mn) lower bound on the running time of any path-comparison-based algorithm for the all-pairs shortest paths problem is proved