Finding the Hidden Path: Time Bounds for All-Pairs Shortest Paths

We investigate the all-pairs shortest paths problem in weighted graphs. We present an algorithm|the Hidden Paths Algorithm|that nds these paths in time O(m n+n 2 log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's algorithm. We argue that m is likely to be small in practice, since m = O(n logn) with high probability for many probability distributions on edge weights. We also prove an (mn) lower bound on the running time of any path-comparison based algorithm for the all-pairs shortest paths problem. Path-comparison based algorithms form a natural class containing the Hidden Paths Algorithm, as well as the algorithms of Dijkstra and Floyd. Lastly, we consider generalized forms of the shortest paths problem, and show that many of the standard shortest paths algorithms are eeective in this more general setting.