Speeding Up Dijkstra ’ s Algorithm for All Pairs Shortest Paths ∗

We present a technique for reducing the number of edge scans performed by Dijkstra’s algorithm for computing all pairs shortest paths. The main idea is to restrict path scanning only to locally shortest paths, i.e., paths whose proper subpaths are shortest paths. On a directed graph with n vertices and m edges, the technique we discuss allows it to reduce the number of edge scans from O(mn) to O(lsp), where lsp ≤ mn is the number of locally shortest paths in the graph. Using Fibonacci heaps, this method can be implemented in O(lsp+n2 log n) worst-case time in a graph with non-negative real edge weights. Experimental evidence suggests that lsp << mn for many classes of graphs that arise in practical applications, resulting in substantial speedups especially in the case of dense graphs.