More Algorithms for All-Pairs Shortest Paths in Weighted Graphs

In the first part of the paper, we reexamine the all-pairs shortest path (APSP) problem and present a new algorithm with running time O(n³log³log n/log²n), which improves all known algorithms for general real-weighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n3-(3-ω)/(2d+4)), where ω<2.376; in two dimensions, this is O(n2.922). Our framework greatly extends the previously considered case of small–integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n3-(3-ω)/4)=O(n2.844) time) for APSP in real–vertex-weighted graphs, as well as an improved result (near O(n(3+ω)/2)=O(n2.688) time) for the all-pairs lightest shortest path problem for small–integer-weighted graphs.