More Algorithms for All-Pairs Shortest Paths in Weighted Graphs

In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time O(n3 log3 log n/ log2 n), 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-)/(2d+4) ), where < 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-)/4 ) = O(n2.844 ) time) for APSP in real-vertex-weighted graphs, as well as an improved result (near O(n(3+)/2 ) = O(n2.688 ) time) for the all-pairs lightes...

• Type: Journal
• Year: 2010
• Where: SIAMCOMP
• Authors: Timothy M. Chan
• Keywords: Real-weighted Dense Graphs, Shortest Path, SIAMCOMP 2010, Small-integer-weighted Graphs