In the first part of the paper, we reexamine the all-pairs shortest paths (APSP) problem and present a new algorithm with running time approaching O(n 3 / log 2 n), which improves all known algorithms for general real-weighted dense graphs and is perhaps close to the best result possible without using fast matrix multiplication, modulo a few log log n factors. 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(n 3−(3−ω)/(2d+4) ), where ω < 2.376; in two dimensions, this is O(n 2.922 ). Our framework greatly extends the previously considered case of small-integer-weighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4 ) = O(n 2.844 ) time) for APSP in realvertex-weighted graphs, as well as an improved result (near O(n (3+ω)/2 ) = O(n 2.688 ) time) for the all-pairs lightest shortest path problem for small-integer-weighted graphs.
To our knowledge, the best-known algorithm for the APSP problem takes Oðn 3 = log nÞ time in the worst case REF .
More algorithms for all-pairs shortest paths in weighted graphs
STOC '07
Mathematics Computer Science