Faster Approximation of Distances in Graphs

Let G = (V, E) be an weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating allpairs shortest paths in G. Our first algorithm runs in ˜O(n2 ) time, and for any u, v ∈ V reports distance no greater than 2dG(u, v)+h(u, v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ∈ V reports distance no greater than (1+ǫ)dG(u, v)+2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n2.24+o(1) ǫ−3 log(nǫ−1 )) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radi...

• Type: Conference
• Year: 2007
• Where: WADS
• Authors: Piotr Berman, Shiva Prasad Kasiviswanathan
• Keywords: Algorithms, All-pairs Shortest Path Problem, Boolean Matrix Multiplication, Shortest Path, WADS 2007