All-Pairs Almost Shortest Paths

Abstract

Let G = (V; E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of Aingworth, Chekuri and Motwani, we describe an ~ O(minfn 3=2 m 1=2 ; n 7=3 g) time algorithm APASP 2 for computing all distances in G with an additive one-sided error of at most 2. Algorithm APASP 2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k ? 2, we describe an ~ O(minfn 2\\Gamma 2 k+2 m 2 k+2 ; n 2+ 2 3k\\Gamma2 g) time algorithm APASP k for computing all distances in G with an additive one-sided error of at most k. We also give an ~ O(n 2 ) time algorithm APASP1 for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in ~ O(n 2 ) time. We say that a weighte...