All Pairs Almost Shortest Paths

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(minfn3=2m1=2; n7=3g) time algorithmAPASP2 for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix multiplication algorithm. Furthermore, for every even k > 2, we describe an ~ O(minfn2 2 k+2m 2 k+2 ; n2+ 2 3k 2 g) time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an ~ O(n2) 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(n2) time. We say that a weighted graph F = (V;E0) k-emulates an unweighted graph G = (V;E) if for every u; v 2 V we have G(u; v) F (u; v) G(u; v) + k. We show that every unweighted graph on n vertices has a 2-emulator with ~ O(n3=2) edges and a 4-emulator with ~ O(n4=3) edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with ~ O(n3=2) edges and that such a 3-spanner can be built in ~ O(mn1=2) time. We also describe an ~ O(n(m2=3 + n)) time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3.