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 D. Aingworth et al. (1996), we describe an O/spl tilde/(min{n/sup 3/2/m/sup 1/2/,n/sup 7/3/}) time algorithm APASP/sub 2/ for computing all distances in G with an additive one-sided error of at most 2. The algorithm APASP/sub 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/spl tilde/(min{n/sup 2-(2)/(k+2)/m/sup (2)/(k+2)/, n/sup 2+(2)/(3k-2)/}) time algorithm APASP/sub k/ for computing all distances in G with an additive one-sided error of at most k. We also give an O/spl tilde/(n/sup 2/) time algorithm APASP/sub /spl infin// for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in O/spl tilde/(n/sup 2/) time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every u, v/spl isin/V we have /spl delta//sub G/(u,v)/spl les//spl delta//sub F/(u,v)/spl les//spl delta//sub G/(u,v)+k. We show that every unweighted graph on n vertices has a 2-emulator with O/spl tilde/(n/sup 3/2/) edges and a 4-emulator with O/spl tilde/(n/sup 4/3/) edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with O/spl tilde/(n/sup 3/2/) edges and that such a 3-spanner can be built in O/spl tilde/(mn/sup 1/2/) time. We also describe an O/spl tilde/(n(m/sup 2/3/+n)) time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3.