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 et al. [SIAM J. Comput., 28 (1999), pp. 1167--1181], we describe an $\Ot(\min\{n^{3/2}m^{1/2},n^{7/3}\})$-time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. 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 ${\tilde{O}}(\min\{n^{2-{2}/{(k+2)}}m^{{2}/{(k+2)}}, n^{2+{2}/{(3k-2)}}\})$-time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an ${\tilde{O}}(n^2)$-time algorithm ${\bf APASP}_\infty$ for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in ${\tilde{O}}(n^2)$ time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every $u,v\in V$ we have $\delta_G(u,v)\le \delta_F(u,v)\le \delta_G(u,v)+k$. We show that every unweighted graph on n vertices has a 2-emulator with ${\tilde{O}}(n^{3/2})$ edges and a 4-emulator with ${\tilde{O}}(n^{4/3})$ edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with ${\tilde{O}}(n^{3/2})$ edges and that such a 3-spanner can be built in ${\tilde{O}}(mn^{1/2})$ time. We also describe an ${\tilde{O}}(n(m^{2/3}+n))$-time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3.