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 D. Aingworth et al. (1996), we describe an O˜(min{n^3/2m^1/2,n^7/3 }) time algorithm APASP₂ 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˜(min{n^2-(2)(k+2)/m⁽²⁾(k+2)/, n²⁺⁽²⁾ (3k-2)/}) 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˜(n²) time algorithm APASP_∞ 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,E´) k-emulates an unweighted graph G=(V,E) if for every u, v∈V we have δ_G(u,v)⩽δ_F (u,v)⩽δ_G(u,v)+k. We show that every unweighted graph on n vertices has a 2-emulator with O˜(n^3/2 ) edges and a 4-emulator with 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 O˜(n ^3/2) edges and that such a 3-spanner can be built in O˜(mn^1/2) time. We also describe an 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