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. 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. 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), n^2+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^HLF) 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.