All-pairs shortest paths with a sublinear additive error

We show that for every 0 ≤ p ≤ 1 there is an algorithm with running time of O(n^{2.575-p/(7.4-2.3p)}) that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u,v in the graph to within an additive error δ^p (u,v), where δ(u,v) is the exact length of the shortest path between u and v. This algorithm runs faster than the fastest algorithm for computing exact shortest paths for any 0 < p ≤ 1. Previously the only way to "bit" the running time of the exact shortest path algorithms was by applying an algorithm of Zwick [FOCS '98] that approximates the shortest path distances within a multiplicative error of (1 + ε). Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error. In fact, the main ingredient we need in order to obtain the above result, which is also interesting in its own right, is an algorithm for computing (1 + ε) multiplicative approximations for the shortest paths, whose running time is faster than the running time of Zwick's approximation algorithm when ε ≪ 1 and the graph has small integer weights.