A linear-size logarithmic stretch path-reporting distance oracle for general graphs

Thorup and Zwick [2001a] proposed a landmark distance oracle with the following properties. Given an n-vertex undirected graph G = (V, E) and a parameter k = 1, 2,...,their oracle has size O(kn^1+1/k), and upon a query (u, v) it constructs a path Π between u and v of length δ(u, v) such that d_G(u,v) ≤ δ(u, v) ≤ (2k-1)d_G(u, v). The query time of the oracle from Thorup and Zwick [2001a] is O(k) (in addition to the length of the returned path), and it was subsequently improved to O(1) [Wulff-Nilsen 2012; Chechik 2014]. A major drawback of the oracle of Thorup and Zwick [2001a] is that its space is Ω(n · log n). Mendel and Naor [2006] devised an oracle with space O(n^1+1/k) and stretch O(k), but their oracle can only report distance estimates and not actual paths. In this article, we devise a path-reporting distance oracle with size O(n^1+1/k), stretch O(k), and query time O(n^∈), for an arbitrarily small constant ∈ > 0. In particular, for k = log n, our oracle provides logarithmic stretch using linear size. Another variant of our oracle has size O(n log log n), polylogarithmic stretch, and query time O(log log n). For unweighted graphs, we devise a distance oracle with multiplicative stretch O(1), additive stretch O(β(k)), for a function β(·), space O(n^1+1/k), and query time O(n^∈), for an arbitrarily small constant ∈ > 0. The tradeoff between multiplicative stretch and size in these oracles is far below Erdo{combining double acute accent}s's girth conjecture threshold (which is stretch 2k - 1 and size O(n^1+1/k)). Breaking the girth conjecture tradeoff is achieved by exhibiting a tradeoff of different nature between additive stretch β(k) and size O(n^1+1/k). A similar type of tradeoff was exhibited by a construction of (1 + ∈, β)-spanners due to Elkin and Peleg [2001]. However, so far (1 + ∈, β)-spanners had no counterpart in the distance oracles' world. An important novel tool that we develop on the way to these results is a distance-preserving path-reporting oracle. We believe that this oracle is of independent interest.