A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs

Abstract

In a seminal paper [27] for any n-vertex undirected graph G = (V,E) and a parameter k = 1, 2, …, Thorup and Zwick constructed a distance oracle of size O(kn1+1/k) which upon a query (u, v) constructs a path Π between u and ν of length δ(u, v) such that dG(u,ν) ≤ δ(u,v) ≤ (2k–1)dG(u,v). The query time of the oracle from [27] is O(k) (in addition to the length of the returned path), and it was subsequently improved to O(1) [29, 11]. A major drawback of the oracle of [27] is that its space is Ω(n · log n). Mendel and Naor [18] devised an oracle with space O(n1+1/k) and stretch O(k), but their oracle can only report distance estimates and not actual paths. In this paper we devise a path-reporting distance oracle with size O(n1+1/k), stretch O(k) and query time O(nε), for an arbitrarily small ε > 0. In particular, for k = log n our oracle provides logarithmic stretch using linear size. Another variant of our oracle has linear size, 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(n1+1/k · β), and query time O(nε), for an arbitrarily small constant ε > 0. The tradeoff between multiplicative stretch and size in these oracles is far below Erdös'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(n1+1/k). A similar type of tradeoff was exhibited by a construction of (1 + ε, β)-spanners due to Elkin and Peleg [16]. However, so far (1 + ε, 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.