Preprocess, Set, Query!

Abstract

Thorup and Zwick (J. ACM 52(1):1–24, 2005 and STOC’01) in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer k≥1 it is possible to preprocess the graph in $\tilde {O}(mn^{1/k})$ time and generate a compact data structure of size O(kn 1+1/k ). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k−1 in O(k) time. For k=2 this gives an oracle of O(n 1.5) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtrascu and Roditty (In: Proc. of 51st FOCS, 2010) broke the theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n 5/3) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time in unweighted undirected graphs. We present a data structure that produces estimated distances with 1+e stretch. The size of the data structure is O(nm 1−e′) and the query time is $\tilde {O}(m^{1-\varepsilon '})$ . Using it for sparse unweighted graphs we can get a data structure of size O(n 1.87) that can supply in O(n 0.87) time estimated distances with multiplicative stretch 1.75.