Preprocess, Set, Query!

Abstract

Thorup and Zwick [J. ACM 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, Pǎtraşcu and Roditty [FOCS’10] broke the long-standing 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. We present a data structure that produces estimated distances with 1 + ε stretch. The size of the data structure is O(nm ^{1 − ε′}) 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 ^1.86) that can supply in O(n ^0.86) time estimated distances with multiplicative stretch 1.75.