Distance Oracles for Sparse Graphs

Years and Authors of Summarized Original Work

• 2005; Thorup, Zwick

• 2012; Pǎtraşcu, Roditty

• 2012; Pǎtraşcu, Roditty, Thorup

Problem Definition

Let G = (V, E) be a weighted undirected graph with n vertices and m edges. A distance oracle is a data structure capable of representing almost shortest paths efficiently, both in terms of space requirement and query time. Thorup and Zwick [7] showed 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 $O(kn^{1+1/k})$ that answers approximate distance queries with 2k − 1 multiplicative stretch in O(k) time. This means that for every u, v ∈ V, it is possible to retrieve an estimate $\hat{d}(u,v)$ to the distance d(u, v) in O(k) time, such that $d(u,v) \leq \hat{ d}(u,v) \leq (2k - 1)d(u,v)$. Recently, [8] showed, using a clever query algorithm, that the query time of Thorup and Zwick can be reduced from O(k) to O(logk). Even more recently, [1] showed...