Approximate distance oracle for unweighted graphs in Õ(n²) time

Let G(V, E) be an undirected weighted graph with |V| = n, |E| = m. Recently Thorup and Zwick introduced a remarkable data-structure that stores all pairs approximate distance information implicitly in o(n2) space, and yet answers any approximate distance query in constant time. They named this data-structure approximate distance oracle because of this feature. Given an integer k < 1, a (2k-1)-approximate distance oracle requires O(kn1+1/k) space and answers a (2k-1)-approximate distance query in O(k) time. Thorup and Zwick showed that a (2k - 1)-approximate distance oracle can be computed in O(kmn1/k) time, and posed the following question : Can (2k - 1)-approximate distance oracle be computed in Õ(n2) time? In this paper, we answer their question in affirmative for unweighted graphs. We present an algorithm that computes (2k -1)-approximate distance oracle for a given unweighted graph in Õ(n2) time. One of the new ideas used in the improved algorithm also leads to the first linear time algorithm for computing an optimal size (2, 1)-spanner of an unweighted graph