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

Let <i>G</i>(<i>V, E</i>) be an undirected weighted graph with |<i>V</i>| = <i>n</i>, |<i>E</i>| = <i>m</i>. Recently Thorup and Zwick introduced a remarkable data-structure that stores all pairs approximate distance information implicitly in <i>o</i>(<i>n</i>²) space, and yet answers any approximate distance query in <i>constant</i> time. They named this data-structure <i>approximate distance oracle</i> because of this feature. Given an integer <i>k</i> < 1, a (2k-1)-approximate distance oracle requires <i>O</i>(<i>kn</i>^1+1/<i>k</i>) space and answers a (2<i>k</i>-1)-approximate distance query in <i>O</i>(<i>k</i>) time. Thorup and Zwick showed that a (2<i>k</i> - 1)-approximate distance oracle can be computed in <i>O</i>(<i>kmn</i>^1/<i>k</i>) time, and posed the following question : <i>Can</i> (2k - 1)-<i>approximate distance oracle be computed in</i> Õ(<i>n</i>²) <i>time</i>?In this paper, we answer their question in affirmative for unweighted graphs. We present an algorithm that computes (2<i>k</i> -1)-approximate distance oracle for a given unweighted graph in Õ(<i>n</i>²) 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.