Approximate distance oracles

Let G = (V,E) be an undirected weighted graph with | V | = n and | E | = m . Let k ≥ 1 be an integer. We show that G = (V,E) can be preprocessed in O ( kmn 1/ k ) expected time, constructing a data structure of size O ( kn 1+1/ k ), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k −1, that is, the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k −1. A 1963 girth conjecture of Erdós, implies that Ω( n 1+1/ k ) space is needed in the worst case for any real stretch strictly smaller than 2 k +1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name "oracle". Previously, data structures that used only O ( n 1+1/ k ) space had a query time of Ω( n 1/ k ).Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs.