Nearly 2-Approximate Distance Oracles in Subquadratic Time

Let G = (V;E) be an unweighted undirected graph on n vertices and m edges. For a fixed pair of real values 1; 0, an distance oracle of G is a space-eficient data structure that answers, in constant time, for any pair of vertices u; v 2 V a distance estimate within the range of [dist(u; v); dist(u; v)+]; here dist denotes distances in the graph G. Two main concerns in designing distance oracles are the approximation ratio (the stretch) and the construction time. A classical result was given in [Baswana, Goyaland and Sen 2005] which builds a (2; 3) distance oracle with ~O (n5=3) space in ~O(n2) time. Recently, [Akav and Roditty, 2020] broke the quadratic running time at the expense of increasing the stretch. More specifically, they obtained an algorithm that constructs a (2 + 5) distance oracle with space ~O(n11=6) in O(m + n2()) time for any constant 2 (0; 1=2). In this paper, we show that one can beat the quadratic running time without compromising on the stretch. More specifically, our algorithm constructs, with high probability, a (2; 3) distance oracle with ~O(n5=3) space in ~O (m+n1:987) time. As a secondary extension, we could further reduce the preprocessing time to ~O(m+n7=4+) by tolerating a (2;O(1=)) stretch, for any constant > 0. Finally, this preprocessing time could be pushed even further to ~O(m + n5=3+) if we allow a stretch of (2 + c), where c = c() is a constant depending exponentially on 1=.