On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

For an undirected unweighted graph G = (V, E) with n vertices and m edges, let d(u, v) denote the distance from u in V to v in V in G. An (alpha, beta)-stretch approximate distance oracle (ADO) for G is a data structure that, given u, v in V, returns in constant time a value d-hat (u, v) such that d(u, v) <= d-hat (u, v) <= alpha * d(u, v) + beta, for some reals alpha > 1, beta. If beta = 0, we say that the ADO has stretch alpha. Thorup and Zwick (2005) showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Patrascu and Roditty (2010) showed that one can obtain stretch 2 using O(m^(1/3)n^(4/3)) space, and so if m is subquadratic in n, then the space usage is also subquadratic. Moreover, Patrascu and Roditty (2010) showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = O-tilde(n), based on the set-intersection hypothesis. In this paper, we investigate the minimum possible stretch achievable by an ADO as a function of the graph's maximum degree, a study motivated by the question of identifying the conditions under which an ADO can be stored with subquadratic space while still ensuring a sub-2 stretch. In particular, we show that if the maximum degree in G is Delta_G <= O(n^(1/k - epsilon)) for some 0 < epsilon <= 1/k, then there exists a (2, 1 - k)-stretch ADO for G that uses O-tilde(n^(2 - (k * epsilon) / 3)) space. For k = 2, this result implies a subquadratic sub-2 stretch ADO for graphs with Delta_G <= O(n^(1/2 - epsilon)). We provide tight lower bounds for the upper bound under the same set intersection hypothesis, showing that if Delta_G = Theta(n^(1/k)), a (2, 1 - k)-stretch ADO requires Omega-tilde(n^2) space. Moreover, we show that for constants epsilon, c > 0, a (2 - epsilon, c)-stretch ADO requires Omega-tilde(n^2) space even for graphs with Delta_G = Theta-tilde(1).