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 ∈ V to v ∈ V in G. An (α, β)-stretch approximate distance oracle (ADO) for G is a data structure that given u, v ∈ V returns in constant (or near constant) time a value d̂(u, v) such that d(u, v) ≤ d̂(u, v) ≤ α · d(u, v) + β, for some reals α > 1, β. Thorup and Zwick [34] showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Pǎtraşcu and Roditty [27] showed that one can obtain stretch 2 using O(m¹/³n⁴/³) space, and so if m is subquadratic in n then the space usage is also subquadratic. Moreover, Pǎtraşcu and Roditty [27] showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = Õ(n), based on the set-intersection hypothesis. In this paper we explore the conditions for which an ADO can beat stretch 2 while using subquadratic space. In particular, we show that if the maximum degree in G is ∆G ≤ O(n¹/k−^ε) for some 0 < ε ≤ 1/k, then there exists an ADO for G that uses Õ(n^{2− kε 3} ) space and has a (2, 1 − k)-stretch. For k = 2 this result implies a subquadratic sub-2 stretch ADO for graphs with ∆G ≤ O(n¹/^2−ε). Moreover, we prove a conditional lower bound, based on the set intersection hypothesis, which states that for any positive integer k ≤ log n, obtaining a sub-^k+2_k stretch for graphs with ∆G = Θ(n^1/k) requires Ω̃(n²) space. Thus, for graphs with maximum degree Θ(n¹/²), obtaining a sub-2 stretch requires Ω̃(n²) space.