An almost 2-approximation for all-pairs of shortest paths in subquadratic time

Let G = (V, E) be an unweighted undirected graph with n vertices and m edges. Dor, Halperin, and Zwick [FOCS 1996, SICOMP 2000] presented an Õ(n²)-time algorithm that computes estimated distances with a multiplicative approximation of 3. Berman and Kasiviswanathan [WADS 2007] improved the approximation of Dor et al. and presented an Õ(n²)time algorithm that produces for every u, v ∈ V an estimate (Equation presented)(u, v) such that: d_G(u, v) ≤ (Equation presented)(u, v) ≤ 2d_G(u, v) + 1. We refer to such an approximation as an (α, β)-approximation, where α is the multiplicative approximation and β is the additive approximation. A prerequisite for an O(n²−^ε)-time algorithm, where ε ∈ (0, 1), is a data structure that uses O(n^2−δ) space, for some δ ≥ ε, and answers queries in constant time. An O(n²−^ε)-time (3, 0)-approximation algorithm became plausible after Thorup and Zwick [STOC 2001, JACM 2005] presented their approximate distance oracles, and in particular an O(n^1.5)-space data structure that reports a (3, 0)-approximate distance in O(1) time. Indeed, using Thorup and Zwick distance oracles together with more ideas, Baswana, Gaur, Sen, and Upadhyay [ICALP 2008] improved the running time of Dor et al., and obtained an O(n²−^ε) time algorithm, at the cost of introducing also an additive approximation. They presented an algorithm that in Õ(m+ n²³/¹²) expected running time constructs an O(n^1.5)-space data structure, that in O(1) time reports a (3, 14)-approximate distance. An O(n²−^ε)-time (2, 1)-approximation algorithm became plausible only after Pǎtraşcu and Roditty [FOCS 2010, SICOMP 2014] presented an O(n⁵/³)-space data structure that reports (2, 1)-approximate distances in O(1) time. However, only few years ago, Sommer [ICALP 2016] obtained an Õ(n²) time algorithm that computes a (2, 1)-distance oracle with Õ(n⁵/³) space. This leads to the following natural question of whether Ω(n²) time is a lower bound for any (3−α, β)-approximation, where α ∈ (0, 1), and β is constant. In this paper we show that this is not the case by presenting an algorithm that for every ε ∈ (0, 1/2) computes in Õ(m) + n²−Ω time an Õ(n^{1 56} )-space data structure that in O(1/ε) time reports, for every u, v ∈ V , an estimate (Equation presented)(u, v) such that: (Equation presented). Our result improves, simultaneously, the running time and the multiplicative approximation of the Õ(n²)-time (3, 0)-approximation algorithm of Dor et al. at the cost of introducing also an additive approximation.