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 (O) over tilde (n(2))-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 (O) over tilde (n(2))time algorithm that produces for every u; v is an element of V an estimate (d) over cap (u; v) such that: d(G)(u; v) <= (d) over cap (u; v) <= 2d(G) (u; v) + 1: We refer to such an approximation as an (alpha, beta)-approximation, where alpha is the multiplicative approximation and beta is the additive approximation. A prerequisite for an O(n(2-epsilon))-time algorithm, where epsilon - (0, 1), is a data structure that uses O (n(2-delta)) space, for some delta >= epsilon, and answers queries in constant time. An O(n(2-epsilon))-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(n(2-epsilon)) time algorithm, at the cost of introducing also an additive approximation. They presented an algorithm that in (O) over tilde (m + n(23/12)) 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(n(2-epsilon))-time (2; 1)-approximation algorithm became plausible only after Patrascu and Roditty [FOCS 2010, SICOMP 2014] presented an O(n(5/3))-space data structure that reports (2; 1)-approximate distances in O (1) time. However, only few years ago, Sommer [ICALP 2016] obtained an (O) over tilde (n(2)) time algorithm that computes a (2; 1)-distance oracle with (O) over tilde (n(5/3)) space. This leads to the following natural question of whether Omega(n(2)) time is a lower bound for any (3-alpha,beta)-approximation, where alpha is an element of (0, 1), and fi is constant. In this paper we show that this is not the case by presenting an algorithm that for every epsilon is an element of (0, 1/2) computes in (O) over tilde (m) + n(2-Omega(epsilon)) time an (O) over tilde (n(15/6))-space data structure that in O(1/epsilon) time reports, for every u, v is an element of V, an estimate (d) over cap (u, v) such that: d(G) (u; v) <= (d) over cap (u, v) <= 2(1 + epsilon)d(G)(u, v) + 5. Our result improves, simultaneously, the running time and the multiplicative approximation of the (O) over tilde (n(2))-time (3, 0)-approximation algorithm of Dor et al. at the cost of introducing also an additive approximation.