New Algorithms for All Pairs Approximate Shortest Paths

Let G = (V, E) be an unweighted undirected graph with n vertices and < edges. Dor, Halperin, and Zwick [FOCS 1996, SICOMP 2000] presented an <(O)over tilde> (min{n(3/2)m(1/2), n(7/3)})-time algorithm that computes estimated distances with an additive approximation of 2 without using Fast Matrix Multiplication (FMM). Recently, Deng, Kirkpatrick, Rong, V. Williams and Zhong [ICALP 2022] improved the running time for dense graphs to (O) over tilde (n(2.29))-time, using FMM, where an exact solution can be computed with FMM in (k) over tilde (n(omega)) time (omega < 2.37286) using Seidel's algorithm. Since an additive 2 approximation is also a multiplicative 2 approximation, computing an additive 2 approximation is at least as hard as computing a multiplicative 2 approximation. Thus, computing a multiplicative 2 approximation might be an easier problem. Nevertheless, more than two decades after the paper of Dor, Halperin, and Zwick was first published, no faster algorithm for computing multiplicative 2 approximation in dense graphs is known, rather then simply computing an additive 2 approximation. In this paper we present faster algorithms for computing a multiplicative 2 approximation without FMM. We show that in (O) over tilde (min{n(1/2)m, n(9/4)}) time it is possible to compute a multiplicative 2 approximation. For distances at least 4 we can get an even faster algorithm that in (O) over tilde (min {n(7/4)m(1/4), n(11/5)}) expected time computes a multiplicative 2 approximation. Our algorithms are obtained by a combination of new ideas that take advantage of a careful new case analysis of the additive approximation algorithms of Dor, Halperin, and Zwick. More specifically, one of the main technical contributions we made is an analysis of the algorithm of Dor, Halperin, and Zwick that reveals certain cases in which their algorithm produces improved additive approximations without any modification. This analysis provides a full characterization of the instances for which it is harder to obtain an improved approximation. Using more ideas we can take care of some of these harder cases and to obtain an improved additive approximation also for them. Our new analysis, therefore, serves as a starting point for future research either on improved upper bounds or on conditional lower bounds.