New methods for approximating shortest paths

Abstract

A spanner H of a graph G is a sparse subgraph that approximates all pairwise distances. Of particular interest are additive spanners on unweighted graphs, which satisfy the following for and two vertices u, v. dist(H, u, v) a L d dist(G, u, v) + f(n), where dist is the distance with respect to the graph H or G and f(n) is a function of the number of vertices of the graph. We study a variety of problems related to additive spanners, and have two new results of significance. For additive spanners with O (n) edges, it is well known that f(n) must be a polynomial function O(n[alpha]) for some 0 < [alpha] < 1. Previously, it was known that the optimal value of [alpha was between 1/13 and 3/7; by combining two previously known methods, our first significant result improves the lower bound from 1/13 to 1/11. The all pairs approximate shortest paths problem takes as input an unweighted graph, and outputs a distance matrix that approximates all pairwise distances. We present a new improvement to the algorithm of Dor, Halperin, and Zwick for the +4 and +6 approximation algorithms. In the +4 approximation algorithm, our new algorithm runs in O(n 15/7) time, an improvement from the previous O(n 11/5), and in the +6 approximation algorithm, out new algorithm runs in O(n 9/9) time, an improvement from the previous O(n 17/8).