Near-Optimal Spanners for General Graphs in (Nearly) Linear Time

Abstract

Let G = (V, E, w) be a weighted undirected graph on |V | = n vertices and |E| = m edges, let k ≥ 1 be any integer, and let < 1 be any parameter. We present the following results on fast constructions of spanners with near-optimal sparsity and lightness, 1 which culminate a long line of work in this area. (By near-optimal we mean optimal under Erdos' girth conjecture and disregarding the -dependencies.)• There are (deterministic) algorithms for constructing (2k − 1)(1 + )-spanners for G with a near-optimal sparsity of O(nThe first algorithm can be implemented in the pointer-machine model within time O(mα(m, n) • log(1/ )/ ) + SORT(m)), where α(•, •) is the two-parameter inverse-Ackermann function and SORT(m) is the time needed to sort m integers. The second algorithm can be implemented in the Word RAM model within time O(m log(1/ )/ )). • There is a (deterministic) algorithm for constructing a (2k − 1)(1 + )-spanner for G that achieves a near-optimal bound of O(n 1/k • poly(1/ )) on both sparsity and lightness. This algorithm can be implemented in the pointer-machine model within time O(mα(m, n) • poly(1/ ) + SORT(m)) and in the Word RAM model within time O(mα(m, n) • poly(1/ )).The previous fastest constructions of (2k−1)(1+ )-spanners with near-optimal sparsity incur a runtime of is O(min{m(n 1+1/k ) + n log n, k • n 2+1/k }), even regardless of the lightness. Importantly, the greedy spanner for stretch 2k − 1 has sparsity O(n 1/k ) -with no -dependence whatsoever, but its runtime is O(m(n 1+1/k + n log n)). Moreover, the state-of-the-art lightness bound of any (2k − 1)-spanner (including the greedy spanner) is poor, even regardless of the sparsity and runtime.1 The sparsity (respectively, lightness) is a normalized notion of size (resp., weight), where we divide the size (resp., weight) by the size n − 1 of a spanning tree (resp., the weight w(MST) of a minimum spanning tree MST).