Light Spanners

TL;DR

This paper introduces a new analysis of the greedy algorithm to construct light spanners with improved weight bounds, advancing the understanding of sparse graph approximations.

Contribution

It provides a novel analysis of the greedy algorithm to produce light spanners with better weight bounds than previous methods.

Findings

Achieves a $(2k-1)(1+\epsilon)$-stretch spanner with weight close to the MST.

Improves previous bounds by a factor of $O(\log k)$.

Uses a novel analysis technique for the classic greedy algorithm.

Abstract

A $t$-spanner of a weighted undirected graph $G=(V,E)$, is a subgraph $H$ such that $d_H(u,v)\le t\cdot d_G(u,v)$ for all $u,v\in V$. The sparseness of the spanner can be measured by its size (the number of edges) and weight (the sum of all edge weights), both being important measures of the spanner's quality -- in this work we focus on the latter. Specifically, it is shown that for any parameters $k\ge 1$ and $\epsilon>0$, any weighted graph $G$ on $n$ vertices admits a $(2k-1)\cdot(1+\epsilon)$-stretch spanner of weight at most $w(MST(G))\cdot O_\epsilon(kn^{1/k}/\log k)$, where $w(MST(G))$ is the weight of a minimum spanning tree of $G$. Our result is obtained via a novel analysis of the classic greedy algorithm, and improves previous work by a factor of $O(\log k)$.