Faster Streaming algorithms for graph spanners

TL;DR

This paper introduces new streaming algorithms for computing graph spanners that achieve near-optimal size-stretch trade-offs, with one algorithm optimized for unweighted graphs in a single pass and another for weighted graphs in multiple passes within the StreamSort model.

Contribution

The paper presents the first single-pass streaming algorithm for unweighted graph spanners with near-optimal size and stretch, and a multi-pass algorithm for weighted graphs in the StreamSort model.

Findings

Single-pass algorithm for unweighted graphs with improved efficiency.

Multi-pass StreamSort algorithm for weighted graphs using minimal memory.

Both algorithms achieve near-optimal size-stretch trade-offs.

Abstract

Given an undirected graph $G=(V,E)$ on $n$ vertices, $m$ edges, and an integer $t\ge 1$, a subgraph $(V,E_S)$, $E_S\subseteq E$ is called a $t$-spanner if for any pair of vertices $u,v \in V$, the distance between them in the subgraph is at most $t$ times the actual distance. We present streaming algorithms for computing a $t$-spanner of essentially optimal size-stretch trade offs for any undirected graph. Our first algorithm is for the classical streaming model and works for unweighted graphs only. The algorithm performs a single pass on the stream of edges and requires $O(m)$ time to process the entire stream of edges. This drastically improves the previous best single pass streaming algorithm for computing a $t$-spanner which requires $\theta(mn^{\frac{2}{t}})$ time to process the stream and computes spanner with size slightly larger than the optimal. Our second algorithm is for {\em StreamSort} model introduced by Aggarwal et al. [FOCS 2004], which is the streaming model augmented with a sorting primitive. The {\em StreamSort} model has been shown to be a more powerful and still very realistic model than the streaming model for massive data sets applications. Our algorithm, which works of weighted graphs as well, performs $O(t)$ passes using $O(\log n)$ bits of working memory only. Our both the algorithms require elementary data structures.