Lower bounds for testing digraph connectivity with one-pass streaming algorithms

TL;DR

This paper establishes lower bounds on the memory required for one-pass streaming algorithms to correctly test key digraph properties such as strong connectivity, acyclicity, and reachability, highlighting fundamental limitations in streaming graph algorithms.

Contribution

It proves that testing these properties in a streaming setting necessitates linear memory in the number of edges, revealing inherent complexity barriers.

Findings

Memory lower bounds of Ω(ε m) for testing properties

Demonstrates fundamental limitations of one-pass streaming algorithms

Applicable to properties like strong connectivity, acyclicity, reachability

Abstract

In this note, we show that three graph properties - strong connectivity, acyclicity, and reachability from a vertex $s$ to all vertices - each require a working memory of $\Omega (\epsilon m)$ on a graph with $m$ edges to be determined correctly with probability greater than $(1+\epsilon)/2$.