Space Lower Bounds for Graph Stream Problems

TL;DR

This paper establishes tight space lower bounds for various graph problems in the streaming model, including shortest path, node depth, min cut, negative cycle detection, and strongly connected components, using communication complexity techniques.

Contribution

It introduces new tight lower bounds for graph problems in the streaming model, extending known results to problems like node depth and other fundamental graph tasks.

Findings

Proves tight single and multipass space lower bounds for node depth in trees.

Extends lower bounds to shortest path, min cut, negative cycle detection, and SCC problems.

Uses communication complexity and hard instance constructions for proofs.

Abstract

This work concerns with proving space lower bounds for graph problems in the streaming model. It is known that computing the length of shortest path between two nodes in the streaming model requires $\Omega(n)$ space, where $n$ is the number of nodes in the graph. We study the problem of finding the depth of a given node in a rooted tree in the streaming model. For this problem we prove a tight single pass space lower bound and a multipass space lower bound. As this is a special case of computing shortest paths on graphs, the above lower bounds also apply to the shortest path problem in the streaming model. The results are obtained by using known communication complexity lower bounds or by constructing hard instances for the problem. Additionally, we apply the techniques used in proving the above lower bound results to prove space lower bounds (single and multipass) for other graph problems like finding min $s-t$ cut, detecting negative weight cycle and finding whether two nodes lie in the same strongly connected component.