Factorial Lower Bounds for (Almost) Random Order Streams

TL;DR

This paper introduces the StreamingCycles problem, a random order streaming problem related to hypermatching, and establishes tight lower bounds on its communication complexity, impacting graph streaming and random walk generation.

Contribution

The paper presents the first tight lower bounds for StreamingCycles, advancing understanding of streaming lower bounds in random order models.

Findings

Established an old lower bound on communication complexity for StreamingCycles.

Provided near-tight lower bounds for component collection in random order graph streams.

Achieved the first exponential space lower bounds for random walk generation.

Abstract

In this paper we introduce and study the \textsc{StreamingCycles} problem, a random order streaming version of the Boolean Hidden Hypermatching problem that has been instrumental in streaming lower bounds over the past decade. In this problem the edges of a graph $G$, comprising $n/\ell$ disjoint length-$\ell$ cycles on $n$ vertices, are partitioned randomly among $n$ players. Every edge is annotated with an independent uniformly random bit, and the players' task is to output the parity of some cycle in $G$ after one round of sequential communication. Our main result is an $\ell^{\Omega(\ell)}$ lower bound on the communication complexity of \textsc{StreamingCycles}, which is tight up to constant factors in $\ell$. Applications of our lower bound for \textsc{StreamingCycles} include an essentially tight lower bound for component collection in (almost) random order graph streams, making progress towards a conjecture of Peng and Sohler [SODA'18] and the first exponential space lower bounds for random walk generation.