Streaming Graph Computations with a Helpful Advisor

TL;DR

This paper explores a streaming graph computation model assisted by annotations, demonstrating that with linear-sized annotations, many graph problems can be solved with minimal memory, including optimal tradeoffs for matrix-vector multiplication.

Contribution

It extends annotation models to graph streaming problems, showing constant hash values suffice for many problems and establishing optimal tradeoffs in matrix-vector multiplication.

Findings

Constant hash values suffice for many graph problems with annotations

Linear-sized annotations enable low-memory algorithms for graph problems

Optimal tradeoffs achieved in matrix-vector multiplication

Abstract

Motivated by the trend to outsource work to commercial cloud computing services, we consider a variation of the streaming paradigm where a streaming algorithm can be assisted by a powerful helper that can provide annotations to the data stream. We extend previous work on such {\em annotation models} by considering a number of graph streaming problems. Without annotations, streaming algorithms for graph problems generally require significant memory; we show that for many standard problems, including all graph problems that can be expressed with totally unimodular integer programming formulations, only a constant number of hash values are needed for single-pass algorithms given linear-sized annotations. We also obtain a protocol achieving \textit{optimal} tradeoffs between annotation length and memory usage for matrix-vector multiplication; this result contributes to a trend of recent research on numerical linear algebra in streaming models.