Streaming Algorithms with Few State Changes
Rajesh Jayaram, David P. Woodruff, Samson Zhou
TL;DR
This work introduces and analyzes streaming algorithms that minimize internal state changes (memory writes) while preserving sublinear space, addressing central problems such as $F_p$ moment estimation and $L_p$-heavy hitters in insertion-only streams. The authors develop a suite of techniques, including reservoir-based SampleAndHold with Morris counters, grouped counter maintenance, and the FullSampleAndHold framework, to achieve near-optimal state-change bounds $O(n^{1-1/p})$ (and $O(n^{1-2/p})$ space for certain regimes) for heavy hitters and, via a Subsampling-Indyk–Woodruff framework, for $F_p$ estimation as well. They also extend these ideas to entropy estimation, leveraging a reduction to moment estimation at multiple exponents $p_i$ and polynomial interpolation. A tight lower bound shows that any $(2- ext{const})$-approximation to $F_p$ requires $ ilde{ heta}ig(n^{1-1/p}ig)$ state changes, demonstrating that the proposed algorithms achieve near-optimal state-change performance without sacrificing space. The results collectively show that one can attain near-optimal state-change budgets while maintaining near-optimal space, enabling practical, write-efficient streaming computations for fundamental data-analysis tasks.
Abstract
In this paper, we study streaming algorithms that minimize the number of changes made to their internal state (i.e., memory contents). While the design of streaming algorithms typically focuses on minimizing space and update time, these metrics fail to capture the asymmetric costs, inherent in modern hardware and database systems, of reading versus writing to memory. In fact, most streaming algorithms write to their memory on every update, which is undesirable when writing is significantly more expensive than reading. This raises the question of whether streaming algorithms with small space and number of memory writes are possible. We first demonstrate that, for the fundamental $F_p$ moment estimation problem with $p\ge 1$, any streaming algorithm that achieves a constant factor approximation must make $Ω(n^{1-1/p})$ internal state changes, regardless of how much space it uses. Perhaps surprisingly, we show that this lower bound can be matched by an algorithm that also has near-optimal space complexity. Specifically, we give a $(1+\varepsilon)$-approximation algorithm for $F_p$ moment estimation that uses a near-optimal $\widetilde{\mathcal{O}}_\varepsilon(n^{1-1/p})$ number of state changes, while simultaneously achieving near-optimal space, i.e., for $p\in[1,2]$, our algorithm uses $\text{poly}\left(\log n,\frac{1}{\varepsilon}\right)$ bits of space, while for $p>2$, the algorithm uses $\widetilde{\mathcal{O}}_\varepsilon(n^{1-2/p})$ space. We similarly design streaming algorithms that are simultaneously near-optimal in both space complexity and the number of state changes for the heavy-hitters problem, sparse support recovery, and entropy estimation. Our results demonstrate that an optimal number of state changes can be achieved without sacrificing space complexity.