Turnstile Streaming Algorithms Might (Still) as Well Be Linear Sketches, for Polynomial-Length Streams

2026-04-23

AI Summary

This work resolves the long-standing open question of whether efficient turnstile streaming algorithms for polynomial-length streams are essentially equivalent to linear sketching. By introducing tools from Fourier analysis and additive combinatorics, we establish this equivalence in the practical turnstile model for the first time: any turnstile algorithm using space $S$ can be simulated by a linear sketch requiring only $O(S)$ linear measurements, with total space $O(S \log S)$. Our approach abandons the traditional transition-graph machinery, enabling efficient reconstruction of the final vector via a linear sketch. This yields new lower bounds and, under a natural smoothness assumption, leads to a compact sketch with bounded entries and total space merely $O(S)$.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Data Mining & Knowledge Management: Data Stream Mining
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A fundamental question in streaming complexity is whether every space-efficient turnstile algorithm is implicitly a linear sketch. The landmark work of Li, Nguyen, and Woodruff [LNW14] established an equivalence between the two, but their reduction requires a stream length that is at least doubly exponential in the dimension $n$. In the opposite direction, results by Kallaugher and Price [KP20] demonstrate a separation for streams of linear length, showing that the equivalence does not hold in general. The most natural and practically relevant regime -- polynomial-length streams -- has therefore remained open. We show that polynomial-length turnstile algorithms admit linear-sketch simulations. More precisely, if a turnstile algorithm uses $S$ bits of space and succeeds on all streams of length $\mathrm{poly}(D, n)$, then on final vectors $x$ with $\|x\|_2 \le D$, its output can be recovered from $O(S)$ linear measurements of $x$, using $O(S \log S)$ bits overall. For smooth problems under appropriate input distributions, a mollified version of the reduction yields a bounded-entry sketch with $O(S / \log D)$ measurements and optimal $O(S)$ total space. Our results extend to strict turnstile streams and non-uniform Read-Once Branching Programs (ROBPs). Our proof departs from prior transition-graph based machinery, relying instead on a Fourier-analytic framework and tools from additive combinatorics to extract discrete linear measurements. Our analysis shows that any $S$-bit algorithm can only be sensitive to a low-dimensional lattice of heavy Fourier frequencies, which we then use to construct the rows of the sketching matrix. Consequently, we obtain new lower bounds for polynomial-length streams via existing real sketching and communication lower bounds.

Problem

• turnstile streaming
• linear sketch
• polynomial-length streams
• streaming complexity
• space-efficient algorithms

Innovation

• turnstile streaming
• linear sketch
• Fourier analysis
• additive combinatorics
• polynomial-length streams