Computing over Data Streams using Catalytic Space

2026-07-09

AI Summary

This work investigates the exact computation of frequency moments (such as $F_0$, $F_2$, and $F_3$) and induced subgraph counts for a fixed graph $H$ in the data stream model under extremely limited clean memory. To this end, it introduces the catalytic space model to streaming algorithms for the first time, requiring that any auxiliary space used be restored to its initial state upon termination. The study establishes that catalytic space does not augment computational power in a single pass, thereby delineating its limitations. However, by leveraging multiple passes, the authors design algorithms that achieve exact solutions using only $O(\log m)$ or $O_H(\log n)$ clean space; for instance, triangle counting can be accomplished in three passes with $O(\log n)$ clean memory.

Abstract

We introduce a streaming model with \emph{catalytic memory}, an auxiliary workspace that must be returned to its initial state at the end of the computation. We show that catalytic space yields dramatic space savings for data stream algorithms. We first study the exact computation of frequency moments in insertion-only data streams. For every $k\ge1$, we give an exact four-pass algorithm for computing $\mathbb{F}_{k}$ using $O(k\log m)$ clean space, where $m$ is the stream length. We also present a $(k+1)$-pass algorithm with the same clean-space complexity that uses a factor of $k$ less catalytic space than the four-pass algorithm. For small moments, we obtain stronger results. In particular, we show that $\mathbb{F}_{2}$ and $\mathbb{F}_{3}$ can be computed exactly in two and three passes, respectively, using only $O(\log m)$ clean space. Additionally, we show that exact $\mathbb{F}_{0}$ computation reduces to computing $\mathbb{F}_{k}$ for a suitably chosen large value of $k$, resulting in an exact four-pass algorithm for $\mathbb{F}_{0}$ using only $O(\log m)$ clean space. We further show how our frequency-moment algorithms can be used to exactly count induced occurrences of any fixed graph $H$ in a graph stream, yielding a four-pass algorithm that uses $O_H(\log n)$ clean space, where $n$ is the number of vertices in the graph. As a special case, we obtain an exact three-pass algorithm for triangle counting using $O(\log n)$ clean space. All of our algorithms are multi-pass. We complement these algorithmic results with a matching limitation showing that catalytic memory does not provide additional power in the single-pass setting. Specifically, we prove that every randomized or deterministic single-pass streaming algorithm using $s$ bits of clean memory and catalytic space can be simulated in the standard streaming model, without catalytic memory, using $O(s)$ space.

Problem

• data streams
• catalytic space
• frequency moments
• graph streaming
• space complexity

Innovation

• catalytic space
• data streams
• frequency moments
• multi-pass algorithms
• subgraph counting