Tight Streaming Lower Bounds for Deterministic Approximate Counting

TL;DR

This paper establishes a tight lower bound of a(k \u2212     n/k) bits for deterministic approximate counting in streaming models, confirming the optimality of existing algorithms like Misra-Gries.

Contribution

It provides the first non-trivial lower bounds for deterministic approximate counting in streaming, using a novel potential function analysis.

Findings

Lower bound of a(k     n/k) bits for deterministic algorithms

Misra-Gries algorithm is space-optimal for heavy hitters

Introduces a new potential function technique for streaming lower bounds

Abstract

We study the streaming complexity of $k$-counter approximate counting. In the $k$-counter approximate counting problem, we are given an input string in $[k]^n$, and we are required to approximate the number of each $j$'s ($j\in[k]$) in the string. Typically we require an additive error $\leq\frac{n}{3(k-1)}$ for each $j\in[k]$ respectively, and we are mostly interested in the regime $n\gg k$. We prove a lower bound result that the deterministic and worst-case $k$-counter approximate counting problem requires $\Omega(k\log(n/k))$ bits of space in the streaming model, while no non-trivial lower bounds were known before. In contrast, trivially counting the number of each $j\in[k]$ uses $O(k\log n)$ bits of space. Our main proof technique is analyzing a novel potential function. Our lower bound for $k$-counter approximate counting also implies the optimality of some other streaming algorithms. For example, we show that the celebrated Misra-Gries algorithm for heavy hitters [MG82] has achieved optimal space usage.