Optimal bounds for approximate counting

TL;DR

This paper presents a new simple algorithm for approximate counting that improves space efficiency, provides a refined analysis of Morris Counter, and establishes a tight lower bound, fully resolving the asymptotic space complexity.

Contribution

It introduces a simplified algorithm with improved space bounds, refines analysis of Morris Counter, and proves a matching lower bound, resolving the asymptotic space complexity of approximate counting.

Findings

New simple algorithm with $O(\log\log N + \log(1/\varepsilon) + \log\log(1/\delta))$ bits

Refined analysis showing Morris Counter also achieves improved bounds

Established a tight lower bound within a factor of 3+o(1)

Abstract

Storing a counter incremented $N$ times would naively consume $O(\log N)$ bits of memory. In 1978 Morris described the very first streaming algorithm: the "Morris Counter". His algorithm's space bound is a random variable, and it has been shown to be $O(\log\log N + \log(1/\varepsilon) + \log(1/\delta))$ bits in expectation to provide a $(1+\varepsilon)$-approximation with probability $1-\delta$ to the counter's value. We provide a new simple algorithm with a simple analysis showing that randomized space $O(\log\log N + \log(1/\varepsilon) + \log\log(1/\delta))$ bits suffice for the same task, i.e. an exponentially improved dependence on the inverse failure probability. We then provide a new analysis showing that the original Morris Counter itself, after a minor but necessary tweak, actually also enjoys this same improved upper bound. Lastly, we prove a new lower bound for this task showing optimality of our upper bound. We thus completely resolve the asymptotic space complexity of approximate counting. Furthermore all our constants are explicit, and our lower bound and tightest upper bound differ by a multiplicative factor of at most $3+o(1)$.