On the amortized complexity of approximate counting

TL;DR

This paper investigates whether maintaining multiple approximate counters can be more memory-efficient on average than maintaining them individually, concluding that no such amortized savings are possible across most parameters.

Contribution

It proves a lower bound showing that amortized memory savings do not exist for multiple approximate counters, extending prior single-counter results.

Findings

No asymptotic memory benefit for multiple counters via amortization

Lower bounds established for a wide parameter range

Utilizes information cost techniques from streaming algorithms

Abstract

Naively storing a counter up to value $n$ would require $\Omega(\log n)$ bits of memory. Nelson and Yu [NY22], following work of [Morris78], showed that if the query answers need only be $(1+\epsilon)$-approximate with probability at least $1 - \delta$, then $O(\log\log n + \log\log(1/\delta) + \log(1/\epsilon))$ bits suffice, and in fact this bound is tight. Morris' original motivation for studying this problem though, as well as modern applications, require not only maintaining one counter, but rather $k$ counters for $k$ large. This motivates the following question: for $k$ large, can $k$ counters be simultaneously maintained using asymptotically less memory than $k$ times the cost of an individual counter? That is to say, does this problem benefit from an improved {\it amortized} space complexity bound? We answer this question in the negative. Specifically, we prove a lower bound for nearly the full range of parameters showing that, in terms of memory usage, there is no asymptotic benefit possible via amortization when storing multiple counters. Our main proof utilizes a certain notion of "information cost" recently introduced by Braverman, Garg and Woodruff in FOCS 2020 to prove lower bounds for streaming algorithms.