Multi-dimensional Approximate Counting

TL;DR

This paper extends approximate counting to multiple dimensions, introducing an optimal counter for Euclidean error that outperforms naive methods, with simple proofs for bounds.

Contribution

It presents a simple, optimal multi-dimensional approximate counter for Euclidean error, improving over naive solutions and providing matching lower bounds.

Findings

Optimal d-dimensional counter for Euclidean error with space complexity matching lower bounds.

Naive d Morris counters are sub-optimal for Euclidean mean squared error.

Simple proofs using variable-length encoding and sphere covering for bounds.

Abstract

The celebrated Morris counter uses $\log_2\log_2 n + O(\log_2 \sigma^{-1})$ bits to count up to $n$ with a relative error $\sigma$, where if $\hat{\lambda}$ is the estimate of the current count $\lambda$, then $\mathbb{E}|\hat{\lambda}-\lambda|^2 <\sigma^2\lambda^2$. A natural generalization is \emph{multi-dimensional} approximate counting. Let $d\geq 1$ be the dimension. The count vector $x\in \mathbb{N}^d$ is incremented entry-wisely over a stream of coordinates $(w_1,\ldots,w_n)\in [d]^n$, where upon receiving $w_k\in[d]$, $x_{w_k}\gets x_{w_k}+1$. A \emph{$d$-dimensional approximate counter} is required to count $d$ coordinates simultaneously and return an estimate $\hat{x}$ of the count vector $x$. Aden-Ali, Han, Nelson, and Yu \cite{aden2022amortized} showed that the trivial solution of using $d$ Morris counters that track $d$ coordinates separately is already optimal in space, \emph{if each entry only allows error relative to itself}, i.e., $\mathbb{E}|\hat{x}_j-x_j|^2<\sigma^2|x_j|^2$ for each $j\in [d]$. However, for another natural error metric -- the \emph{Euclidean mean squared error} $\mathbb{E} |\hat{x}-x|^2$ -- we show that using $d$ separate Morris counters is sub-optimal. In this work, we present a simple and optimal $d$-dimensional counter with Euclidean relative error $\sigma$, i.e., $\mathbb{E} |\hat{x}-x|^2 <\sigma^2|x|^2$ where $|x|=\sqrt{\sum_{j=1}^d x_j^2}$, with a matching lower bound. The upper and lower bounds are proved with ideas that are strikingly simple. The upper bound is constructed with a certain variable-length integer encoding and the lower bound is derived from a straightforward volumetric estimation of sphere covering.