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arXiv 2411.03071cs.DS
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多维近似计数

Multi-dimensional Approximate Counting

Dingyu Wang

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AI总结：

针对多维近似计数问题，提出一种基于变长整数编码的简单最优d维计数器，实现欧几里得相对误差σ，并通过对球覆盖的体积估计证明匹配下界，优于d个独立Morris计数器方案。

摘要：

著名的Morris计数器使用log₂log₂ n + O(log₂ σ⁻¹)比特来计数至n，相对误差为σ，其中若λ̂是对当前计数λ的估计，则E|λ̂-λ|² < σ²λ²。一个自然的推广是\u201c多维\u201d近似计数。设d≥1为维度。计数向量x∈ℕᵈ在坐标流(w₁,…,wₙ)∈[d]ⁿ上逐项递增，其中收到wₖ∈[d]时，x_{wₖ}←x_{wₖ}+1。一个\u201cd维近似计数器\u201d需要同时计数d个坐标并返回计数向量x的估计x̂。Aden-Ali、Han、Nelson和Yu证明了使用d个分别跟踪d个坐标的Morris计数器的平凡解在空间上已经是最优的，\u201c如果每个条目仅允许相对于自身的误差\u201d，即对每个j∈[d]，E|x̂ⱼ-xⱼ|² < σ²|xⱼ|²。然而，对于另一种自然的误差度量——\u201c欧几里得均方误差\u201dE|x̂-x|²——我们证明使用d个独立的Morris计数器是次优的。在这项工作中，我们提出了一个简单且最优的d维计数器，具有欧几里得相对误差σ，即E|x̂-x|² < σ²|x|²，其中|x|=√(∑_{j=1}^d xⱼ²)，并给出了匹配的下界。上界和下界的证明使用了极其简单的思想。上界通过某种变长整数编码构造，下界则通过对球覆盖的直接体积估计推导得出。

英文摘要：

The celebrated Morris counter uses $\log_2\log_2 n + O(\log_2 σ^{-1})$ bits to count up to $n$ with a relative error $σ$, where if $\hatλ$ is the estimate of the current count $λ$, then $\mathbb{E}|\hatλ-λ|^2 <σ^2λ^2$. 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., $\mathbb{E}|\hat{x}_j-x_j|^2<σ^2|x_j|^2$ 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 $σ$, i.e., $\mathbb{E} |\hat{x}-x|^2 <σ^2|x|^2$ 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.

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