Multi-dimensional Approximate Counting
Dingyu Wang

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 bits to count up to with a relative error , where if is the estimate of the current count , then . A natural generalization is \emph{multi-dimensional} approximate counting. Let be the dimension. The count vector is incremented entry-wisely over a stream of coordinates , where upon receiving , . A \emph{-dimensional approximate counter} is required to count coordinates simultaneously and return an estimate of the count vector . Aden-Ali, Han, Nelson, and Yu \cite{aden2022amortized} showed that the trivial solution of using Morris counters that track coordinates separately is already optimal in space,…
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Taxonomy
TopicsData Management and Algorithms · Bayesian Modeling and Causal Inference · Machine Learning and Algorithms
