Approximate counting with a floating-point counter

TL;DR

This paper introduces a floating-point probabilistic counter that efficiently estimates large counts using minimal memory, improving upon Morris's original approximate counting method with a new unbiased estimator and detailed performance analysis.

Contribution

It presents a novel floating-point counter design with a simple unbiased estimator, extending Morris's approximate counting technique with enhanced accuracy and practical formulas for performance assessment.

Findings

Uses d + log log n bits for counting

Achieves an unbiased estimate with standard deviation ~0.6 * n * 2^{-d/2}

Provides a general performance analysis framework

Abstract

Memory becomes a limiting factor in contemporary applications, such as analyses of the Webgraph and molecular sequences, when many objects need to be counted simultaneously. Robert Morris [Communications of the ACM, 21:840--842, 1978] proposed a probabilistic technique for approximate counting that is extremely space-efficient. The basic idea is to increment a counter containing the value $X$ with probability $2^{-X}$. As a result, the counter contains an approximation of $\lg n$ after $n$ probabilistic updates stored in $\lg\lg n$ bits. Here we revisit the original idea of Morris, and introduce a binary floating-point counter that uses a $d$-bit significand in conjunction with a binary exponent. The counter yields a simple formula for an unbiased estimation of $n$ with a standard deviation of about $0.6\cdot n2^{-d/2}$, and uses $d+\lg\lg n$ bits. We analyze the floating-point counter's performance in a general framework that applies to any probabilistic counter, and derive practical formulas to assess its accuracy.