Approximate counting: a detailed analysis. (English) Zbl 0562.68027

Approximate counting is a probabilistic algorithm proposed by R. Morris [Commun. ACM 21, 840-842 (1978; Zbl 0386.68035)] that allows the storage of (many) large counts in small counters. The algorithm allows counting up to some integer n in space \(\approx \log_ 2\log_ 2n+\delta\) with a constant expected relative accuracy that is \(O(2^{- \delta /2})\). For instance, using only 8 bits, one can count up to \(2^{16}=65536\) with an accuracy of about 15 %. The paper presents a complete analysis of the algorithm which is equivalent to a pure birth process with discrete time and birth probabilities of the form \(2^{- k}\). The probability distribution of the approximate result is characterized exactly and is also shown to tend to a limiting distribution. Mean and variance of the result are estimated asymptotically using a combination of: (i) combinatorial identities in the theory of integer partitions; (2) Mellin transform techniques. The paper concludes with a comparison of approximate counting with direct sampling methods. One should also note that related methods appear in the space-efficient simulation of deterministic machines by probabilistic machines (Freivalds, Gill).


68Q25 Analysis of algorithms and problem complexity
68W99 Algorithms in computer science
68N25 Theory of operating systems


Zbl 0386.68035
Full Text: DOI


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