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**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).

### MSC:

68Q25 | Analysis of algorithms and problem complexity |

68W99 | Algorithms in computer science |

68N25 | Theory of operating systems |

### Keywords:

analysis of algorithms; combinatorial analysis; estimation methods; probabilistic algorithm; birth process; Mellin transform; approximate counting; direct sampling### Citations:

Zbl 0386.68035### References:

[1] | L. Comtel,L’Analyse Combinatoire, 2 vol., P.U.F., Paris (1970). |

[2] | G. Doetsch,Handbuch der Laplace Transformation, Birkhauser Verlag, Basel (1955). · Zbl 0065.34001 |

[3] | P. Flajolet and N. Martin,Probabilistic counting, in Proc. 24th Annual Symp. on Foundations of Comp. Sc., Tucson, Arizona (1984), pp. 76–82. |

[4] | R. G. Gallager,Variations on a theme by Huffmann, IEEE Trans. IT, 24 (1978) pp. 669–674. · Zbl 0399.94012 |

[5] | L. Kleinrock,Queuing Systems, Wiley Interscience, New York (1976). · Zbl 0361.60082 |

[6] | D. E. Knuth,The Art of Computer Programming: Sorting and Searching, Addison-Wesley, Reading (1973). · Zbl 0302.68010 |

[7] | G. Langdon and J. Rissanen,Compression of black white images with binary arithmetic coding, IEEE Trans. on Communications (1981). · Zbl 0456.94009 |

[8] | R. Morris,Counting large numbers of events in small registers, Comm. ACM, 21 (1978), pp. 840–842. · Zbl 0386.68035 |

[9] | S. Todd, N. Martin, G. Langdon and D. Helman,Dynamic statistics collection for compression coding, Unpublished manuscript, 12 p. (1981). |

[10] | E. T. Whittaker and G. N. Watson,A Course in Modern Analysis, (1907); 4th edition, Cambridge Univ. Press, 1927. · JFM 45.0433.02 |

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