## Approximate counting: A detailed analysis (1985)

Venue: | BIT |

Citations: | 41 - 2 self |

### BibTeX

@ARTICLE{Flajolet85approximatecounting:,

author = {Philippe Flajolet},

title = {Approximate counting: A detailed analysis},

journal = {BIT},

year = {1985},

volume = {25},

pages = {113--134}

}

### Years of Citing Articles

### OpenURL

### Abstract

Approximate counting is an algorithm proposed by R. Morris which makes it possible to keep approximate counts of large numbers in small counters. The algorithm is useful for gathering statistics of a large number of events as well as for applications related to data compression (Todd et al.). We provide here a complete analysis of approximate counting which establishes good convergence properties of the algorithm and allows to quantify precisely complexity-accuracy tradeoffs. Introduction. As shown by an easy information-theoretic argument, maintaining a counter whose values may range in the interval 1 to M essentially necessitates log,M bits. This lower bound is of course achieved by a 1 standard binary counter. R. Morris [8] has proposed a probabilistic algorithm that maintains an

### Citations

931 |
A Course of Modern Analysis
- Whittaker, Watson
- 1927
(Show Context)
Citation Context ...erm is unform in n and 1.s120 PHILIPPE FLAJOLET PROOF. The proof proceeds by stages using the previously mentioned approximations. . (i) Truncation of the sum: let r = r(n) = 2(10g2 n)ll2. We set - r =-=(10)-=- Obviously PA,, = c (-1Y2 - j(j - 1 112 Q,: 1 ~ - $ - j( 1 - 2 -(l-~')y j= 0 (ii) Simplification of the denominators : define using the fact that IQ-Ql-l-jl = 0(2-'+'+j) since the sum of (r+ 1) terms:... |

214 |
The Art of Computer Programming: Sorting and Searching
- Knuth
- 1973
(Show Context)
Citation Context ...(- 1)Q-j(j-1)/2~,:1~-1 (5) cj - l-l-j, where for all k: k Qk = n (1-2-i), and Qo = 1. i=l ' Now from (4), there immediately follows an expression for the coefficients pn,l of Hl(x) : whence with (5), =-=(6)-=-: I - 1 PROPOSITION 1. The probability pn, I of having counter value 1 after n increments is I - 1 i 1-1-j This expression permits an easy numerical calculation of the probabilities involved -. - in a... |

72 |
Probabilistic counting
- Flajolet, Martin
- 1983
(Show Context)
Citation Context ...xact counts. There are other cases like data base systems where probabilistic counting methods prove useful. We mention a related algorithm; called ‘‘Probabilistic Counting” that has been proposed in =-=[3]-=-. This algorithm makes it possible to determine the approximate value of the number of distinct elements in a file in a single pass using a few operations per element and only O(1) additional storage.... |

59 |
Counting large numbers of events in small registers
- Morris
- 1978
(Show Context)
Citation Context ...tic argument, maintaining a counter whose values may range in the interval 1 to M essentially necessitates log,M bits. This lower bound is of course achieved by a 1 standard binary counter. R. Morris =-=[8]-=- has proposed a probabilistic algorithm that maintains an approximate count using only about log,log,M bits. This paper is devoted to a detailed analysis of characteristic parameters of that algorithm... |

18 |
Analyse Combinatoire
- Comtet
- 1970
(Show Context)
Citation Context ...as been proposed in [3]. This algorithm makes it possible to determine the approximate value of the number of distinct elements in a file in a single pass using a few operations per element and only O=-=(1)-=- additional storage. Received October 1982. Revised August 1984.s114 PHILIPPE FLAJOLET The plan of the paper is as follows. We start with a simple version of the algorithm: approximate counting with b... |

9 |
Handbuch der Laplace Transformation I, Birkhauser Verlag
- Doetsch
- 1971
(Show Context)
Citation Context ...ualities imply Lemma 1. a We are thus left with estimating the behaviour of F(x),as given by (24). To that purpose, we use the Meffin integral transform which for a real function f is defined by (see =-=[2]-=-): l; (25) f*(s) = Jl[f(x); s] = f(x)X”- ‘dx. This transform is useful for studying harmonic sums like (24) : from the obvious functional property (26) Jl[f(ax);s] = a--”f*(s), a > 0, J it follows for... |

3 |
Queueing systems. Wiley Interscience
- Kleinrock
- 1975
(Show Context)
Citation Context ... . (1 - a* - &*)- 1 which after simplification using (a1 - aj) = 2-j( 1 - 2 - 9 gives : * 1 -1 Co = (1 -r) (1 - 1 /4)- . . . ( 1 - 2-(' - ")- '. Similarly, we find in general - (- 1)Q-j(j-1)/2~,:1~-1 =-=(5)-=- cj - l-l-j, where for all k: k Qk = n (1-2-i), and Qo = 1. i=l ' Now from (4), there immediately follows an expression for the coefficients pn,l of Hl(x) : whence with (5), (6): I - 1 PROPOSITION 1. ... |

1 |
Variations on u theme by Huffmann
- Gallager
- 1978
(Show Context)
Citation Context ... a large number of events in a storage efficient way [SI. It was proposed for applications to data compression [9] when building an adaptive encoding scheme to represent ~~~~~- random” data (see e.g. =-=[4]-=- for adaptive Huffman codes and [7] for arithmetic coding); there, typically a large number of counters need to be maintained to gather statistics on the data to be compressed, but high accuracy of ea... |

1 |
Compression of black white images with binary arithmetic coding
- Langdon, Rissanen
- 1981
(Show Context)
Citation Context ...age efficient way [SI. It was proposed for applications to data compression [9] when building an adaptive encoding scheme to represent ~~~~~- random” data (see e.g. [4] for adaptive Huffman codes and =-=[7]-=- for arithmetic coding); there, typically a large number of counters need to be maintained to gather statistics on the data to be compressed, but high accuracy of each counter is not a critical factor... |

1 |
Dynamic statistics collection for compression coding, Unpublished manuscript
- Todd, Martin, et al.
- 1981
(Show Context)
Citation Context ... trade-offs can be quantified. The algorithm itself is useful for gathering statistics on a large number of events in a storage efficient way [SI. It was proposed for applications to data compression =-=[9]-=- when building an adaptive encoding scheme to represent ~~~~~- random” data (see e.g. [4] for adaptive Huffman codes and [7] for arithmetic coding); there, typically a large number of counters need to... |