ALCOMFT-TR-03-18

Marianne Durand and Philippe Flajolet
Loglog Counting of Large Cardinalities
INRIA. Work packages 1 and 5. July 2003.

Abstract: Using an auxiliary memory smaller than the size of this abstract, the LogLog algorithm makes it possible to estimate in a single pass and within a few percents the number of different words in the whole of Shakespeare's works. In general the LogLog algorithm makes use of m ``small bytes'' of auxiliary memory in order to estimate in a single pass the number of distinct elements (the ``cardinality'') in a file, and it does so with an accuracy that is of the order of 1/\sqrt{m}. The ``small bytes'' to be used in order to count cardinalities till N_max comprise about loglog N_max bits, so that cardinalities well in the range of billions can be determined using one or two kilobytes of memory only. The basic version of the LogLog algorithm is validated by a complete analysis. An optimized version, super-LogLog, is also engineered and tested on real-life data. The algorithm parallelizes optimally.

Postscript file: ALCOMFT-TR-03-18.ps.gz (221 kb).