An optimal cardinality estimation algorithm based on order statistics and its full analysis

Building on the ideas of Flajolet and Martin (1985), Alon et al. (1987), Bar-Yossef et al. (2002), Giroire (2005), we develop a new algorithm for cardinality estimation, based on order statistics which, according to Chassaing and Gerin (2006), is optimal among similar algorithms. This algorithm has a remarkably simple analysis that allows us to take its fine-tuning and the characterization of its properties further than has been done until now. We prove that, asymptotically, it is strictly unbiased (contrarily to Probabilistic Counting, Loglog, Hyperloglog), we verify that its relative precision is about 1/ √ m− 2 when m words of storage are used, and we fully characterize the limit law of the estimates it provides, in terms of gamma distribution—this is the first such algorithm for which the limit law has been established. We also develop a Poisson analysis for the pre-asymptotic regime. In this way, we are able to devise a complete algorithm, covering all cardinalities ranges from 0 to very large.