# How to Count Quickly and Accurately: A Unified Analysis of Probabilistic Counting and Other Related Problems

Abstract. We consider a class of probabilistic counting algorithms parameterized by an integer $d\geq 0$ that estimate the number of elements $N$ in a large set. Our algorithms generalize an idea of Flajolet and Martin who limited themselves to the case $d=0$. As noted by Brassard and Bratley it is far from obvious how to carry out a more precise analysis of the unbiased estimate of $N$ \dots ''. We present a novel and complete analysis of these new counting algorithms that -- to the best of our knowledge -- cannot be obtained by an extension of the analysis by Flajolet and Martin. We present results concerning the average value, the variance and the limiting generating function of an estimate of $N$. Moreover, our novel approach is {\it not} limited to probabilistic counting algorithms, and it can be applied in the investigation of several other splitting algorithms" such as selecting the loser within a group of people, estimating the number of questions necessary to identify the number of distinct objects, searching algorithms based on digital tries, approximate counting, electing $d$ finalists in a contest (cf. polling system), and so forth.

helmut@gauss.cam.wits.ac.za,

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