Analysis of a splitting process arising in probabilistic counting and other related algorithms

Abstract

We present an analytical method of analyzing a class of "splitting algorithms" that include probabilistic counting, selecting the leader, estimating the number of questions necessary to identify distinct objects, searching algorithms based on digital tries, approximate counting, and so forth. In our discussion we concentrate on the analysis of a generalized probabilistic counting algorithm. Our technique belongs to the toolkit of the analytical analysis of algorithms, and it involves solutions of functional equations, analytical poissonization and depoissonization as well as Mellin transform. In particular, we deal with an instance of the functional equation g ( z ) = pa(z)g(z/2) + b(z), where a(z) and b ( z ) are 379 380 KIRSCHENHOFER, PRODINGER, A N D SZPANKOWSKI given functions and p < 1 is a constant. With respect to our generalized probabilistic counting algorithm, we obtain asymptotic expansions of the first two moments of an estimate of the cardinality of a set that is computed by the algorithm. We also derive the asymptotic distribution of this estimate, and observe that it actually fluctuates, leading to a conclusion that its limiting distribution does not exist.

A Binomial Splitting Process in Connection with Corner Parking Problems

“…We focus on deriving an asymptotic approximation to the probability P(X n = k), which then leads to an effective estimate for the corresponding distribution functions. The method of proof we use here relies on the same analytic de-Poissonization procedure we used for the first two moments, and requires a uniform estimate with respect to k; see [14,20] for a similar analysis. We begin by consideringÃ…”

Flexible approximate counting

“…Mellin integrals and Euler's formula are central. Later, they [11] extended the analysis technique to many familiar probabilistic algorithms besides approximate counting. Errors are bounded but a limiting distribution of values does not exist.…”

On the asymptotic behavior of some algorithms

“…Surprisingly, this convergence does not hold. With calculus on some alternating series, Mellin transforms and complex analysis methods, Flajolet [8] has shown that the variable L(t) − log 1/a (t) exhibits an asymptotic oscillating behavior with respect to t. For an extension of these results, see also Kirschenhofer et al [14].…”