The binomial transform and its application to the analysis of skip lists

Abstract

To any sequence of real numbers 〈a _n 〉_n≥0, we can associate another sequence 〈â _s 〉 _s ≥0, called its binomial transform. This transform is defined through the rule

$$\hat a_s = \mathcal{B}_s a_n = \sum\limits_n {( - 1)^n \left( {\begin{array}{*{20}c}s \\n \\\end{array} } \right)a_n .}$$

We study the properties of this transform, obtaining rules for its manipulation and a table of transforms, that allow us to invert many transforms by inspection.

We use these methods to perform a detailed analysis of skip lists, a probabilistic data structure introduced by W. Pugh as an alternative to balanced trees. In particular, we obtain the mean and variance for the cost of searching for the first or the last element in the list (confirming results obtained previously by other methods), and also for the cost of searching for a random element (whose variance was not known). We obtain exact (albeit sometimes complicated) expressions for all n≥0, and from them we find the corresponding asymptotic expressions.

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant No. A-8237, the Information Technology Research Centre of Ontario, and FONDECYT(Chile) under grant 1940271. Part of this work was done while the first author was on sabbatical and the third author a post doc at the University of Waterloo.