A Limit Theory for Random Skip Lists

The skip list was introduced by Pugh in 1989 as a data structure for dictionary operations. Using a binary tree representation of skip lists, we obtain the limit law for the path lengths of the leaves in the skip list. We also show that the height (maximal path length) of a skip list holding $n$ elements is in probability asymptotic to $c \log_{1/p} n$, where $c$ is the unique solution greater than 1 of the equation $\log(1 - p) = \log(c - 1) - \lbrack c/(c - 1) \rbrack \log c$, and $p \in (0, 1)$ is a design parameter of the skip list.