Abstract
The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.
Highlights
The random split tree introduced by Devroye (1999) is a general tree model which for special choices of its parameters covers various random trees that are fundamental in Computer Science for their use as data structures, e.g. binary search trees, quadtrees, m-ary search trees, simplex trees, tries etc
In the probabilistic analysis of algorithms the asymptotic behavior of such quantities is studied for this reason
Neininger (2002) mentioned that a limit theorem for the Wiener index of the general split tree can be proved in a similar way after determining the asymptotic expansion of its expectation sufficiently well. We prove this asymptotic expansion and use the contraction method to obtain the limit theorem for the Wiener index of random split trees which fulfil the general assumption
Summary
The random split tree introduced by Devroye (1999) is a general tree model which for special choices of its parameters covers various random trees that are fundamental in Computer Science for their use as data structures, e.g. binary search trees, quadtrees, m-ary search trees, simplex trees, tries etc. Let Pn denote the internal path length in a random split tree of size n with branching factor b where the one-dimensional marginal distribution V of the splitting vector fulfills the general assumption. Let Wn denote the Wiener index in a random split tree of size n with branching factor b where the one-dimensional marginal distribution V of the splitting vector fulfills the general assumption. Let (Wn, Pn) denote the vector consisting of the Wiener index and the internal path length of a random split tree of size n with branching factor b where the one-dimensional marginal distribution of the splitting vector For some constant α > 0 and ε > 0, Holmgren (2010) showed that Theorem 1.1 implies the similar asymptotic behavior for the internal path length for the nodes in that random split tree.
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