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A weakly 1-stable distribution for the number of random records and cuttings in split trees

Part of: Combinatorial probability Discrete mathematics in relation to computer science Algorithms - Computer Science Limit theorems Theory of data Graph theory

Published online by Cambridge University Press:  01 July 2016

Cecilia Holmgren
Cecilia Holmgren*
Affiliation:
Uppsala University
*
∗ Current address: Algorithms Project, INRIA Rocquencourt, F-78153 Le Chesnay, France. Email address: cecilia@math.uu.se
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Abstract

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In this paper we study the number of random records in an arbitrary split tree (or, equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for the convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree in Holmgren (2010), which is one specific case of split trees. Other important examples of split trees include m-ary search trees, quad trees, medians of (2k + 1)-trees, simplex trees, tries, and digital search trees.

Keywords

Random treesplit treecutrecordstable distributioninfinitely divisible distribution

MSC classification

Primary: 05C05: Trees 05C80: Random graphs 68W40: Analysis of algorithms 68P10: Searching and sorting
Secondary: 68R10: Graph theory (including graph drawing) 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 68P05: Data structures
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 151 - 177
Copyright
Copyright © Applied Probability Trust 2011 

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