Advances in Applied Probability

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

Cecilia Holmgren

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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.

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Adv. in Appl. Probab., Volume 43, Number 1 (2011), 151-177.

First available in Project Euclid: 15 March 2011

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Zentralblatt MATH identifier

Primary: 05C05: Trees 05C80: Random graphs [See also 60B20] 68W40: Analysis of algorithms [See also 68Q25] 68P10: Searching and sorting
Secondary: 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35] 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 68P05: Data structures

Random tree split tree cut record stable distribution infinitely divisible distribution


Holmgren, Cecilia. A weakly 1-stable distribution for the number of random records and cuttings in split trees. Adv. in Appl. Probab. 43 (2011), no. 1, 151--177. doi:10.1239/aap/1300198517.

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