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QuickSelect Tree Process Convergence, With an Application to Distributional Convergence for the Number of Symbol Comparisons Used by Worst-Case Find

Published online by Cambridge University Press:  09 July 2014

JAMES ALLEN FILL  and
JASON MATTERER
JAMES ALLEN FILL
Affiliation:
Department of Applied Mathematics and Statistics, The Johns Hopkins University, 34th and Charles Streets, Baltimore, MD 21218-2682, USA (e-mail: jimfill@jhu.edu, jtmatterer@gmail.com)
JASON MATTERER
Affiliation:
Department of Applied Mathematics and Statistics, The Johns Hopkins University, 34th and Charles Streets, Baltimore, MD 21218-2682, USA (e-mail: jimfill@jhu.edu, jtmatterer@gmail.com)
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Abstract

We define a sequence of tree-indexed processes closely related to the operation of the QuickSelect search algorithm (also known as Find) for all the various values of n (the number of input keys) and m (the rank of the desired order statistic among the keys). As a ‘master theorem’ we establish convergence of these processes in a certain Banach space, from which known distributional convergence results as n → ∞ about

  1. (1) the number of key comparisons required

    are easily recovered

    1. (a) when m/n → α ∈ [0, 1], and

    2. (b) in the worst case over the choice of m.

    From the master theorem it is also easy, for distributional convergence of

  2. (2) the number of symbol comparisons required,

both to recover the known result in the case (a) of fixed quantile α and to establish our main new result in the case (b) of worst-case Find.

Our techniques allow us to unify the treatment of cases (1) and (2) and indeed to consider many other cost functions as well. Further, all our results provide a stronger mode of convergence (namely, convergence in Lp or almost surely) than convergence in distribution. Extensions to MultipleQuickSelect are discussed briefly.

Keywords

Primary 60F25Secondary 68W4060F15
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 5: Honouring the Memory of Philippe Flajolet - Part 1 , September 2014 , pp. 805 - 828
Copyright
Copyright © Cambridge University Press 2014 

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