Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-27T00:10:40.509Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], [email protected])
JASON MATTERER
Affiliation:
Department of Applied Mathematics and Statistics, The Johns Hopkins University, 34th and Charles Streets, Baltimore, MD 21218-2682, USA (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Billingsley, P. (1995) Probability and Measure, third edition, Wiley Interscience.Google Scholar
[2]Devroye, L. (1984) Exponential bounds for the running time of a selection algorithm. J. Comput. System Sci. 29 1–7.CrossRefGoogle Scholar
[3]Devroye, L. (2001) On the probabilistic worst-case time of ‘Find’. Algorithmica 31 291–303.CrossRefGoogle Scholar
[4]Devroye, L. and Fawzi, O. (2010) Simulating the Dickman distribution. Statist. Probab. Lett. 80 242–247.CrossRefGoogle Scholar
[5]Fill, J. A. (2013) Distributional convergence for the number of symbol comparisons used by QuickSort. Ann. Appl. Probab. 23 1129–1147.CrossRefGoogle Scholar
[6]Fill, J. A. and Huber, M. L. (2010) Perfect simulation of Vervaat perpetuities. Electron. J. Probab. 15 96–109.CrossRefGoogle Scholar
[7]Fill, J. A. and Nakama, T. (2010) Analysis of the expected number of bit comparisons required by Quickselect. Algorithmica 58 730–769.CrossRefGoogle Scholar
[8]Fill, J. A. and Nakama, T. (2013) Distributional convergence for the number of symbol comparisons used by QuickSelect. Adv. Appl. Probab. 45, 425–450.CrossRefGoogle Scholar
[9]Grübel, R. (1998) Hoare's selection algorithm: A Markov chain approach. J. Appl. Probab. 35 36–45.CrossRefGoogle Scholar
[10]Grübel, R. and Rösler, U. (1996) Asymptotic distribution theory for Hoare's selection algorithm. Adv. Appl. Probab. 28 252–269.CrossRefGoogle Scholar
[11]Hoare, C. A. R. (1961) Find (algorithm 65). Commun. Assoc. Comput. Mach. 4 321–322.Google Scholar
[12]Hwang, H. and Tsai, T. (2002) Quickselect and the Dickman function. Combin. Probab. Comput. 11 353–371.CrossRefGoogle Scholar
[13]Knuth, D. E. (1972) Mathematical analysis of algorithms. In Information Processing 71: Proc. IFIP Congress, Ljubljana, 1971, North-Holland, pp. 19–27.Google Scholar
[14]Kodaj, B. and Móri, T. F. (1997) On the number of comparisons in Hoare's algorithm ‘FIND’. Studia Sci. Math. Hungar. 33 185–207.Google Scholar
[15]Lent, J. and Mahmoud, H. M. (1996) Average-case analysis of multiple Quickselect: An algorithm for finding order statistics. Statist. Probab. Lett. 28 299–310.CrossRefGoogle Scholar
[16]Mahmoud, H. M. and Smythe, R. T. (1998) Probabilistic analysis of multiple Quickselect. Algorithmica 22 569–584.CrossRefGoogle Scholar
[17]Mahmoud, H. M., Modarres, R. and Smythe, R. T. (1995) Analysis of Quickselect: An algorithm for order statistics. RAIRO Informatique Théorique et Applications 29 255–276.CrossRefGoogle Scholar
[18]Mitzenmacher, M. and Upfal, E. (2005) Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge University Press.CrossRefGoogle Scholar
[19]Prodinger, H. (1995) Multiple Quickselect: Hoare's Find algorithm for several elements. Inform. Process. Lett. 56 123–129.CrossRefGoogle Scholar
[20]Rosenthal, H. P. (1970) On the subspaces of Lp (p > 2) spanned by sequences of independent random variables. Israel J. Math. 8 273–303.CrossRefGoogle Scholar
[21]Vallée, B., Clément, J., Fill, J. A. and Flajolet, P. (2009) The number of symbol comparisons in QuickSort and QuickSelect. In 36th International Colloquium on Automata, Languages and Programming: ICALP 2009, Part I (Albers, S.et al., ed.), Vol. 5555 of Lecture Notes in Computer Science, Springer, pp. 750–763.CrossRefGoogle Scholar
[22]Williams, D. (1991) Probability with Martingales, Cambridge University Press.CrossRefGoogle Scholar