Electronic Journal of Probability

A Gaussian limit process for optimal FIND algorithms

Henning Sulzbach, Ralph Neininger, and Michael Drmota

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We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to $c \cdot n^\alpha$ are chosen, where $0 < \alpha \leq \frac{1}{2}$, $c > 0$ and $n$ is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as $n \to \infty$, which depends on $\alpha$. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 3, 28 pp.

Accepted: 6 January 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 68P10: Searching and sorting 60G15: Gaussian processes 60C05: Combinatorial probability 68Q25: Analysis of algorithms and problem complexity [See also 68W40]

FIND algorithm Quickselect complexity key comparisons functional limit theorem contraction method Gaussian process

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Sulzbach, Henning; Neininger, Ralph; Drmota, Michael. A Gaussian limit process for optimal FIND algorithms. Electron. J. Probab. 19 (2014), paper no. 3, 28 pp. doi:10.1214/EJP.v19-2933. https://projecteuclid.org/euclid.ejp/1465065645

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  • Adler, Robert J. An introduction to continuity, extrema, and related topics for general Gaussian processes. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 12. Institute of Mathematical Statistics, Hayward, CA, 1990. x+160 pp. ISBN: 0-940600-17-X
  • Anderson, D. H.; Brown, R. Combinatorial aspects of C. A. R. Hoare's FIND algorithm. Australas. J. Combin. 5 (1992), 109–119.
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
  • Blanchet, Jose H.; Sigman, Karl. On exact sampling of stochastic perpetuities. J. Appl. Probab. 48A (2011), New frontiers in applied probability: a Festschrift for Soren Asmussen, 165–182. ISBN: 0-902016-08-3
  • Boucheron, S., Lugosi, G. and Massart, P.: Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, (2013).
  • Dadoun, B. and Neininger, R.: A statistical view on exchanges in Quickselect. Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO), (2014), 40–51. arXiv:1307.8403
  • Devroye, Luc. Exponential bounds for the running time of a selection algorithm. J. Comput. System Sci. 29 (1984), no. 1, 1–7.
  • Devroye, Luc. Simulating perpetuities. Methodol. Comput. Appl. Probab. 3 (2001), no. 1, 97–115.
  • Devroye, L. On the probabilistic worst-case time of "find". Mathematical analysis of algorithms. Algorithmica 31 (2001), no. 3, 291–303.
  • Devroye, Luc; Fawzi, Omar. Simulating the Dickman distribution. Statist. Probab. Lett. 80 (2010), no. 3-4, 242–247.
  • Devroye, L. and James, L.: The double CFTP method. ACM Trans. Model. Comput. Simul. 21, (2011), 1–20.
  • Drmota, Michael; Janson, Svante; Neininger, Ralph. A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18 (2008), no. 1, 288–333.
  • Dudley, R. M. Sample functions of the Gaussian process. Ann. Probability 1 (1973), no. 1, 66–103.
  • Eickmeyer, Kord; Rüchendorf, Ludger. A limit theorem for recursively defined processes in $L^ p$. Statist. Decisions 25 (2007), no. 3, 217–235.
  • Fill, James Allen; Huber, Mark Lawrence. Perfect simulation of Vervaat perpetuities. Electron. J. Probab. 15 (2010), no. 4, 96–109.
  • Fill, James Allen; Nakama, Takéhiko. Analysis of the expected number of bit comparisons required by Quickselect. Algorithmica 58 (2010), no. 3, 730–769.
  • Fill, James Allen; Nakama, Takehiko. Distributional convergence for the number of symbol comparisons used by QuickSelect. Adv. in Appl. Probab. 45 (2013), no. 2, 425–450.
  • Grabner, Peter J.; Prodinger, Helmut. On a constant arising in the analysis of bit comparisons in quickselect. Quaest. Math. 31 (2008), no. 4, 303–306.
  • Grübel, Rudolf. Hoare's selection algorithm: a Markov chain approach. J. Appl. Probab. 35 (1998), no. 1, 36–45.
  • Grübel, Rudolf. On the median-of-$k$ version of Hoare's selection algorithm. Theor. Inform. Appl. 33 (1999), no. 2, 177–192.
  • Grübel, Rudolf; Rösler, Uwe. Asymptotic distribution theory for Hoare's selection algorithm. Adv. in Appl. Probab. 28 (1996), no. 1, 252–269.
  • Hoare, C.A.R.: Algorithm 65, FIND. Comm. Assoc. Comput. Mach. 4, (1961), 321–322.
  • Hwang, Hsien-Kuei; Tsai, Tsung-Hsi. Quickselect and the Dickman function. Combin. Probab. Comput. 11 (2002), no. 4, 353–371.
  • Kirschenhofer, Peter; Prodinger, Helmut. Comparisons in Hoare's Find algorithm. Combin. Probab. Comput. 7 (1998), no. 1, 111–120.
  • Kirschenhofer, P.; Prodinger, H.; Martínez, C. Analysis of Hoare's FIND algorithm with median-of-three partition. Average-case analysis of algorithms (Dagstuhl, 1995). Random Structures Algorithms 10 (1997), no. 1-2, 143–156.
  • Knape, Margarete; Neininger, Ralph. Approximating perpetuities. Methodol. Comput. Appl. Probab. 10 (2008), no. 4, 507–529.
  • Knape, Margarete; Neininger, Ralph. Appendix to "Approximating perpetuities" []. Methodol. Comput. Appl. Probab. 15 (2013), no. 3, 707–712.
  • Knof, D.: Struktur von Fixpunkten aus Prozess gleichungen. Dissertation, Christian-Albrechts-Universität zu Kiel, (2007). Electronically available via urn:nbn:de:gbv:8-diss-23169
  • Knof, Diether; Roesler, Uwe. The analysis of find and versions of it. Discrete Math. Theor. Comput. Sci. 14 (2012), no. 2, 129–154.
  • Knuth, Donald E. Mathematical analysis of algorithms. Information processing 71 (Proc. IFIP Congress, Ljubljana, 1971), Vol. 1: Foundations and systems, pp. 19–27. North-Holland, Amsterdam, 1972.
  • Kodaj, B.; Móri, T. F. On the number of comparisons in Hoare's algorithm "FIND”. Studia Sci. Math. Hungar. 33 (1997), no. 1-3, 185–207.
  • Mahmoud, Hosam M. Average-case analysis of moves in Quick Select. ANALCO09—Workshop on Analytic Algorithmics and Combinatorics, 35–40, SIAM, Philadelphia, PA, 2009.
  • Mahmoud, Hosam M. Distributional analysis of swaps in Quick Select. Theoret. Comput. Sci. 411 (2010), no. 16-18, 1763–1769.
  • Mahmoud, Hosam M.; Modarres, Reza; Smythe, Robert T. Analysis of QUICKSELECT: an algorithm for order statistics. RAIRO Inform. Théor. Appl. 29 (1995), no. 4, 255–276.
  • Martínez, Conrado; Roura, Salvador. Optimal sampling strategies in quicksort and quickselect. SIAM J. Comput. 31 (2001/02), no. 3, 683–705 (electronic).
  • Martínez, Conrado; Panario, Daniel; Viola, Alfredo. Adaptive sampling strategies for quickselect. ACM Trans. Algorithms 6 (2010), no. 3, Art. 53, 46 pp.
  • Mörters, Peter; Peres, Yuval. Brownian motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. xii+403 pp. ISBN: 978-0-521-76018-8
  • Neininger, Ralph; Rüschendorf, Ludger. A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 (2004), no. 1, 378–418.
  • Neininger, Ralph; Rüschendorf, Ludger. Analysis of algorithms by the contraction method: additive and max-recursive sequences. Interacting stochastic systems, 435–450, Springer, Berlin, 2005.
  • Neininger, R. and Sulzbach, H.: On a functional contraction method. arXiv:1202.1370
  • Paulsen, Volkert. The moments of FIND. J. Appl. Probab. 34 (1997), no. 4, 1079–1082.
  • Ragab, M.: Partial Quicksort and weighted branching processes. Dissertation, Christian-Albrechts-Universität zu Kiel, (2011). Electronically available via urn:nbn:de:gbv:8-diss-73701
  • Ragab, Mahmoud; Roesler, Uwe. The Quicksort process. Stochastic Process. Appl. 124 (2014), no. 2, 1036–1054.
  • Rösler, Uwe. A limit theorem for "Quicksort". RAIRO Inform. Théor. Appl. 25 (1991), no. 1, 85–100.
  • Rösler, U.: Quickselect revisited. J. Iranian Stat. Soc. 3, (2004), 271–296.
  • Rösler, U.; Rüschendorf, L. The contraction method for recursive algorithms. Average-case analysis of algorithms (Princeton, NJ, 1998). Algorithmica 29 (2001), no. 1-2, 3–33.
  • Rüschendorf, Ludger. On stochastic recursive equations of sum and max type. J. Appl. Probab. 43 (2006), no. 3, 687–703.
  • Sulzbach, H.: On a Functional Contraction Method. Dissertation, Goethe-Universität Frankfurt am Main, (2012). Electronically available via urn:nbn:de:hebis:30:3-248587
  • Talagrand, Michel. Regularity of Gaussian processes. Acta Math. 159 (1987), no. 1-2, 99–149.
  • Vallée, Brigitte; Clément, Julien; Fill, James Allen; Flajolet, Philippe. The number of symbol comparisons in QuickSort and QuickSelect. Automata, languages and programming. Part I, 750–763, Lecture Notes in Comput. Sci., 5555, Springer, Berlin, 2009.
  • Wild, S., Nebel, M.E. and Mahmoud, H.M.: Analysis of Quickselect under Yaroslavskiy's Dual-Pivoting Algorithm. ARXIV1306.3819.