The Annals of Applied Probability

A functional limit theorem for the profile of search trees

Michael Drmota, Svante Janson, and Ralph Neininger

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Abstract

We study the profile Xn,k of random search trees including binary search trees and m-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}X_{n,k}$ for k=⌊αlogn⌋ in a certain range of α.

A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 1 (2008), 288-333.

Dates
First available in Project Euclid: 9 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1199890024

Digital Object Identifier
doi:10.1214/07-AAP457

Mathematical Reviews number (MathSciNet)
MR2380900

Zentralblatt MATH identifier
1143.68019

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

Keywords
Functional limit theorem search trees profile of trees random trees analysis of algorithms

Citation

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. doi:10.1214/07-AAP457. https://projecteuclid.org/euclid.aoap/1199890024


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References

  • Bentkus, Yu. V. and Rachkauskas, A. (1984). Estimates for the distance between sums of independent random elements in Banach spaces. Teor. Veroyatnost. i Primenen. 29 49--64. [English transl. Theory Probab. Appl. (1984) 50--65.]
    Mathematical Reviews (MathSciNet): MR0739500
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR0233396
    Zentralblatt MATH: 0172.21201
  • Bousquet-Mélou, M. and Janson, S. (2006). The density of the ISE and local limit laws for embedded trees. Ann. Appl. Probab. 16 1597--1632.
    Mathematical Reviews (MathSciNet): MR2260075
    Digital Object Identifier: doi:10.1214/105051606000000213
    Project Euclid: euclid.aoap/1159804993
    Zentralblatt MATH: 1132.60009
  • Chauvin, B., Drmota, M. and Jabbour-Hattab, J. (2001). The profile of binary search trees. Ann. Appl. Probab. 11 1042--1062.
    Mathematical Reviews (MathSciNet): MR1878289
    Digital Object Identifier: doi:10.1214/aoap/1015345394
    Project Euclid: euclid.aoap/1015345394
    Zentralblatt MATH: 1012.60038
  • Chauvin, B. and Drmota, M. (2006). The random multisection problem, travelling waves, and the distribution of the height of $m$-ary search trees. Algorithmica 46 299--327.
    Mathematical Reviews (MathSciNet): MR2291958
    Digital Object Identifier: doi:10.1007/s00453-006-0107-7
    Zentralblatt MATH: 1117.68095
  • Chauvin, B., Klein, T., Marckert, J.-F. and Rouault, A. (2005). Martingales and profile of binary search trees. Electron. J. Probab. 10 420--435.
    Mathematical Reviews (MathSciNet): MR2147314
  • Chauvin, B. and Rouault, A. (2004). Connecting yule process, bisection and binary search trees via martingales. J. Iranian Statistical Society 3 89--116.
  • Chern, H.-H. and Hwang, H.-K. (2001). Transitional behaviors of the average cost of Quicksort with median-of-$(2t+1)$. Algorithmica 29 44--69.
    Mathematical Reviews (MathSciNet): MR1887298
    Digital Object Identifier: doi:10.1007/BF02679613
    Zentralblatt MATH: 0967.68048
  • Chern, H.-H., Hwang, H.-K. and Tsai, T.-H. (2002). An asymptotic theory for Cauchy--Euler differential equations with applications to the analysis of algorithms. J. Algorithms 44 177--225.
    Mathematical Reviews (MathSciNet): MR1933199
    Digital Object Identifier: doi:10.1016/S0196-6774(02)00208-0
    Zentralblatt MATH: 1030.68114
  • Devroye, L. and Hwang, H.-K. (2006). Width and mode of the profile for some random trees of logarithmic height. Ann. Appl. Probab. 16 886--918.
    Mathematical Reviews (MathSciNet): MR2244436
    Digital Object Identifier: doi:10.1214/105051606000000187
    Project Euclid: euclid.aoap/1151592254
    Zentralblatt MATH: 1128.60008
  • Drmota, M. and Hwang, H.-K. (2005). Profiles of random trees: Correlation and width of random recursive trees and binary search trees. Adv. in Appl. Probab. 37 321--341.
    Mathematical Reviews (MathSciNet): MR2144556
    Digital Object Identifier: doi:10.1239/aap/1118858628
    Project Euclid: euclid.aap/1118858628
    Zentralblatt MATH: 1073.60006
  • Drmota, M. and Hwang, H.-K. (2005). Bimodality and phase transitions in the profile variance of random binary search trees. SIAM J. Discrete Math. 19 19--45.
    Mathematical Reviews (MathSciNet): MR2178182
    Digital Object Identifier: doi:10.1137/S0895480104440134
    Zentralblatt MATH: 1086.68037
  • Flajolet, P. and Odlyzko, A. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216--240.
    Mathematical Reviews (MathSciNet): MR1039294
    Digital Object Identifier: doi:10.1137/0403019
    Zentralblatt MATH: 0712.05004
  • Flajolet, P. and Sedgewick, R. (2008). Analytic Combinatorics. Cambridge Univ. Press. To appear. Available at http://algo.inria.fr/flajolet/Publications/books.html.
    Mathematical Reviews (MathSciNet): MR2483235
  • Fuchs, M., Hwang, H.-K. and Neininger, R. (2006). Profiles of random trees: Limit theorems for random recursive trees and binary search trees. Algorithmica 46 367--407.
    Mathematical Reviews (MathSciNet): MR2291961
    Digital Object Identifier: doi:10.1007/s00453-006-0109-5
    Zentralblatt MATH: 1106.68083
  • Giné, E. and León, J. R. (1980). On the central limit theorem in Hilbert space. Stochastica 4 43--71.
    Mathematical Reviews (MathSciNet): MR0573725
    Zentralblatt MATH: 0432.60011
  • Hwang, H.-K. (2007). Profiles of random trees: Plane-oriented recursive trees. Random Structures Algorithms 30 380--413.
    Mathematical Reviews (MathSciNet): MR2309623
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR0959133
    Zentralblatt MATH: 0635.60021
  • Janson, S. (1994). Orthogonal Decompositions and Functional Limit Theorems for Random Graph Statistics. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR1219708
  • Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Applications. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR0488211
    Zentralblatt MATH: 0352.60001
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1876169
    Zentralblatt MATH: 0996.60001
  • Krantz, S. G. (1982). Function Theory of Several Complex Variables. Wiley, New York, 1982.
    Mathematical Reviews (MathSciNet): MR0635928
    Zentralblatt MATH: 0471.32008
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1102015
    Zentralblatt MATH: 0748.60004
  • Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space $D(0, \infty)$. J. Appl. Probab. 10 109--121.
    Mathematical Reviews (MathSciNet): MR0362429
    Digital Object Identifier: doi:10.2307/3212499
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0258.60008
  • Lynch, W. (1965). More combinatorial problems on certain trees. Comput. J. 7 299--302.
    Mathematical Reviews (MathSciNet): MR0172492
  • Mahmoud, H. M. (1992). Evolution of Random Search Trees. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR1140708
    Zentralblatt MATH: 0762.68033
  • Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378--418.
    Mathematical Reviews (MathSciNet): MR2023025
    Digital Object Identifier: doi:10.1214/aoap/1075828056
    Project Euclid: euclid.aoap/1075828056
    Zentralblatt MATH: 1041.60024
  • Rudin, W. (1991). Functional Analysis, 2nd ed. McGraw-Hill, New York.
    Mathematical Reviews (MathSciNet): MR1157815
    Zentralblatt MATH: 0867.46001
  • Trèves, F. (1967). Topological Vector Spaces, Distributions and Kernels. Academic Press, New York.
    Mathematical Reviews (MathSciNet): MR0225131
  • Zolotarev, V. M. (1977). Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces. Teor. Veroyatnost. i Primenen. 21 741--758. [Erratum (1977) 22 901. English transl. Theory Probab. Appl. (1976) 21 721--737; (1977) 22 881.]
    Mathematical Reviews (MathSciNet): MR0517338
  • Zolotarev, V. M. (1977). Ideal metrics in the problem of approximating the distributions of sums of independent random variables. Teor. Veroyatnost. i Primenen. 22 (1977) 449--465. [English transl. Theory Probab. Appl. (1977) 22 433--449.]
    Mathematical Reviews (MathSciNet): MR0455066

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