The Annals of Applied Probability

A functional limit theorem for the profile of search trees

Michael Drmota, Svante Janson, and Ralph Neininger

Full-text: Open access

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.]
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • 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.
  • Chauvin, B., Drmota, M. and Jabbour-Hattab, J. (2001). The profile of binary search trees. Ann. Appl. Probab. 11 1042--1062.
  • 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.
  • Chauvin, B., Klein, T., Marckert, J.-F. and Rouault, A. (2005). Martingales and profile of binary search trees. Electron. J. Probab. 10 420--435.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Flajolet, P. and Odlyzko, A. (1990). Singularity analysis of generating functions. SIAM J. Discrete Math. 3 216--240.
  • Flajolet, P. and Sedgewick, R. (2008). Analytic Combinatorics. Cambridge Univ. Press. To appear. Available at http://algo.inria.fr/flajolet/Publications/books.html.
  • 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.
  • Giné, E. and León, J. R. (1980). On the central limit theorem in Hilbert space. Stochastica 4 43--71.
  • Hwang, H.-K. (2007). Profiles of random trees: Plane-oriented recursive trees. Random Structures Algorithms 30 380--413.
  • Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • Janson, S. (1994). Orthogonal Decompositions and Functional Limit Theorems for Random Graph Statistics. Amer. Math. Soc., Providence, RI.
  • Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Applications. Wiley, New York.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • Krantz, S. G. (1982). Function Theory of Several Complex Variables. Wiley, New York, 1982.
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Isoperimetry and Processes. Springer, Berlin.
  • Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space $D(0, \infty)$. J. Appl. Probab. 10 109--121.
  • Lynch, W. (1965). More combinatorial problems on certain trees. Comput. J. 7 299--302.
  • Mahmoud, H. M. (1992). Evolution of Random Search Trees. Wiley, New York.
  • Neininger, R. and Rüschendorf, L. (2004). A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 378--418.
  • Rudin, W. (1991). Functional Analysis, 2nd ed. McGraw-Hill, New York.
  • Trèves, F. (1967). Topological Vector Spaces, Distributions and Kernels. Academic Press, New York.
  • 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.]
  • 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.]