The Annals of Applied Probability

A general limit theorem for recursive algorithms and combinatorial structures

Ralph Neininger and Ludger Rüschendorf

Full-text: Open access

Abstract

Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer theorem is derived which allows us to establish a limit law on the basis of the recursive structure and the asymptotics of the first and second moments of the sequence. In particular, a general asymptotic normality result is obtained by this theorem which typically cannot be handled by the more common $\ell_2$ metrics. As applications we derive quite automatically many asymptotic limit results ranging from the size of tries or $m$-ary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by different methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proofs of these results we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric, which is the main tool in this article.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 1 (2004), 378-418.

Dates
First available in Project Euclid: 3 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1075828056

Digital Object Identifier
doi:10.1214/aoap/1075828056

Mathematical Reviews number (MathSciNet)
MR2023025

Zentralblatt MATH identifier
1041.60024

Subjects
Primary: 60F05: Central limit and other weak theorems 68Q25: Analysis of algorithms and problem complexity [See also 68W40]
Secondary: 68P10: Searching and sorting

Keywords
Contraction method multivariate limit law asymptotic normality random trees recursive algorithms divide-and-conquer algorithm random recursive structures Zolotarev metric

Citation

Neininger, Ralph; Rüschendorf, Ludger. A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab. 14 (2004), no. 1, 378--418. doi:10.1214/aoap/1075828056. https://projecteuclid.org/euclid.aoap/1075828056


Export citation

References

  • Aho, A. V., Hopcroft, J. E. and Ullman, J. D. (1983). Data Structures and Algorithms. Addison--Wesley, Reading, MA.
  • Baeza-Yates, R. A. (1987). Some average measures in $m$-ary search trees. Inform. Process. Lett. 25 375--381.
  • Bai, Z.-D., Hwang, H.-K., Liang, W.-Q. and Tsai, T.-H. (2001). Limit theorems for the number of maxima in random samples from planar regions. Electron. J. Probab. 6.
  • Bai, Z.-D., Hwang, H.-K. and Tsai, T.-H. (2002). Berry--Esseen bounds for the number of maxima in planar regions. Electron J. Probab. 8.
  • Burton, R. and Rösler, U. (1995). An $L_2$ convergence theorem for random affine mappings. J. Appl. Probab. 32 183--192.
  • Chen, W.-M., Hwang, H.-K. and Chen, G.-H. (1999). The cost distribution of queue-mergesort, optimal mergesorts, and power-of-2 rules. J. Algorithms 30 423--448.
  • Chern, H.-H. and Hwang, H.-K. (2001). Phase changes in random $m$-ary search trees and generalized quicksort. Random Structures Algorithms 19 316--358.
  • 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.
  • Cramer, M. (1997). Stochastic analysis of the merge--sort algorithm. Random Structures Algorithms 11 81--96.
  • Cramer, M. and Rüschendorf, L. (1995). Analysis of recursive algorithms by the contraction method. Athens Conference on Applied Probability and Time Series Analysis. Lecture Notes in Statist. 114 18--33. Springer, New York.
  • Devroye, L. (1991). Limit laws for local counters in random binary search trees. Random Structures Algorithms 2 303--315.
  • Devroye, L. (2003). Limit laws for sums of functions of subtrees of random binary search trees. SIAM J. Comput. 32 152--171.
  • Dobrow, R. P. and Fill, J. A. (1999). Total path length for random recursive trees. Combin. Probab. Comput. 8 317--333.
  • Fill, J. A. (1996). On the distribution of binary search trees under the random permutation model. Random Structures Algorithms 8 1--25.
  • Flajolet, P. and Golin, M. (1994). Mellin transforms and asymptotics. The mergesort recurrence. Acta Inform. 31 673--696.
  • Flajolet, P., Gourdon, X. and Martínez, C. (1997). Patterns in random binary search trees. Random Structures Algorithms 11 223--244.
  • Geiger, J. (2000). A new proof of Yaglom's exponential limit law. In Algorithms, Trees Combinatorics and Probability (D. Gardy and A. Mokkadem, eds.) 245--249. Birkhäuser, Basel.
  • Grübel, R. and Rösler, U. (1996). Asymptotic distribution theory for Hoare's selection algorithm. Adv. in Appl. Probab. 28 252--269.
  • Hubalek, F., Hwang, H.-K., Lew, W., Mahmoud, H. M. and Prodinger, H. (2002). A multivariate view of random bucket digital search trees. J. Algorithms 44 121--158.
  • Hwang, H.-K. (1996). Asymptotic expansions of the mergesort recurrences. Acta Inform. 35 911--919.
  • Hwang, H.-K. (1998). Limit theorems for mergesort. Random Structures Algorithms 8 319--336.
  • Hwang, H.-K. (2001). Lectures on asymptotic analysis given at McGill Univ., Montreal.
  • Hwang, H.-K. and Neininger, R. (2002). Phase change of limit laws in the quicksort recurrence under varying toll functions. SIAM J. Comput. 31 1687--1722.
  • Hwang, H.-K. and Tsai, T.-H. (2002). Quickselect and Dickman function. Combin. Probab. Comput. 11 353--371.
  • Jacquet, P. and Régnier, M. (1988). Normal limiting distribution of the size and the external path length of tries. Technical Report RR-0827, INRIA-Rocquencourt.
  • Jacquet, P. and Régnier, M. (1988). Normal limiting distribution of the size of tries. In Performance'87 209--223. North-Holland, Amsterdam.
  • Jacquet, P. and Szpankowski, W. (1995). Asymptotic behavior of the Lempel--Ziv parsing scheme and digital search trees. Theoret. Comput. Sci. 144 161--197.
  • Kirschenhofer, P., Prodinger, H. and Szpankowski, W. (1989). On the balance property of Patricia tries: External path length viewpoint. Theoret. Comput. Sci. 68 1--17.
  • Kirschenhofer, P., Prodinger, H. and Szpankowski, W. (1989). On the variance of the external path length in a symmetric digital trie. Discrete Appl. Math. 25 129--143.
  • Kirschenhofer, P., Prodinger, H. and Szpankowski, W. (1994). Digital search trees again revisited: The internal path length perspective. SIAM J. Comput. 23 598--616.
  • Knuth, D. E. (1973). The Art of Computer Programming 3. Addison--Wesley, Reading, MA.
  • Kodaj, B. and Móri, T. F. (1997). On the number of comparisons in Hoare's algorithm ``FIND.'' Studia Sci. Math. Hungar. 33 185--207.
  • Lew, W. and Mahmoud, H. M. (1994). The joint distribution of elastic buckets in multiway search trees. SIAM J. Comput. 23 1050--1074.
  • Mahmoud, H. M. (1992). Evolution of Random Search Trees. Wiley, New York.
  • Mahmoud, H. M. (2000). Sorting. A Distribution Theory. Wiley, New York.
  • Mahmoud, H. M. and Pittel, B. (1989). Analysis of the space of search trees under the random insertion algorithm. J. Algorithms 10 52--75.
  • Mahmoud, H. M. and Smythe, R. T. (1991). On the distribution of leaves in rooted subtrees of recursive trees. Ann. Appl. Probab. 1 406--418.
  • Mahmoud, H. M. and Smythe, R. T. (1992). Asymptotic joint normality of outdegrees of nodes in random recursive trees. Random Structures Algorithms 3 255--266.
  • Mahmoud, H. M., Modarres, R. and Smythe, R. T. (1995). Analysis of quickselect: An algorithm for order statistics. RAIRO Inform. Théor. Appl. 29 255--276.
  • Mahmoud, H. M., Smythe, R. T. and Szymański, J. (1993). On the structure of random plane-oriented recursive trees and their branches. Random Structures Algorithms 4 151--176.
  • Neininger, R. (2001). On a multivariate contraction method for random recursive structures with applications to Quicksort. Random Structures Algorithms 19 498--524.
  • Neininger, R. and Rüschendorf, L. (1999). On the internal path length of $d$-dimensional quad trees. Random Structures Algorithms 15 25--41.
  • Neininger, R. and Rüschendorf, L. (2002). Rates of convergence for Quicksort. J. Algorithms 44 52--62.
  • Pittel, B. (1999). Normal convergence problem? Two moments and a recurrence may be the clues. Ann. Appl. Probab. 9 1260--1302.
  • Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. Wiley, New York.
  • Rachev, S. T. and Rüschendorf, L. (1994). On the rate of convergence in the CLT with respect to the Kantorovich metric. In Probability in Banach Spaces (J. Hoffmann-Jorgensen, J. Kuelbs and M. B. Marcus, eds.) 9 193--207. Birkhäuser, Boston.
  • Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. in Appl. Probab. 27 770--799.
  • Rösler, U. (1991). A limit theorem for ``Quicksort.'' RAIRO Inform. Théor. Appl. 25 85--100.
  • Rösler, U. (1992). A fixed point theorem for distributions. Stochastic Process. Appl. 42 195--214.
  • Rösler, U. (2001). On the analysis of stochastic divide and conquer algorithms. Algorithmica 29 238--261.
  • Rösler, U. and Rüschendorf, L. (2001). The contraction method for recursive algorithms. Algorithmica 29 3--33.
  • Schachinger, W. (2001). Asymptotic normality of recursive algorithms via martingale difference arrays. Discrete Math. Theor. Comput. Sci. 4 363--397.
  • Senatov, V. V. (1980). Some uniform estimates of the convergence rate in the multidimensional central limit theorem. Theory Probab. Appl. 25 745--759.
  • Smythe, R. T. and Mahmoud, H. M. (1994). A survey of recursive trees. Teor. Īmovīr. ta Mat. Statist. 51 1--29.
  • Szpankowski, W. (2001). Average Case Analysis of Algorithms on Sequences. Wiley, New York.
  • Tenenbaum, G. (1995). Introduction to Analytic and Probabilistic Number Theory. (C. B. Thomas, transl.). Cambridge Univ. Press.
  • Zolotarev, V. M. (1976). Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces. Theory Probab. Appl. 21 721--737.
  • Zolotarev, V. M. (1977). Ideal metrics in the problem of approximating distributions of sums of independent random variables. Theory Probab. Appl. 22 433--449.