Second phase changes in random m-ary search trees and generalized quicksort: Convergence rates

Annals of Probability

Volumn 31, Issue 2, 2003, Pages 609-629

  Hwang, Hsien Kuei ^a  

^a INSTITUTE OF STATISTICAL SCIENCE   (Taiwan)

Author keywords

Asymptotic normality; Convergence rates; Local limit theorems; Method of moments; Phase change; Quicksort; Search trees

Indexed keywords

EID: 0038336151     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1048516530     Document Type: Article

References (25)

1. ALDOUS, D. (1996). Probability distributions on cladograms. In Random Discrete Structures (D. Aldous and R. Pemantle, eds.) 1-18. Springer, New York.
2. BARLOW, M. T., PEMANTLE, R. and PERKINS, E. A. (1997). Diffusion-limited aggregation on a tree. Probab. Theory Related Fields 107 1-60.
3. BEN-NAIM, E., KRAPIVSKY, P. L. and MAJUMDAR, S. N. (2001). Extremal properties of random trees. Phys. Rev. E 64.
4. BERRY, A. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49 122-136.
5. 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.
6. DEVROYE, L. (1999). Universal limit laws for depths in random trees. SIAM J. Comput. 28 409-432.
7. DEVROYE, L. (2002). Limit laws for sums of functions of subtrees of random binary search trees. SIAM J. Comput. 32 152-171.
8. DONGARRA, J. and SULLIVAN, S. (2000). The top 10 algorithms. Computing in Science and Engineering January/February 22-23.
9. ESSEEN, C.-G. (1945). Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math. 77 1-125.
10. FILL, J. A. and JANSON, S. (2002). Quicksort asymptotics. J. Algorithms 44 4-28.
11. GONNET, G. H. (1991). Handbook of Algorithms and Data Structures (in Pascal and C). Addison-Wesley, Reading, MA.
12. HENNEQUIN, P. (1989). Combinatorial analysis of quicksort algorithm. RAIRO Inform. Théor. Appl. 23 317-333.
13. HENNEQUIN, P. (1991). Analyse en moyenne d'algorithme, tri rapide et arbres de recherche. Ph.D. thesis, Ecole Polytechnique. Available at pauillac.inria.fr/algo/AofA/Research/src/Hennequin.PhD.html.
14. HOARE, C. A. R. (1962). Quicksort. Comput. J. 5 10-15.
15. HWANG, H.-K. (1996). Large deviations for combinatorial distributions. I. Central limit theorems. Ann. Appl. Probab. 6 297-319.
16. HWANG, H.-K. and NEININGER, R. (2002). Phase changes of the limit laws in the quicksort recurrence under varying toll functions. SIAM J. Comput. 31 1687-1722.
17. LEW, W. and MAHMOUD, H. M. (1994). The joint distribution of elastic buckets in multiway search trees. SIAM J. Comput. 23 1050-1074.
18. MAHMOUD, H. M. (1992). Evolution of Random Search Trees. Wiley, New York.
19. MAHMOUD, H. M. (2002). The size of random bucket trees via urn models. Acta Inform. 38 813-383.
20. MAHMOUD, H. M. and PITTEL, B. (1984). On the most probable shape of a search tree grown from random permutations. SIAM Journal on Algebraic and Discrete Methods 1 69-81.
21. MAHMOUD, H. M. and PITTEL, B. (1989). Analysis of the space of search trees under the random insertion algorithm. J. Algorithms 10 52-75.
22. NEININGER, R. and RÜSCHENDORF, L. (2001). A general contraction theorem and asymptotic normality in combinatorial structures. Available at www.stochastik.uni-freiburg.de/homepages/neininger.
23. PETROV, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
24. PITTEL, B. (1999). Normal convergence problem? Two moments and a recurrence may be the clues. Ann. Applied Probab. 9 1260-1302.
25. SEDGEWICK, R. (1980). Quicksort. Garland, New York.