Bimodality and phase transitions in the profile variance of random binary search trees

SIAM Journal on Discrete Mathematics

Volumn 19, Issue 1, 2005, Pages 19-45

  Drmota, Michael ^a   Hwang, Hsien Kuei ^b  

^a INSTITUT FÜR DISKRETE MATHEMATIK UND GEOMETRIE   (Austria)

^b INSTITUTE OF STATISTICAL SCIENCE   (Taiwan)

Author keywords

Asymptotic bimodality; Bessel functions; Binary search trees; Profile; Saddle point method; Singularity analysis; Stirling numbers of the first kind

Indexed keywords

APPROXIMATION THEORY; ASYMPTOTIC STABILITY; DATA STRUCTURES;

ASYMPTOTIC BIMODALITY; BINARY SEARCH TREES; PROFILE; SINGULARITY ANALYSIS;

PHASE TRANSITIONS;

EID: 22244444622     PISSN: 08954801     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895480104440134     Document Type: Article

References (30)

1. L. COMTET (1974), Advanced Combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, The Netherlands.
2. L. DEVROYE (1987), Branching processes in the analysis of the heights of trees, Acta Inform., 24, pp. 277-298.
3. L. DEVROYE.(1988), Applications of the theory of records in the study of random trees, Acta Inform., 26, pp. 123-130.
4. L. DEVROYE (1999), Universal limit laws for depths in random trees, SIAM J. Comput., 28, pp. 409-432.
5. L. DEVROYE (2003), Limit laws for sums of functions of subtrees of random binary search trees, SIAM J. Comput., 32, pp. 152-171.
6. L. DEVROYE AND J. M. ROBSON (1995), On the generation of random binary search trees, SIAM J. Comput., 24, pp. 1141-1156.
7. M. DRMOTA AND B. GITTENBERGER (1997), On the profile of random trees, Random Structures Algorithms, 10, pp. 421-451.
8. S. R. FINCH (2003), Mathematical Constants, Cambridge University Press, Cambridge, UK.
9. P. FLAJOLET AND A. M. ODLYZKO (1990), Singularity analysis of generating functions, SIAM J. Discrete Math., 3, pp. 216-240.
10. M. FUCHS, H.-K. HWANG, AND R. NEININGER (2004), Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees, preprint.
11. G. H. GONNET AND R. BAEZA-YATES, Handbook of Algorithms and Data Structures, 2nd ed., Addison-Wesley, Wokingham, UK, 1989.
12. J. M. HAMMERSLEY (1951), The sum of products of the natural numbers, Proc. London Math. Soc., 1, pp. 435-452.
13. H.-K. HWANG (1995), Asymptotic expansions for the Stirling numbers of the first kind, J. Combin. Theory, Ser. A, 71, pp. 343-351.
14. H.-K. HWANG AND R. NEININGER (2002), Phase change of limit laws in the quicksort recurrence under varying toll functions, SIAM J. Comput., 31, pp. 1687-1722.
15. J. JABBOUR-HATTAB (2001), Martingales and large deviations for binary search trees, Random Structures Algorithms, 19, pp. 112-127.
16. G. KERSTING (1998), On the Height Profile of a Conditioned Galton-Watson Tree, preprint.
17. D.E. KNUTH (1998), The Art of Computer Programming, Volume III, Sorting and Searching, 2nd ed., Addison-Wesley, Reading, MA.
18. G. LOUCHARD (1987), Exact and asymptotic distributions in digital and binary search trees, RAIRO Inform. Théor. Appl., 21, pp. 479-495.
19. G. LOUCHARD AND W. SZPANKOWSKI (1995), Average profile and limiting distribution for a phrase size in the Lempel-Ziv parsing algorithm, IEEE Trans. Inform. Theory, 41, pp. 478-488.
20. W. C. LYNCH (1965), More combinatorial properties of certain trees, Comput. J., 7, pp. 299-302.
21. H. M. MAHMOUD (1992), Evolution of Random Search Trees, John Wiley & Sons, New York.
22. H. MAHMOUD AND B. PITTEL (1984), On the most probable shape of a binary search tree grown from a random permutation, SIAM J. Algebraic Discrete Methods, 5, pp. 69-81.
23. S. N. MAJUMDAR AND P. L. KRAPIVSKY (2003), Extreme value statistics and traveling fronts: Various applications, Phys. A, 318, pp. 161-170.
24. A. MEIR AND J. W. MOON (1978), On the altitude of nodes in random trees, Canad. J. Math., 30, pp. 997-1015.
25. J. PITMAN (1999), The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest, Ann. Probab., 27, pp. 261-283.
26. B. G. PITTEL (1984), On growing random binary trees, J. Math. Anal. Appl., 103, pp. 461-480.
27. R. T. SMYTHE AND H. M. MAHMOUD (1995), A survey of recursive trees, Theory Probab. Math. Stat., 51, pp. 1-27.
28. N. M. TEMME (1993), Asymptotic estimates of Stirling numbers, Stud. Appl. Math., 89, pp. 233-243.
29. R. VAN DER HOFSTAD, G. HOOGHIEMSTRA, AND P. VAN MIEGHEM (2002), On the covariance of the level sizes in random recursive trees, Random Structures Algorithms, 20, pp. 519-539.
30. R. WONG (2001), Asymptotic Approximations of Integrals, SIAM, Philadelphia.