TY - JOUR

T1 - Mixed Poisson approximation of node depth distributions in random binary search trees

AU - Grübel, Rudolf

AU - Stefanoski, Nikolče

PY - 2005/2

Y1 - 2005/2

N2 - We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and compounded standard distributions, we derive approximations by Poisson and mixed Poisson distributions; these lead to asymptotic normality results. We are particularly interested in the influence of the key value on the distribution of the node depth. Methodologically our message is that the explicit representation may provide additional insight if compared to the standard approach that is based on the recursive structure of the trees. Further, in order to exhibit the influence of the key on the distributional asymptotics, a suitable choice of distance of probability distributions is important. Our results are also applicable in connection with the number of recursions needed in Hoare's [Comm. ACM 4 (1961) 321-322] selection algorithm FIND.

AB - We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and compounded standard distributions, we derive approximations by Poisson and mixed Poisson distributions; these lead to asymptotic normality results. We are particularly interested in the influence of the key value on the distribution of the node depth. Methodologically our message is that the explicit representation may provide additional insight if compared to the standard approach that is based on the recursive structure of the trees. Further, in order to exhibit the influence of the key on the distributional asymptotics, a suitable choice of distance of probability distributions is important. Our results are also applicable in connection with the number of recursions needed in Hoare's [Comm. ACM 4 (1961) 321-322] selection algorithm FIND.

KW - Asymptotic normality

KW - Hoare's selection algorithm

KW - Mixed Poisson distributions

KW - Poisson approximation

KW - Random permutations

KW - Randomized algorithms

UR - http://www.scopus.com/inward/record.url?scp=14544301012&partnerID=8YFLogxK

U2 - 10.1214/105051604000000611

DO - 10.1214/105051604000000611

M3 - Article

AN - SCOPUS:14544301012

VL - 15

SP - 279

EP - 297

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 1 A

ER -