Asymptotic variance of random symmetric digital search trees

Hsien-Kuei Hwang (1) , Michael Fuchs (2) , Vytas Zacharovas (1)

1 STAT Sinica - Institute of Statistical Science
2 Department of Applied Mathematics [Hsinchu]

Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n (logn)(2)-variance for certain notions of total path-length is also clarified.

Mots clés: Digital search trees; Poisson generating functions; Poissonization; Laplace transform; Mellin transform; saddle-point method; Colless index; weighted path-length

Domaines: Mathématique discrète [cs.DM]

Hsien-Kuei Hwang, Michael Fuchs, Vytas Zacharovas. Asymptotic variance of random symmetric digital search trees. Discrete Mathematics and Theoretical Computer Science, 2010, Vol. 12 no. 2 (2), pp.103-165. ⟨10.46298/dmtcs.498⟩. ⟨hal-00990457⟩