A functional limit theorem for the profile of search trees

Abstract

We study the profile $X_{n,k}$ of random search trees including binary search trees and $m$-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}X_{n,k}$ for $k=\lfloor\alpha\log n\rfloor$ in a certain range of $\alpha$. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.Comment: Published in at http://dx.doi.org/10.1214/07-AAP457 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Partial match queries in random quadtrees

“…In the context of the analysis of performance of algorithms, the method was first employed by Rösler [29] who proved convergence in distribution for the rescaled total cost of the randomized version of quicksort. The method was then further developed by Rachev and Rüschendorf [27], Rösler [30], and later on in [5,9,22,24,25,31] and has permitted numerous analyses in distribution for random discrete structures. So far, the method has mostly been used to analyze random variables taking real values, though a few applications on functions spaces have been made, see [5,9,16].…”

“…The method was then further developed by Rachev and Rüschendorf [27], Rösler [30], and later on in [5,9,22,24,25,31] and has permitted numerous analyses in distribution for random discrete structures. So far, the method has mostly been used to analyze random variables taking real values, though a few applications on functions spaces have been made, see [5,9,16]. Here we are interested in the function space D[0, 1] with the uniform topology, but the main idea persists: (1) devise a recursive equation for the quantity of interest (here the process(C n (s), s ∈ [0, 1])), and (2) prove that a properly rescaled version of the quantity converges to a fixed point of a certain map related to the recursive equation ; (3) if the map is a contraction in a certain metric space, then a fixed point is unique and may be obtained by iteration.…”

Mass distribution exponents for growing trees

“…In this respect, a promising direction of investigation is provided by the approach of the problem via continuous time branching processes [8][9][10]. Indeed, within this framework, it is possible to relate the mass distribution to other asymptotic large time distributions which are themeselves determined by coupled integral equations.…”

“…These problems have been analyzed using continuous time branching processes [8][9][10] and we believe that the mass distribution exponents can in principle be deduced using those methods. However, we find it illuminating to study the model directly using discrete time and elementary methods.…”

On martingale tail sums for the path length in random trees

“…An alternative proof could be given relying on a distributional recurrence for the sequence W n (z). This approach was taken by Drmota, Janson and Neininger [11] studying the profile of m-ary search trees including the case of BSTs. As for the martingale approach, this methodology can be worked out for any β ∈ N 0 or β = −1.…”