6 research outputs found
Width and mode of the profile for some random trees of logarithmic height
We propose a new, direct, correlation-free approach based on central moments
of profiles to the asymptotics of width (size of the most abundant level) in
some random trees of logarithmic height. The approach is simple but gives
precise estimates for expected width, central moments of the width and almost
sure convergence. It is widely applicable to random trees of logarithmic
height, including recursive trees, binary search trees, quad trees,
plane-oriented ordered trees and other varieties of increasing trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000187 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Pivot Sampling in Dual-Pivot Quicksort
The new dual-pivot Quicksort by Vladimir Yaroslavskiy - used in Oracle's Java
runtime library since version 7 - features intriguing asymmetries in its
behavior. They were shown to cause a basic variant of this algorithm to use
less comparisons than classic single-pivot Quicksort implementations. In this
paper, we extend the analysis to the case where the two pivots are chosen as
fixed order statistics of a random sample and give the precise leading term of
the average number of comparisons, swaps and executed Java Bytecode
instructions. It turns out that - unlike for classic Quicksort, where it is
optimal to choose the pivot as median of the sample - the asymmetries in
Yaroslavskiy's algorithm render pivots with a systematic skew more efficient
than the symmetric choice. Moreover, the optimal skew heavily depends on the
employed cost measure; most strikingly, abstract costs like the number of swaps
and comparisons yield a very different result than counting Java Bytecode
instructions, which can be assumed most closely related to actual running time.Comment: presented at AofA 2014 (http://www.aofa14.upmc.fr/
A functional limit theorem for the profile of search trees
We study the profile of random search trees including binary search
trees and -ary search trees. Our main result is a functional limit theorem
of the normalized profile for in a certain range of . 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