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

A limit process for partial match queries in random quadtrees and 22-d trees

We consider the problem of recovering items matching a partially specified pattern in multidimensional trees (quadtrees and $k$-d trees). We assume the traditional model where the data consist of independent and uniform points in the unit square. For this model, in a structure on $n$ points, it is known that the number of nodes $C_n(\xi )$ to visit in order to report the items matching a random query $\xi$, independent and uniformly distributed on $[0,1]$, satisfies $\mathbf {E}[{C_n(\xi )}]\sim\kappa n^{\beta}$, where $\kappa$ and $\beta$ are explicit constants. We develop an approach based on the analysis of the cost $C_n(s)$ of any fixed query $s\in[0,1]$, and give precise estimates for the variance and limit distribution of the cost $C_n(x)$. Our results permit us to describe a limit process for the costs $C_n(x)$ as $x$ varies in $[0,1]$; one of the consequences is that $\mathbf {E}[{\max_{x\in[0,1]}C_n(x)}]\sim \gamma n^{\beta}$; this settles a question of Devroye [Pers. Comm., 2000].Comment: Published in at http://dx.doi.org/10.1214/12-AAP912 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1107.223

On a functional contraction method

In den letzten zwanzig Jahren hat sich die Kontraktionsmethode als ein wesentlicher Zugang zu Problemen der Konvergenz in Verteilung von Folgen von Zufallsvariablen, die additiven Rekurrenzen genügen, herausgestellt. Dabei beschränkten sich ihre Anwendungen zunächst auf reellwertige Zufallsvariablen, in den letzten Jahren wurde die Methode allerdings auch für komplexere Wertebereiche, wie etwa Hilberträume entwickelt. Basierend auf der Klasse der Zolotarev-Metriken, die in den siebziger Jahren eingeführt wurden, entwickeln wir die Methode im Rahmen von Banachräumen und präzisieren sie in den Fällen von stetigen resp. cadlag Funktionen auf dem Einheitsintervall. Wir formulieren ausreichende Bedingungen an die unter Betrachtung stehende Folge und deren möglichen Grenzwert, welcher eine stochastische Fixpunktgleichung erfüllt, die es erlauben, in Anwendungen funktionale Grenzwertsätze zu beweisen. Im Weiteren präsentieren wir als Anwendung zunächst einen neuen Beweis vom klassischen Invarianzprinzip nach Donsker, der auf additiven Rekursionen beruht. Außerdem wenden wir die Methode zur Analyse der Komplexität von partiellen Suchproblemen in zweidimensionalen Quadrantenbäumen und 2-d Bäumen an. Diese grundlegenden Datenstrukturen werden seit ihrer Einführung in den siebziger Jahren viel studiert. Unsere Ergebnisse liefern Antworten auf Fragen, die seit den Pionierarbeiten von Flajolet et al. in den achtziger und neunziger Jahren auf diesem Gebiet unbeantwortet blieben. Wir erwarten, dass die von uns entwickelte funktionale Kontraktionsmethode in den nächsten Jahren zur Lösung weiterer Fragen des asymptotischen Verhaltens von Zufallsgrößen, die additive Rekursionen erfüllen, beitragen wird.Within the last twenty years, the contraction method has turned out to be a fruitful approach to distributional convergence of sequences of random variables which obey additive recurrences. It was mainly invented for applications in the real-valued framework; however, in recent years, more complex state spaces such as Hilbert spaces have been under consideration. Based upon the family of Zolotarev metrics which were introduced in the late seventies, we develop the method in the context of Banach spaces and work it out in detail in the case of continuous resp. cadlag functions on the unit interval. We formulate sufficient conditions for both the sequence under consideration and its possible limit which satisfies a stochastic fixed-point equation, that allow to deduce functional limit theorems in applications. As a first application we present a new and considerably short proof of the classical invariance principle due to Donsker. It is based on a recursive decomposition. Moreover, we apply the method in the analysis of the complexity of partial match queries in two-dimensional search trees such as quadtrees and 2-d trees. These important data structures have been under heavy investigation since their invention in the seventies. Our results give answers to problems that have been left open in the pioneering work of Flajolet et al. in the eighties and nineties. We expect that the functional contraction method will significantly contribute to solutions for similar problems involving additive recursions in the following years

AN ANALYSIS OF RANDOM d-DIMENSIONAL QUAD TREES* LUC DEVRDYEt AND LOUISE LAFDREST$

Abstract. It is shown that the depth of the last node inserted in a random quad tree constructed from independent uniform [Q, 11 d random vectors is in probability asymptotic to (2/d) log n, where log denotes the natural logarithm. In addition, for d =2, exact values are obtained for all the moments of the depth of the last node. Key words. average time analysis, probability ineq ' alities, random quad tree, multidimensional data structures, search tree, expected behavior, analysis of a;orithm