31 research outputs found
A limit field for orthogonal range searches in two-dimensional random point search trees
We consider the cost of general orthogonal range queries in random quadtrees.
The cost of a given query is encoded into a (random) function of four variables
which characterize the coordinates of two opposite corners of the query
rectangle. We prove that, when suitably shifted and rescaled, the random cost
function converges uniformly in probability towards a random field that is
characterized as the unique solution to a distributional fixed-point equation.
We also state similar results for -d trees. Our results imply for instance
that the worst case query satisfies the same asymptotic estimates as a typical
query, and thereby resolve an old question of Chanzy, Devroye and Zamora-Cura
[\emph{Acta Inf.}, 37:355--383, 2000]Comment: 24 pages, 8 figure
The dual tree of a recursive triangulation of the disk
In the recursive lamination of the disk, one tries to add chords one after
another at random; a chord is kept and inserted if it does not intersect any of
the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011)
2224-2270] have proved that the set of chords converges to a limit
triangulation of the disk encoded by a continuous process . Based
on a new approach resembling ideas from the so-called contraction method in
function spaces, we prove that, when properly rescaled, the planar dual of the
discrete lamination converges almost surely in the Gromov-Hausdorff sense to a
limit real tree , which is encoded by . This confirms
a conjecture of Curien and Le Gall.Comment: Published in at http://dx.doi.org/10.1214/13-AOP894 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On martingale tail sums in affine two-color urn models with multiple drawings
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and
arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn
schemes with multiple drawings. We show that, in large-index urns (urn index
between and ) and triangular urns, the martingale tail sum for the
number of balls of a given color admits both a Gaussian central limit theorem
as well as a law of the iterated logarithm. The laws of the iterated logarithm
are new even in the standard model when only one ball is drawn from the urn in
each step (except for the classical Polya urn model). Finally, we prove that
the martingale limits exhibit densities (bounded under suitable assumptions)
and exponentially decaying tails. Applications are given in the context of node
degrees in random linear recursive trees and random circuits.Comment: 17 page
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 genĂŒgen, herausgestellt. Dabei beschrĂ€nkten sich ihre Anwendungen zunĂ€chst auf reellwertige Zufallsvariablen, in den letzten Jahren wurde die Methode allerdings auch fĂŒr komplexere Wertebereiche, wie etwa HilbertrĂ€ume entwickelt. Basierend auf der Klasse der Zolotarev-Metriken, die in den siebziger Jahren eingefĂŒhrt wurden, entwickeln wir die Methode im Rahmen von BanachrĂ€umen und prĂ€zisieren sie in den FĂ€llen 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 erfĂŒllt, die es erlauben, in Anwendungen funktionale GrenzwertsĂ€tze zu beweisen. Im Weiteren prĂ€sentieren wir als Anwendung zunĂ€chst einen neuen Beweis vom klassischen Invarianzprinzip nach Donsker, der auf additiven Rekursionen beruht. AuĂerdem wenden wir die Methode zur Analyse der KomplexitĂ€t von partiellen Suchproblemen in zweidimensionalen QuadrantenbĂ€umen und 2-d BĂ€umen an. Diese grundlegenden Datenstrukturen werden seit ihrer EinfĂŒhrung 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 nĂ€chsten Jahren zur Lösung weiterer Fragen des asymptotischen Verhaltens von ZufallsgröĂen, die additive Rekursionen erfĂŒllen, 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
On a functional contraction method
Methods for proving functional limit laws are developed for sequences of
stochastic processes which allow a recursive distributional decomposition
either in time or space. Our approach is an extension of the so-called
contraction method to the space of continuous functions
endowed with uniform topology and the space of
c\`{a}dl\`{a}g functions with the Skorokhod topology. The contraction method
originated from the probabilistic analysis of algorithms and random trees where
characteristics satisfy natural distributional recurrences. It is based on
stochastic fixed-point equations, where probability metrics can be used to
obtain contraction properties and allow the application of Banach's fixed-point
theorem. We develop the use of the Zolotarev metrics on the spaces
and in this context. Applications are
given, in particular, a short proof of Donsker's functional limit theorem is
derived and recurrences arising in the probabilistic analysis of algorithms are
discussed.Comment: Published at http://dx.doi.org/10.1214/14-AOP919 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On martingale tail sums for the path length in random trees
For a martingale converging almost surely to a random variable ,
the sequence is called martingale tail sum. Recently, Neininger
[Random Structures Algorithms, 46 (2015), 346-361] proved a central limit
theorem for the martingale tail sum of R{\'e}gnier's martingale for the path
length in random binary search trees. Gr{\"u}bel and Kabluchko [to appear in
Annals of Applied Probability, (2016), arXiv 1410.0469] gave an alternative
proof also conjecturing a corresponding law of the iterated logarithm. We prove
the central limit theorem with convergence of higher moments and the law of the
iterated logarithm for a family of trees containing binary search trees,
recursive trees and plane-oriented recursive trees.Comment: Results generalized to broader tree model; convergence of moments in
the CL
On weighted depths in random binary search trees
Following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg.
Inform. Sci. 21 (2007) 133-141], the weighted depth of a node in a labelled
rooted tree is the sum of all labels on the path connecting the node to the
root. We analyze weighted depths of nodes with given labels, the last inserted
node, nodes ordered as visited by the depth first search process, the weighted
path length and the weighted Wiener index in a random binary search tree. We
establish three regimes of nodes depending on whether the second order
behaviour of their weighted depths follows from fluctuations of the keys on the
path, the depth of the nodes, or both. Finally, we investigate a random
distribution function on the unit interval arising as scaling limit for
weighted depths of nodes with at most one child
A limit process for partial match queries in random quadtrees and -d trees
We consider the problem of recovering items matching a partially specified
pattern in multidimensional trees (quadtrees and -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 points, it is known that
the number of nodes to visit in order to report the items matching
a random query , independent and uniformly distributed on ,
satisfies , where and
are explicit constants. We develop an approach based on the analysis of
the cost of any fixed query , and give precise estimates
for the variance and limit distribution of the cost . Our results
permit us to describe a limit process for the costs as varies in
; one of the consequences is that ; 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
The profile of binary search trees
BinĂ€rsuchbĂ€ume sind eine wichtige Datenstruktur, die in der Informatik vielfach Anwendung finden. Ihre Konstruktion ist deterministisch, zur Analyse ihrer Eigenschaften wird aber eine rein zufĂ€llige Eingabe zugrundegelegt. Viele GröĂe, wie z.B. Tiefe, Höhe und PfadlĂ€nge werden seit Jahren viel untersucht. Als besonders interessant hat sich die Analyse des Profils, der Anzahl Knoten einer bestimmten Tiefe herausgestellt. In dieser Arbeit wird ein funktionaler Grenzwertsatz fĂŒr das am Erwartungswert normierte Profil vorgestellt. Dazu werden unterschiedliche ZugĂ€nge gewĂ€hlt, die hauptsĂ€chlich auf dem sogenannten Profil-Polynom beruhen. ZunĂ€chst wird ein klassischer Zugang mit Hilfe von Martingalen besprochen. Der diskrete Prozess wird dazu auf kanonische Weise in ein zeitstetiges Modell (Yule-Prozess) eingebettet. Ergebnisse im kontinuierlichen Prozess werden dann durch Stoppen auf den diskreten ĂŒbertragen. Zudem wird ein neuerer Zugang vorgestellt, der auf der Kontraktionsmethode in BanachrĂ€umen unter Verwendung der Zolotarev-Metrik beruht.Binary search trees are an important data structure in Computer Science. Their construction is deterministic, for analyzing their typical behavior a random input is assumed. Many quantities like depth, height and path length have been studied for years. The analysis of the profile, the number of nodes on a certain level has turned out to be quite interesting. In this work a functional limit theorem for the profile, normalized by its expectation, is presented. Two different approaches, both based upon the so-called profile-polynomial, are presented. A classical approach based upon martingales is discussed. The discrete process is embedded in a time-continuous model (Yule-process) in a canonical way. Results for the continuous process are transferred to the discrete one by making use of stopping times. Secondly, a new approach based on to the contraction method in Banach spaces using the Zolotarev metric is presented