88 research outputs found
A statistical view on exchanges in Quickselect
In this paper we study the number of key exchanges required by Hoare's FIND
algorithm (also called Quickselect) when operating on a uniformly distributed
random permutation and selecting an independent uniformly distributed rank.
After normalization we give a limit theorem where the limit law is a perpetuity
characterized by a recursive distributional equation. To make the limit theorem
usable for statistical methods and statistical experiments we provide an
explicit rate of convergence in the Kolmogorov--Smirnov metric, a numerical
table of the limit law's distribution function and an algorithm for exact
simulation from the limit distribution. We also investigate the limit law's
density. This case study provides a program applicable to other cost measures,
alternative models for the rank selected and more balanced choices of the pivot
element such as median-of- versions of Quickselect as well as further
variations of the algorithm.Comment: Theorem 4.4 revised; accepted for publication in Analytic
Algorithmics and Combinatorics (ANALCO14
The CLT Analogue for Cyclic Urns
A cyclic urn is an urn model for balls of types where in each
draw the ball drawn, say of type , is returned to the urn together with a
new ball of type . The case is the well-known Friedman urn.
The composition vector, i.e., the vector of the numbers of balls of each type
after steps is, after normalization, known to be asymptotically normal for
. For the normalized composition vector does not
converge. However, there is an almost sure approximation by a periodic random
vector. In this paper the asymptotic fluctuations around this periodic random
vector are identified. We show that these fluctuations are asymptotically
normal for all . However, they are of maximal dimension only when
does not divide . For being a multiple of the fluctuations are
supported by a two-dimensional subspace.Comment: Extended abstract to be replaced later by a full versio
On the contraction method with degenerate limit equation
A class of random recursive sequences (Y_n) with slowly varying variances as
arising for parameters of random trees or recursive algorithms leads after
normalizations to degenerate limit equations of the form X\stackrel{L}{=}X.
For nondegenerate limit equations the contraction method is a main tool to
establish convergence of the scaled sequence to the ``unique'' solution of the
limit equation. In this paper we develop an extension of the contraction method
which allows us to derive limit theorems for parameters of algorithms and data
structures with degenerate limit equation. In particular, we establish some new
tools and a general convergence scheme, which transfers information on mean and
variance into a central limit law (with normal limit). We also obtain a
convergence rate result. For the proof we use selfdecomposability properties of
the limit normal distribution which allow us to mimic the recursive sequence by
an accompanying sequence in normal variables.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000017
Polya urns via the contraction method
We propose an approach to analyze the asymptotic behavior of P\'olya urns
based on the contraction method. For this, a new combinatorial discrete time
embedding of the evolution of the urn into random rooted trees is developed. A
decomposition of these trees leads to a system of recursive distributional
equations which capture the distributions of the numbers of balls of each
color. Ideas from the contraction method are used to study such systems of
recursive distributional equations asymptotically. We apply our approach to a
couple of concrete P\'olya urns that lead to limit laws with normal limit
distributions, with non-normal limit distributions and with asymptotic periodic
distributional behavior.Comment: minor revision; accepted for publication in Combinatorics,
Probability & Computing (Special issue dedicated to the memory of Philippe
Flajolet
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
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
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