1,040 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
Exact L^2-distance from the limit for QuickSort key comparisons (extended abstract)
Using a recursive approach, we obtain a simple exact expression for the
L^2-distance from the limit in R\'egnier's (1989) classical limit theorem for
the number of key comparisons required by QuickSort. A previous study by Fill
and Janson (2002) using a similar approach found that the d_2-distance is of
order between n^{-1} log n and n^{-1/2}, and another by Neininger and
Ruschendorf (2002) found that the Zolotarev zeta_3-distance is of exact order
n^{-1} log n. Our expression reveals that the L^2-distance is asymptotically
equivalent to (2 n^{-1} ln n)^{1/2}
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
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
Quicksort asymptotics
The number of comparisons X_n used by Quicksort to sort an array of n
distinct numbers has mean mu_n of order n log n and standard deviation of order
n. Using different methods, Regnier and Roesler each showed that the normalized
variate Y_n := (X_n - mu_n) / n converges in distribution, say to Y; the
distribution of Y can be characterized as the unique fixed point with zero mean
of a certain distributional transformation.
We provide the first rates of convergence for the distribution of Y_n to that
of Y, using various metrics. In particular, we establish the bound 2 n^{- 1 /
2} in the d_2-metric, and the rate O(n^{epsilon - (1 / 2)}) for
Kolmogorov-Smirnov distance, for any positive epsilon.Comment: 23 pages. See also http://www.mts.jhu.edu/~fill/ and
http://www.math.uu.se/~svante/ . To be submitted for publication in May, 200
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