Random Permutation Model Research Articles

An almost sure result for path lengths in binary search trees

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variableZ. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜mof finite binary trees with at mostmnodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector timesZ.

Asymptotic analysis of (3, 2, 1)-shell sort

We analyze the (3, 2, 1)-Shell Sort algorithm under the usual random permutation model. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 59–75, 2002

Stochastic Analysis of Shell Sort

We analyze the Shell Sort algorithm under the usual random permutation model. Using empirical distribution functions, we recover Louchard's result that the running time of the 1-stage of.2; 1/-Shell Sort has a limiting distribution given by the area under the absolute Brownian bridge. The analysis extends to .h; 1/- Shell Sort where we find a limiting distribution given by the sum of areas under correlated absolute Brownian bridges. A variation of.h; 1/-Shell Sort which is slightly more efficient is presented and its asymptotic behavior analyzed. 1. Introduction. Shell Sort is an algorithm that generalizes the method of sorting by insertion. It is essentially several stages of insertion sort. The algorithm was proposed in Shell (1959). The method received considerable attention over the past quarter of a century after Knuth's 1973 book popularized it (Knuth, 1973). From a practical point of view, the interest in Shell Sort stems from the fact that it is a rather practicable method of in situ sorting with little overhead and can be implemented with ease. From a theoretical standpoint, the interest is that insertion sort has an average of 2.n 2 / running time to sort n random keys, whereas the appropriate choice of the parameters of the stages of Shell Sort can bring down the order of magnitude. For instance, by a certain choice of the structure of the stages, a 2-stage Shell Sort can sort in O.n 5=3 / average running time. Ultimately, an optimized choice of the parameter can come close to the information-theoretic lower bound of O.n ln n/ average case. The analysis of Shell Sort has stood as a formidable challenge. Most research on Shell Sort has gone in the direction of making good choices for the parameters of the stages to obtain good worst-case behavior (see, for example, the review paper of Sedgewick (1996)). We propose here to take the research along a different axis and to analyze the stochastic structure of the algorithm. We rederive the limit result of Louchard (1986) for .2; 1/-Shell Sort, which he proved by essentially combinatoric arguments, and we generalize the approach to the analysis of .h; 1/-Shell Sort. The integrated alsolute value of the Brownian bridge appears in the limiting distributions; the moments of the distribution of this random variable were found by Shepp (1982), and Johnson and Killeen (1983) gave an explicit characterization of the distribution function.

Random walks and random permutations

A connection is made between the random-turns model of vicious walkers and random permutations indexed by their increasing subsequences. Consequently the scaled distribution of the maximum displacements in a particular asymmeteric version of the model can be determined to be the same as the scaled distribution of the eigenvalues at the soft edge of the GUE (random Hermitian matrices). The scaling of the distribution gives the maximum mean displacement µ after t time steps as µ = (2t)1/2 with standard deviation proportional to µ1/3. The exponent 1/3 is typical of a large class of two-dimensional growth problems.

Multiway Trees of Maximum and Minimum Probability under the Random Permutation Model

Multiway trees, also known as m–ary search trees, are data structures generalising binary search trees. A common probability model for analysing the behaviour of these structures is the random permutation model. The probability mass function Q on the set of m–ary search trees under the random permutation model is the distribution induced by sequentially inserting the records of a uniformly random permutation into an initially empty m–ary search tree. We study some basic properties of the functional Q, which serves as a measure of the ‘shape’ of the tree. In particular, we determine exact and asymptotic expressions for the maximum and minimum values of Q and identify and count the trees achieving those values.

On the distribution of binary search trees under the random permutation model

We study the distribution Q on the set Bn of binary search trees over a linearly ordered set of n records under the standard random permutation model. This distribution also arises as the stationary distribution for the move-to-root (MTR) Markov chain taking values in Bn when successive requests are independent and identically distributed with each record equally likely. We identify the minimum and maximum values of the functional Q and the trees achieving those values and argue that Q is a crude measure of the “shape” of the tree. We study the distribution of Q(T) for two choices of distribution for random trees T; uniform over Bn and Q. In the latter case, we obtain a limiting normal distribution for −ln Q(T). © 1996 John Wiley & Sons, Inc.

On the Generation of Random Binary Search Trees

We consider the computer generation of random binary search trees with n nodes for the standard random permutation model. The algorithms discussed here output the number of external nodes at each level, but not the shape of the tree. This is important, for example, when one wishes to simulate the height of the binary search tree. Various paradigms are proposed, including depth-first search with pruning, incremental methods in which the tree grows with random-sized jumps, and a tree growing procedure gleaned from birth-and-death processes. The last method takes $O(\log^{4} n)$ expected time.

On the completeness of certain classes of two stage sampling strategies

In this paper the problem of estimation of the population mean, for a finite two statge population, under the assumptions of non-informativeness of labels of distinguishable units has been studied. Completeness of certain classes of strategies, for estimating the population mean, under a two stage random permutation model, has been established.

On the efficiency of prediction estimators in two-stage sampling

A random permutation model appropriate to two-stage cluster sampling is presented and the optimal model-unbiased (prediction) estimator of a finite population mean is obtained. The prediction estimator is compared, in an empirical study with a variety of real populations, to the classical estimator used in PPS (probability proportional to size) sampling of the clusters. The study has shown that the prediction estimator may be only marginally better, in terms of efficiency, than the classical estimator in PPS sampling. The special cases of stratified sampling and uni-stage cluster sampling are also studied.

Estimation under two stage random permutation models

Estimation of the population mean under assumptions of non-informativeness of labels in a two stage finite population of distinguishable units has been studied. Under the random permutation model, for the two stage set up, sample mean, the natural estimator, is found to be the best.

Some aspects of random permutation models in finite population sampling theory

We first consider Neyman's optimum allocation of sample size to strata in the light of available auxiliary information for which a suitable random permutation model is assumed. For a special case of this model the allocation of the sample size reduces to the same as when a certain superpopulation regression model is assumed. Motivated by this, more generally, we discuss some optimality results under random permutation models and compare them with the corresponding results when a superpopulation regression model is assumed.

Optimum Estimation of Finite Population Variance Under Generalised Random Permutation Models

Considering certain generalised random permutation models we have derived optimal estimators of finite population variance within a general class of quadratic estimators. The class of non-negative unbiased polynomial estimators of variance has also been examined and an optimal estimator within that class has been derived under this model.

Linear Models for Randomized Response Designs

Abstract Linear models, based on random permutation models, are developed to include a wide class of one-sample randomized response designs. The general form of the “optimal” estimator of the finite population mean or proportion is obtained. Most of the conventional randomized response estimators are seen to be optimal, in terms of minimum average mean squared error, within their associated designs. The optimality results are obtained for sampling without replacement and include extensions of randomized response designs to unequal probability sampling.

Optimal estimation of a finite population mean under generalized random permutation models

Abstract Employing certain generalized random permutation models and a general class of linear estimators of a finite population mean, it is shown that many of the conventional estimators are “optimal” in the sense of minimum average mean square error. Simple proofs are provided by using a well-known theorem on UMV estimation. The results also cover certain simple response error situations.