Quicksort Algorithm Research Articles

Refined quicksort asymptotics

AbstractThe complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of n data, permuted uniformly at random, the appropriately normalized complexity Yn is known to converge almost surely to a non‐degenerate random limit Y. This assumes a natural embedding of all Yn on one probability space, e.g., via random binary search trees. In this note a central limit theorem for the error term in the latter almost sure convergence is shown: urn:x-wiley:10429832:media:rsa20497:rsa20497-math-0001 where denotes a standard normal random variable. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 346–361, 2015

Performance assessment of multithreaded quicksort algorithm on simultaneous multithreaded architecture

Sorting huge amounts of datasets have become essential in many computer applications, such as search engines, database and web-based applications, in order to improve searching performance. Moreove...

DUAL PIVOT QUICKSORT

In this paper, we analyze the dual pivot Quicksort, a variant of the standard Quicksort algorithm, in which two pivots are used for the partitioning of the array. We are solving recurrences of the expected number of key comparisons and exchanges performed by the algorithm, obtaining the exact and asymptotic total average values contributing to its time complexity. Further, we compute the average number of partitioning stages and the variance of the number of key comparisons. In terms of mean values, dual pivot Quicksort does not appear to be faster than ordinary algorithm.

Smoothed analysis of left-to-right maxima with applications

A left-to-right maximum in a sequence of n numbers s 1 , …, s n is a number that is strictly larger than all preceding numbers. In this article we present a smoothed analysis of the number of left-to-right maxima in the presence of additive random noise. We show that for every sequence of n numbers s i ∈ [0,1] that are perturbed by uniform noise from the interval [-ϵ,ϵ], the expected number of left-to-right maxima is Θ(√ n /ϵ + log n ) for ϵ>1/ n . For Gaussian noise with standard deviation σ we obtain a bound of O ((log 3/2 n )/σ + log n ). We apply our results to the analysis of the smoothed height of binary search trees and the smoothed number of comparisons in the quicksort algorithm and prove bounds of Θ(√ n /ϵ + log n ) and Θ( n /ϵ+1√ n /ϵ + n log n ), respectively, for uniform random noise from the interval [-ϵ,ϵ]. Our results can also be applied to bound the smoothed number of points on a convex hull of points in the two-dimensional plane and to smoothed motion complexity, a concept we describe in this article. We bound how often one needs to update a data structure storing the smallest axis-aligned box enclosing a set of points moving in d -dimensional space.

Experiments in program verification using Event-B

Abstract The Event-B method can be used to model all sorts of discrete event systems, among them sequential programs. In this article we describe our experiences with using Event-B by way of two examples. We present a simple model of a factorial program, explaining the method, and a more intricate model of the Quicksort algorithm, providing some insights into strengths and weaknesses of Event-B. The two models are interspersed with our observations and some suggestions of how, we believe, Event-B could evolve. This evaluation of Event-B is intended to serve for determining directions for the evolution of Event-B and judging progress. It is our hope that the observations and suggestions can also be put to use for similar modelling formalisms, such as Z, ASM or VDM.

IQSelect Algorithms for Finding the Kth Value Based on Incomplete Quick Sort Algorithm

IQSelect Algorithms for Finding the Kth Value Based on Incomplete Quick Sort Algorithm

A Subpath Kernel for Rooted Unordered Trees

Kernel method is one of the promising approaches to learning with tree-structured data, and various efficient tree kernels have been proposed to capture informative structures in trees. In this paper, we propose a new tree kernel function based on ``subpath sets'' to capture vertical structures in tree-structured data, since tree-structures are often used to code hierarchical information in data. We also propose a simple and efficient algorithm for computing the kernel by extending the Multikey quicksort algorithm used for sorting strings. The time complexity of the algorithm is O((|T_1|+|T_2|)log(|T_1|+|T_2|)) time on average, and the space complexity is O({|T_1|+|T_2|)}, where |T_1| and |T_2| are the numbers of nodes in two trees T_1 and T_2. We apply the proposed kernel to two supervised classification tasks, XML classification in web mining and glycan classification in bioinformatics. The experimental results show that the predictive performance of the proposed kernel is competitive with that of the existing efficient tree kernel proposed by Vishwanathan et al., and is also empirically faster than the existing kernel.

Multi-thread quicksort algorithm based on partitioning

Firstly, the improved method based on Java was introduced. A reasonable experimental threshold had been obtained, which improved the efficiency of quicksort when sorting a small dataset. Secondly, quicksort was optimized by using the multi-thread technology, and experiments were conducted to compare the performance of the single-thread and multi-thread algorithms. The results show that the multi-thread algorithm is several times faster than the single-thread one on multi-core computers. Finally, theoretical analysis was made on multi-thread quicksort algorithm, and the theoretical upper limit of the algorithm was obtained on multi-core computers.

The complexity probabilistic quasi-metric space

We introduce and study a probabilistic quasi-metric on the set of complexity functions, which provides an efficient framework to measure the distance from a complexity function f to another one g in the case that f is asymptotically more efficient than g. In this context we also obtain a version of the Banach fixed point theorem which allows us to show that the functionals associated both to Divide and Conquer algorithms and Quicksort algorithms have a unique fixed point.

A Novel Image Median Filtering Algorithm based on Incomplete Quick Sort Algorithm

Quick sort algorithm is studied thoroughly, a new incomplete quick sort algorithm is designed, and then a novel image median filtering algorithm based on incomplete quick sort algorithm is proposed to improve the filtering speed. The new algorithm considers in detail the characteristic of image median filtering (In median filtering algorithm, the sorting operation of all pixel values is not necessary, the median value can be given by many other methods), and can give the median value by only sorting part of the pixels value in the neighborhood, thus it can reduce many data move operations, and then greatly improve the speed of image median filtering. Algorithm analysis and a lot of experiment results show that, the new algorithm greatly improves the speed of image median filtering, and can keep the edge, outline, texture and much other information to a great extent.

Guest editorial: special issue on learning theory

This special issue collects some of the most notable learning theory papers of 2008, which were chosen from the Conference on Learning Theory (COLT). The diversity of problems addressed in these papers reflects the significant progress being made in our theoretical understanding of the foundations of learning at both a computational and statistical level— topics include active learning, ranking, stability, graphical models, computational learning, to name a few. The best student paper, by Balcan, Hanneke, and Wortman, provides a novel analysis of active learning, showing how in a variety of settings substantial asymptotic improvements are always possible under active learning. Shalev-Shwartz and Singer utilize an elegant minimax interpretation of boosting to provide more efficient and robust boosting algorithms. In a regret minimization setting, Hazan and Kale provide an algorithm which achieves even less regret when the variation in experts is low (thus making another connection to our concentration inequalities, particular those which utilize variance properties of random variables). Ailon and Mohri provide an efficient reduction of ranking with a classification algorithm: the simple algorithm cleverly uses binary classifiers with a QuickSort algorithm. Shamir and Tishby provide a general characterization of the stability of the widely-used k-means algorithm, whose analysis identifies the factors which influence the stability of clustering algorithms. Kleinberg, Niculescu-Mizil, and Sharma provide online algorithms where the action set (of the experts) is time varying. These algorithms are shown to be optimal in that they match the information theoretic lower bounds. Blais, O’Donnell, and Wimmer

Ranked algorithm to fill sinks in digital elevation model based on quicksort

A new ranked algorithm to fill sinks in Digital Elevation Models (DEM) was introduced,including an improved quicksort algorithm to rank DEM data in pieces and a filling algorithm first developed by Olivier Planchon and Frederic Darboux in 2001 (The idea was first suggested by Moran and Vezina,so it's called the MV algorithm). Since removing the depressions of a DEM is usually a time-consuming procedure,the new method keeps the integrity and continuity of the drainage networks extracted by the pits-removed DEM and well reduces the depth of seeking path in iteration to accelerate pits-removing process. Different from the constant time complexity of traditional algorithms,the time complexity of the new method is variable,and the best performance for a natural terrain appears to be in 200 to 500 divided-pieces. The application of the new method in 6 study areas shows a time complexity of about O(n1.29) on average,which is more efficient than the MV algorithm and Arcgis 9.2 (improved Jenson Domingue algorithm).

GPU-Quicksort

In this article, we describe GPU-Quicksort, an efficient Quicksort algorithm suitable for highly parallel multicore graphics processors. Quicksort has previously been considered an inefficient sorting solution for graphics processors, but we show that in CUDA, NVIDIA's programing platform for general-purpose computations on graphical processors, GPU-Quicksort performs better than the fastest-known sorting implementations for graphics processors, such as radix and bitonic sort. Quicksort can thus be seen as a viable alternative for sorting large quantities of data on graphics processors.

InterviewAn interview with C.A.R. Hoare

C.A.R. Hoare, developer of the Quicksort algorithm and a lifelong contributor to the theory and design of programming languages, discusses the practical application of his theoretical ideas.

Moves and displacements of particular elements in Quicksort

In this research note we investigate the number of moves and the displacement of particular elements during the execution of the well-known quicksort algorithm. This type of analysis is useful if the costs of data moves were dependent on the source and target locations, and possibly the moved element itself. From the mathematical point of view, the analysis of these quantities turns out to be related to the analysis of quickselect, a selection algorithm which is a variant of quicksort that finds the i -th smallest element of n given elements, without sorting them. Our results constitute thus a novel application of M. Kuba’s machinery [M. Kuba, On quickselect, partial sorting and multiple quickselect, Inform. Process. Lett. 99(5) (2006) 181–186] for the solution of general quickselect recurrences.

Quicksort Algorithms: Application of Fixed Point Theorem in Probabilistic Quasi-Metric Spaces at Domain of Words

Quicksort Algorithms: Application of Fixed Point Theorem in Probabilistic Quasi-Metric Spaces at Domain of Words

Contraction Maps on Ifqm-spaces with Application to Recurrence Equations of Quicksort

Recently, C. Alaca, D. Turkoglu and C. Yildiz [Chaos, Solitons and Fractals, 2006], have proved intuitionistic fuzzy versions of the celebrated Banach fixed point theorem and Edelstein fixed point theorem respectively, by means of a notion of intuitionistic fuzzy metric space which is based on the concept of fuzzy metric space due to I. Kramosil and J. Michalek [Kybernetika, 1975]. In this paper we generalize the notions of intuitionistic fuzzy metric space by Alaca, Turkoglu and Yildiz to the quasi-metric setting and we present an intuitionistic fuzzy quasi-metric version of the Banach contraction principle. We apply this approach to deduce the existence of solution for the recurrence equations associated to the analysis of Quicksort algorithm in the framework of intuitionistic fuzzy quasi-metric spaces (ifqm-spaces, in short).

On sorting and load balancing on GPUs

In this paper we take a look at GPU-Quicksort, an efficient Quicksort algorithm suitable for the highly parallel multi-core graphics processors. Quicksort had previously been considered an inefficient sorting solution for graphics processors, but GPU-Quicksort often performs better than the fastest known sorting implementations for graphics processors, such as radix and bitonic sort. Quicksort can thus be seen as a viable alternative for sorting large quantities of data on graphics processors. We also take look at a comparison of different load balancing schemes. To get maximum performance on the many-core graphics processors it is important to have an even balance of the workload so that all processing units contribute equally to the task at hand. This can be hard to achieve when the cost of a task is not known beforehand and when new sub-tasks are created dynamically during execution. With the recent advent of scatter operations and atomic hardware primitives it is now possible to bring some of the more elaborate dynamic load balancing schemes from the conventional SMP systems domain to the graphics processor domain.

Quicksort algorithm: Application of a fixed point theorem in intuitionistic fuzzy quasi-metric spaces at a domain of words

In this paper, we apply an intuitionistic fuzzy quasi-metric version of a fixed point theorem, to obtain the existence of solution for a recurrence equation associated with the analysis of Quicksort algorithms.

Average-case analysis of QuickSort and Binary Insertion Tree height using incompressibility

We study the Kolmogorov complexity of a Binary Insertion Tree, and present a succinct encoding scheme for Binary Insertion Trees produced from incompressible permutations. Based on the encoding scheme, we obtain a simple incompressibility argument that yields an asymptotic analysis of the average height of a Binary Insertion Tree. This argument further implies that the QuickSort algorithm sorts a permutation of n elements in Θ ( n lg n ) comparisons on average.