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Optimal Sampling Strategies in Quicksort and Quickselect
 PROC. OF THE 25TH INTERNATIONAL COLLOQUIUM (ICALP98), VOLUME 1443 OF LNCS
, 1998
"... It is well known that the performance of quicksort can be substantially improved by selecting the median of a sample of three elements as the pivot of each partitioning stage. This variant is easily generalized to samples of size s = 2k + 1. For large samples the partitions are better as the median ..."
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Cited by 28 (4 self)
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It is well known that the performance of quicksort can be substantially improved by selecting the median of a sample of three elements as the pivot of each partitioning stage. This variant is easily generalized to samples of size s = 2k + 1. For large samples the partitions are better as the median of the sample makes a more accurate estimate of the median of the array to be sorted, but the amount of additional comparisons and exchanges to find the median of the sample also increases. We show that the optimal sample size to minimize the average total cost of quicksort (which includes both comparisons and exchanges) is s = a \Delta p n + o( p n ). We also give a closed expression for the constant factor a, which depends on the medianfinding algorithm and the costs of elementary comparisons and exchanges. The result above holds in most situations, unless the cost of an exchange exceeds by far the cost of a comparison. In that particular case, it is better to select not the median of...
On a multivariate contraction method for random recursive structures with applications to Quicksort
, 2001
"... The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an appl ..."
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Cited by 28 (15 self)
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The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an application the asymptotic correlations and a bivariate limit law for the number of key comparisons and exchanges of medianof(2t + 1) Quicksort is given. Moreover, for the Quicksort programs analyzed by Sedgewick the exact order of the standard deviation and a limit law follow, considering all the parameters counted by Sedgewick.
Improving Memory Performance of Sorting Algorithms
 ACM J. Exp. Algorithmics
, 2000
"... ing with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept, ACM Inc., 1515 Broadway, New York, N ..."
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Cited by 24 (4 self)
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ing with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works, requires prior specific permission and/or a fee. Permissions may be requested from Publications Dept, ACM Inc., 1515 Broadway, New York, NY 10036 USA, fax +1 (212) 8690481, or permissions@acm.org. 2 \Delta Li Xiao, Xiaodong Zhang, and Stefan A. Kubricht isting restructured algorithms (e.g., [4]) mainly attempt to reduce capacity misses on directmapped caches. In this paper, we report substantial performance improvement obtained by further exploiting memory locality to reduce other types of cache misses, such as conflict misses and TLB misses. We present several restructured mergesort and quicksort algorithms and their implementations by fully using existing processor hardware facilities (such as cache associativity and TLB), by integrating tiling and padding techniques, and by properly partitioning the data set for cache op...
Optimizing Sorting with Genetic Algorithms
 In The International Symposium on Code Generation and Optimization
, 2005
"... 1 ..."
A Practical Quicksort Algorithm for Graphics Processors
, 2008
"... In this paper we present GPUQuicksort, an efficient Quicksort algorithm suitable for highly parallel multicore graphics processors. Quicksort has previously been considered as an inefficient sorting solution for graphics processors, but we show that GPUQuicksort often performs better than the fa ..."
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Cited by 17 (0 self)
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In this paper we present GPUQuicksort, an efficient Quicksort algorithm suitable for highly parallel multicore graphics processors. Quicksort has previously been considered as an inefficient sorting solution for graphics processors, but we show that GPUQuicksort 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.
Transitional Behaviors of the Average Cost of Quicksort With Medianof(2t + 1)
, 2001
"... A fine analysis is given of the transitional behavior of the average cost of quicksort with medianofthree. Asymptotic formulae are derived for the stepwise improvement of the average cost of quicksort when iterating medianofthree k rounds for all possible values of k. The methods used are genera ..."
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Cited by 11 (6 self)
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A fine analysis is given of the transitional behavior of the average cost of quicksort with medianofthree. Asymptotic formulae are derived for the stepwise improvement of the average cost of quicksort when iterating medianofthree k rounds for all possible values of k. The methods used are general enough to apply to quicksort with medianof(2t + 1) and to explain in a precise manner the transitional behaviors of the average cost from insertion sort to quicksort proper. Our results also imply nontrivial bounds on the expected height, "saturation level", and width in a random locally balanced binary search tree.
Density Approximation and Exact Simulation of Random Variables that are Solutions of FixedPoint Equations
 Adv. Appl. Probab
, 2002
"... An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic an ..."
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Cited by 10 (6 self)
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An algorithm is developed for the exact simulation from distributions that are defined as fixedpoints of maps between spaces of probability measures. The fixedpoints of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. Approximating sequences for the densities of the fixedpoints with explicit error bounds are constructed. The sampling algorithm relies on a modified rejection method. AMS subject classifications. Primary: 65C10; secondary: 65C05, 68U20, 11K45.
Impact of PCIBus Load on Applications in a PC Architecture
, 2003
"... Any data exchanged between the processor and main memory uses the memory bus, sharing it with data exchanged between I/O devices and main memory. If the processor and a device try to transfer data at the same time, an impact can be seen on the processor as well as on the device. As a result, the exe ..."
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Cited by 10 (0 self)
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Any data exchanged between the processor and main memory uses the memory bus, sharing it with data exchanged between I/O devices and main memory. If the processor and a device try to transfer data at the same time, an impact can be seen on the processor as well as on the device. As a result, the execution time of an application on the processor may increase due to the memorybus load generated by I/O devices. In realtime environments, this impact can result in missed deadlines and a behavior that is different to that intended by the designer of the system. This paper
On the adaptiveness of quicksort
 IN: WORKSHOP ON ALGORITHM ENGINEERING & EXPERIMENTS, SIAM
, 2005
"... Quicksort was first introduced in 1961 by Hoare. Many variants have been developed, the best of which are among the fastest generic sorting algorithms available, as testified by the choice of Quicksort as the default sorting algorithm in most programming libraries. Some sorting algorithms are adapti ..."
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Cited by 8 (1 self)
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Quicksort was first introduced in 1961 by Hoare. Many variants have been developed, the best of which are among the fastest generic sorting algorithms available, as testified by the choice of Quicksort as the default sorting algorithm in most programming libraries. Some sorting algorithms are adaptive, i.e. they have a complexity analysis which is better for inputs which are nearly sorted, according to some specified measure of presortedness. Quicksort is not among these, as it uses Ω(n log n) comparisons even when the input is already sorted. However, in this paper we demonstrate empirically that the actual running time of Quicksort is adaptive with respect to the presortedness measure Inv. Differences close to a factor of two are observed between instances with low and high Inv value. We then show that for the randomized version of Quicksort, the number of element swaps performed is provably adaptive with respect to the measure Inv. More precisely, we prove that randomized Quicksort performs expected O(n(1+log(1+ Inv/n))) element swaps, where Inv denotes the number of inversions in the input sequence. This result provides a theoretical explanation for the observed behavior, and gives new insights on the behavior of the Quicksort algorithm. We also give some empirical results on the adaptive behavior of Heapsort and Mergesort.
A Simple, Fast Parallel Implementation of Quicksort and its Performance Evaluation on SUN Enterprise 10000
"... This paper looks into the behavior of a simple, finegrain parallel extension of Quicksort for cachecoherent shared address space multiprocessors. Quicksoft has many nice properties: i) it is fast and general purpose; it is widely believed that Quicksoft is the fastest generalpurpose sorting algor ..."
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Cited by 6 (1 self)
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This paper looks into the behavior of a simple, finegrain parallel extension of Quicksort for cachecoherent shared address space multiprocessors. Quicksoft has many nice properties: i) it is fast and general purpose; it is widely believed that Quicksoft is the fastest generalpurpose sorting algorithm, on average, and for a large number of elelnents [Blelloch et al. 1991; Dusseau et al. 1996; Helman et al. 1996b; Sohn and Kodama 1998], ii) it is inplace, iii) it exhibits good cache pcrforinance and iv) it is simple to inlplelnent. The new generation of hardwarecoherent, shared address space multiprocessor systems with their already donfinant position on the tightlycoupled nmltiprocessor systems are our target systems. The implementation of the parallel Quicksort algorithm utilizes the capabilities that these new systems have to or and uses the fbllowing algorithmic techniques: Cacheqcieni:. Each processor tries to use all keys when sequentially passing through the keys of a cachedblock from the key array