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40
Expected Time Bounds for Selection
, 1975
"... A new selection algorithm is presented which is shown to be very efficient on the average, both theoretically and practically. The number of comparisons used to select the ith smallest of n numbers is n q min(i,ni) q o(n). A lower bound within 9 percent of the above formula is also derived. ..."
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Cited by 466 (4 self)
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A new selection algorithm is presented which is shown to be very efficient on the average, both theoretically and practically. The number of comparisons used to select the ith smallest of n numbers is n q min(i,ni) q o(n). A lower bound within 9 percent of the above formula is also derived.
Fast computation of database operations using graphics processors
 Proc. of ACM SIGMOD
, 2004
"... We present new algorithms for performing fast computation of several common database operations on commodity graphics processors. Specifically, we consider operations such as conjunctive selections, aggregations, and semilinear queries, which are essential computational components of typical databa ..."
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Cited by 113 (15 self)
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We present new algorithms for performing fast computation of several common database operations on commodity graphics processors. Specifically, we consider operations such as conjunctive selections, aggregations, and semilinear queries, which are essential computational components of typical database, data warehousing, and data mining applications. While graphics processing units (GPUs) have been designed for fast display of geometric primitives, we utilize the inherent pipelining and parallelism, single instruction and multiple data (SIMD) capabilities, and vector processing functionality of GPUs, for evaluating boolean predicate combinations and semilinear queries on attributes and executing database operations efficiently. Our algorithms take into account some of the limitations of the programming model of current GPUs and perform no data rearrangements. Our algorithms have been implemented on a programmable GPU (e.g. NVIDIA’s GeForce FX 5900) and applied to databases consisting of up to a million records. We have compared their performance with an optimized implementation of CPUbased algorithms. Our experiments indicate that the graphics processor available on commodity computer systems is an effective coprocessor for performing database operations.
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 34 (5 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...
Analysis of Hoare's Find Algorithm with Medianofthree partition. Random Structures & Algorithms
 Random Structures & Algorithms
, 1997
"... ABSTRACT: Hoare’s FIND algorithm can be used to select the jth element out of a file of n elements. It bears a remarkable similarity to Quicksort; in each pass of the algorithm, a pivot element is used to split the file into two subfiles, and recursively the algorithm proceeds with the subfile that ..."
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Cited by 24 (2 self)
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ABSTRACT: Hoare’s FIND algorithm can be used to select the jth element out of a file of n elements. It bears a remarkable similarity to Quicksort; in each pass of the algorithm, a pivot element is used to split the file into two subfiles, and recursively the algorithm proceeds with the subfile that contains the sought element. As in Quicksort, different strategies for selecting the pivot are reasonable. In this paper, we consider the Medianofthree version, where the pivot element is chosen as the median of a random sample of three elements. Establishing some hypergeometric differential equations, we find explicit formulae for both the average number of passes and comparisons. We compare these results with the corresponding ones for the basic partition strategy. � 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 143�156 Ž 1997. 1.
An improved master theorem for divideandconquer recurrences
 In Automata, languages and programming
, 1997
"... Abstract. This paper presents new theorems to analyze divideandconquer recurrences, which improve other similar ones in several aspects. In particular, these theorems provide more information, free us almost completely from technicalities like floors and ceilings, and cover a wider set of toll fun ..."
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Cited by 15 (2 self)
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Abstract. This paper presents new theorems to analyze divideandconquer recurrences, which improve other similar ones in several aspects. In particular, these theorems provide more information, free us almost completely from technicalities like floors and ceilings, and cover a wider set of toll functions and weight distributions, stochastic recurrences included.
On the probabilistic worstcase time of "FIND"
 ALGORITHMICA
, 2001
"... We analyze the worstcase number of comparisons Tn of Hoare’s selection algorithm find when the input is a random permutation, and worst case is measured with respect to the rank k. We give a new short proof that Tn/n tends to a limit distribution, and provide new bounds for the limiting distributi ..."
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Cited by 13 (0 self)
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We analyze the worstcase number of comparisons Tn of Hoare’s selection algorithm find when the input is a random permutation, and worst case is measured with respect to the rank k. We give a new short proof that Tn/n tends to a limit distribution, and provide new bounds for the limiting distribution.
PERFECT SIMULATION OF VERVAAT PERPETUITIES
, 908
"... Abstract. We use coupling into and from the past to sample perfectly in a simple and provably fast fashion from the Vervaat family of perpetuities. The family includes the Dickman distribution, which arises both in number theory and in the analysis of the Quickselect algorithm, which was the motivat ..."
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Cited by 11 (0 self)
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Abstract. We use coupling into and from the past to sample perfectly in a simple and provably fast fashion from the Vervaat family of perpetuities. The family includes the Dickman distribution, which arises both in number theory and in the analysis of the Quickselect algorithm, which was the motivation for our work.
Distributional convergence for the number of symbol comparisons used by QuickSort
, 2012
"... Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the n independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic sourc ..."
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Cited by 10 (4 self)
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Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the n independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic source and that QuickSort operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild “tameness ” condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by n. Additionally, under a condition that grows more restrictive as p increases, we have convergence of moments of orders p and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, i.e., whenever each key is generated as an infinite string of iid symbols. This is somewhat surprising: Even for the classical model that each key is an iid string of unbiased (“fair”) bits, the mean exhibits periodic fluctuations of order n.
Analytic Variations on Bucket Selection and Sorting
, 1998
"... : We provide complete averagecase as well as probabilistic analysis of the cost of bucket selection and sorting algorithms. Two variations of bucketing (and flavors therein) are considered: distributive bucketing (large number of buckets) and radix bucketing (recursive with a small number of bucket ..."
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Cited by 9 (2 self)
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: We provide complete averagecase as well as probabilistic analysis of the cost of bucket selection and sorting algorithms. Two variations of bucketing (and flavors therein) are considered: distributive bucketing (large number of buckets) and radix bucketing (recursive with a small number of buckets, suitable for digital computation). For Distributive Selection a compound Poisson limit is established. For all other flavors of bucket selection and sorting, central limit theorems underlying the cost are derived by asymptotic techniques involving perturbation of Rice's integral and contour integration (saddle point methods). In the case of radix bucketing, periodic fluctuations appear in the moments of both the selection and sorting algorithms. (R'esum'e : tsvp) Unit'e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T'el'ephone : (33) 01 39 63 55 11  T'el'ecopie : (33) 01 39 63 53 Variations analytiques sur la s'election et l...
Towards optimal multiple selection
 32ND INTERNATIONAL COLLOQUIUM, ICALP 2005
, 2005
"... The multiple selection problem asks for the elements of rank r1, r2,..., rk from a linearly ordered set of n elements. Let B denote the information theoretic lower bound on the number of element comparisons needed for multiple selection. We first show that a variant of multiple quickselect — a wel ..."
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Cited by 9 (1 self)
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The multiple selection problem asks for the elements of rank r1, r2,..., rk from a linearly ordered set of n elements. Let B denote the information theoretic lower bound on the number of element comparisons needed for multiple selection. We first show that a variant of multiple quickselect — a well known, simple, and practical generalization of quicksort — solves this problem with B + O(n) expected comparisons. We then develop a deterministic divideandconquer algorithm that solves the problem in O(B) time and B + o(B) +O(n) element comparisons.