R. Sedgewick. The analysis of quicksort programs. Acta Informatica, 7:327--355, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is in its final position, this recursive scheme sorts the input in increasing order. If we assume that each of the n permutations of n elements is equally likely, then the average number of comparisons used by quicksort when given a random permutation, denoted by C n , satisfies C 1 = 0 and (cf. [15, 24, 25, 27]) C n = n c 1 C j (n 2) # This work was done while this author was at Institute of Mathematics, Academia Sinica, Taipei. for some constant c 1 depending on implementation. The solution is easily seen to satisfy C n = 2(n 1)H n 3 n O(1) 1) where H n = 1#j#n j 1 denotes ....

R. Sedgewick, The analysis of Quicksort programs, Acta Informatica, 7, 327--355, 1977. 23

....is in its nal position, this recursive scheme sorts the input in increasing order. If we assume that each of the n permutations of n elements is equally likely, then the average number of comparisons used by quicksort when given a random permutation, denoted by C n , satis es C 1 = 0 and (cf. [15, 24, 25, 27]) C n = n c 1 C j (n 2) This work was done while this author was at Institute of Mathematics, Academia Sinica, Taipei. for some constant c 1 depending on implementation. The solution is easily seen to satisfy C n = 2(n 1)H n 3 n O(1) 1) where H n = denotes the n th ....

R. Sedgewick, The analysis of Quicksort programs, Acta Informatica, 7, 327-355, 1977. 23

....circumstances. Both algorithms are based upon the divide and conquer principle and operate using similar ideas. A brief, but complete description is given in Section 2. Excellent sources for background information and further references on quicksort and quickselect and their analysis include [5, 10, 15, 16, 18, 17, 19]. Contrary to other divide and conquer algorithms, quicksort and quickselect are not guaranteed to divide the problem into subproblems of approximately the same size. Not even there is certainty that the size of the subproblems will be a fraction of the size of the original problem. Hence, their ....

....by theoretical analysis and practical experiments. In this variant of quicksort, we select a pivot in each recursive stage by taking a sample of three elements and using the median of the sample as the pivot. The idea is that it is more likely that no subarray is degenerate after the partitioning [2, 15, 20]. This variant is easily generalized to samples of size s = 2k 1 elements, so that the (k 1) element in the sample is selected as the pivot. Van Emden [21] analyzed this generalized variant of quicksort using wise information theoretic arguments, showing that the average number of ....

R. Sedgewick. The analysis of quicksort programs. Acta Informatica, 7:327-- 355, 1976.

....of the covariance of these parameters. As corollaries limit laws and variances of linear combinations of these parameters are obtained. This would also be in the range of a univariate approach but results here without any work. The analysis covers as well the more complex situation of Sedgewick s [50] cost measure for concrete Quicksort implementations. Here the cost of the algorithm is measured as a linear combination of several parameters of the algorithm. Asymptotically only the number of key comparisons and exchanges matter. Therefore we obtain the exact order of the standard deviation of ....

....the number of key comparisons, key exchanges, partitioning stages, and stack pushes and pops made during the execution of the algorithm. One approach to de ne a univariate cost measure for Quicksort algorithms may be in taking linear combinations to weight the speci c parameters, see Sedgewick [49, 50]. We consider mainly the number of key comparisons C n and key exchanges B n , since these parameters are in the mean of the order n ln n, whereas other parameters are of smaller order. The n denotes the number of items to be sorted and the underlying probabilistic model consists of all ....

[Article contains additional citation context not shown here]

Sedgewick, R. (1977). The analysis of Quicksort programs. Acta Informat. 7, 327-355.

....as limit distributions of parameters considered in the probabilistic analysis of algorithms. In Quicksort, invented by Hoare [18, 21] the number of comparisons C n satis es (C n E C n ) n X where denotes convergence in distribution. For a probabilistic analysis of many variants see Sedgewick [37, 38]. The limit law was rst obtained by Hennequin [19, 20] by a moments method approach and R egnier [32] with a 2 martingal argument, the distribution of X was recently shown by R osler [33] to satisfy the xed point equation X UX (1 U)X 0 1 2 ln(U) 2(1 U) ln(1 U) where U is a ....

Sedgewick, R. (1977). The analysis of Quicksort programs. Acta Informat. 7, 327-355.

....improves the overall performance of the algorithm. Cuto values in the range 6 to 15 do about as well, but n 0 = 9 is the optimal choice for most implementations. Actually, for quicksort, it is typically better to ignore small sub les and perform a single pass of insertion sort over the entire le [7, 8]. In this paper, we tackle the analysis of quick nd with recursion cuto , for both the standard and the median of three variants. Computing the optimal cuto value can be readily done using dynamic programming, solving the recurrences for small values of the parameters. However, this approach ....

....1) 4 Selection jn j(j 1) 2 j Table 2: Cuto functions (j dn=2e) On the other hand, Table 2 lists the average number of comparisons, exchanges, and single data movements made by insertion sort and selection sort. The analysis of these basic algorithms can be found elsewhere; see for instance [6, 7, 9, 10]. Since we should actually use a slight modi cation of selection sort, we brie y justify the values given in Table 2. For each i, 1 i j, the algorithm has already located and sorted the smallest i 1 elements in the sub le. At the i th pass, it has to locate the minimum among the remaining n ....

[Article contains additional citation context not shown here]

Sedgewick, R. The analysis of quicksort programs. Acta Informatica 7 (1976), 327-355.

....distributions of parameters considered in the probabilistic analysis of algorithms. In Quicksort, invented by Hoare [18, 21] the number of comparisons C n satisfies (C n Gamma EC n ) n X where denotes convergence in distribution. For a probabilistic analysis of many variants see Sedgewick [37, 38]. The limit law was first obtained by Hennequin [19, 20] by a moments method approach and R egnier [32] with a martingal argument, the distribution of X was recently shown by Rosler [33] to satisfy the fixed point equation X UX (1 Gamma U)X 0 1 2 ln(U) 2(1 Gamma U) ln(1 Gamma U) ....

Sedgewick, R. (1977). The analysis of Quicksort programs. Acta Informat. 7, 327--355.

....which makes it competitive for large data sets even for vector computers running in sequential mode [13] This motivated the present study on vectorization properties of the kernel of quicksort. Several implementations of quicksort for sequential computers have been investigated to some depth in [7, 8, 16, 14, 15, 10]. The large grain parallelism of quicksort is well known and was used for implementations on parallel computers [1, 11, 12] Fine grain parallelism which is suitable for vector processing has been less studied. An implementation for vector computers was suggested by Stone [17] The basic ....

R. Sedgewick, The analysis of quicksort programs, Acta Informatica, 7 (1977), pp. 327--355.

....we skip a reference, quoted Emden s result as a n = e k n ln n O(n) This is not correct. We provide an example by a special QUICKSORT version. The best results concerning the expectation of the 2k 1 median version of QUICKSORT are in Bertram Kretzberg (1991) 2] See also Sedgewick (1977) [22]. The method uses generating functions and algebraic relations of the exact distribution of Z n;1 . However, the method breaks down even by small disturbance of the distribution. In Section 2 we present a more general method to derive the asymptotic behavior of the expectation a n even under some ....

Sedgewick, R. The analysis of Quicksort programs. Acta Informatica 7, 327-355, 1977. 30

....number of element comparisons made by Hoare s selection routine to find the median of N elements is bounded by 3:39N , and that selecting more extreme elements require fewer element comparisons on an average. The expected number of element exchanges performed is less than 3:39N=6 as shown by Sedgewick [1977]. By inspecting the program, the number of memory reads is seen 21 #include iterator defines iteratortraits template class position, class ordering position unguardedpartition(position first, position pivotposition, position beyond, ordering less) typedef ....

....slow in practice. However, we also experimented with programs, which were better at handling multisets following the ideas in the theoretical parts of Sections 4.3 and 5.3, but our preliminary implementations were not competitive. At the moment we are willing to follow Sedgewick s recommendation [Sedgewick 1977] not to add any overhead to deal with equal elements. 7. CONCLUDING REMARKS The problem of constructing a heap is well studied and well understood. The main goal in this paper has been to understand the execution environment where the heap construction programs are run. Most modern computers ....

Sedgewick, R. 1977. The analysis of Quicksort programs. Acta Informatica 7, 327--355.

....most circumstances. Both algorithms are based upon the divide and conquer principle and operate using similar ideas. A brief, but complete description is given in Section 2. Excellent sources for background information and further references on quicksort and quickselect and their analysis include [5, 10, 15, 16, 18, 17, 19]. Contrary to other divide and conquer algorithms, quicksort and quickselect are not guaranteed to divide the problem into subproblems of approximately the same size. Not even there is certainty that the size of the subproblems will be a fraction of the size of the original problem. Hence, their ....

....by theoretical analysis and practical experiments. In this variant of quicksort, we select a pivot in each recursive stage by taking a sample of three elements and using the median of the sample as the pivot. The idea is that it is more likely that no subarray is degenerate after the partitioning [2, 15, 20]. This variant is easily generalized to samples of size s = 2k 1 elements, so that the (k 1) th element in the sample is selected as the pivot. Van Emden [21] analyzed this generalized variant of quicksort using wise information theoretic arguments, showing that the average number of ....

R. Sedgewick. The analysis of quicksort programs. Acta Informatica, 7:327-- 355, 1976.

....see that the number of comparisons in quicksort is infinitely often at least equal to nh n (nh n 1) 2 for any sequence h n decreasing to zero, and some irrational . Yet, for i.i.d. data drawn from the same nonatomic distribution, the expected number of comparisons is asymptotic to 2n log n (Sedgewick, 1977). Therefore, Weyl sequences are not appropriate for generating test data for sorting algorithms. With a uniform [0; 1] the expected number of comparisons grows as n log n log log n. In fact, we have the following. Proposition 17. Let be uniform [0; 1] For any constant C, with probability ....

R. Sedgewick, "The analysis of quicksort programs," Acta Informatica, vol. 4, pp. 327-- 355, 1977.

....is by van Emden [6] He showed a n n ln n n e K : 1 Gamma R 1 0 (t ln(t) 1 Gamma t) ln(1 Gamma t) g K (t)dt Here g K is the density of the (K 1) th order statistic of 2K 1 independent rvs with a uniform distribution. This is a beta(K 1,K 1) distribution on [0; 1] Other authors [21] [2] 14] quoted van Emden s result as a n = e Kn ln n O(n) This quotation is not correct. We provide an example by a special QUICKSORT version. The best results concerning the expectation of the 2K 1 median version of QUICKSORT are in Bertram Kretzberg [2] See also Sedgewick [21] The ....

....authors [21] 2] 14] quoted van Emden s result as a n = e Kn ln n O(n) This quotation is not correct. We provide an example by a special QUICKSORT version. The best results concerning the expectation of the 2K 1 median version of QUICKSORT are in Bertram Kretzberg [2] See also Sedgewick [21]. The method are generating functions and algebraic relations of the exact distribution of Z n;1 . However, these methods break down even for small disturbance of the distribution. In Section 2 we present a more general method to derive the asymptotic behaviour of the expectation a n even under ....

Sedgewick, R. The analysis of Quicksort programs. Acta Informatica 7, 327-355, 1977.

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R. Sedgewick. The analysis of quicksort programs. Acta Informatica, 7:327--355, 1977.

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Sedgewick, R. (1977). The analysis of Quicksort programs. Acta Informat. 7, 327--355.

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R. Sedgewick, `The analysis of quicksort programs', Acta Informatica, 7, 327-355 (1977).

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R. Sedgewick. The analysis of quicksort programs. Acta Inform., 7:327#355, 1977.

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Sedgewick,R. The analysis of Quicksort programs. Acta Informatica 7, 327-355, 1977. 25

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Sedgewick,R. The analysis of Quicksort programs. Acta Informatica 7, 327-355, 1977.

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R. Sedgewick. The analysis of quicksort programs. Acta Inform., 7:327--355, 1977.

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