C.J.H. McDiarmid, R.B. Hayward, Large deviations for quicksort, J. Algorithms 21 (1996) 476--507.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the potentially worst case of quicksort proper is to take rst a random sample of odd size 1 and then choose the median of this sample as the pivot. So the general cost measures satisfy the recurrence (35) with r = 2t 1 and p t = 1. Quadratic worst case is still possible but less likely; see [71, 84] for more recise results on large deviations. In terms of BSTs, this version of quicksort corresponds to picking the median of (2t 1) elements as the root once the size of the tree exceeds 2t 1. The number of partitions used to nd the median of (2t 1) satis es (36) It is known (see [20] that ....

C. J. H. McDiarmid and R. B. Hayward, Large deviations for quicksort, Journal of Algorithms, 21 (1996), 476-507.

....the potentially worst case of quicksort proper is to take first a random sample of odd size 1 and then choose the median of this sample as the pivot. So the general cost measures satisfy the recurrence (35) with r = 2t 1 and p t = 1. Quadratic worst case is still possible but less likely; see [71, 84] for more recise results on large deviations. In terms of BSTs, this version of quicksort corresponds to picking the median of (2t 1) elements as the root once the size of the tree exceeds 2t 1. The number of partitions used to find the median of (2t 1) satisfies (36) It is known (see [20] ....

C. J. H. McDiarmid and R. B. Hayward, Large deviations for quicksort, Journal of Algorithms, 21 (1996), 476-507.

....than 0. 5, the element x mid will lie in the middle third of the input range, and thus after one iteration of the while loop, the difference hi lo will be at most n 3, thus implying that the expected number of iterations of the while loop is O(log n) This can be shown to hold with high probability [16]. The crucial test in this algorithm is the following operation: Pick a random element lying in the range [lo,hi] Using the two sided depth test, we can extract all elements whose values lie between two endpoints. Once these fragments pass the two tests, they are rendered into the color buffer, ....

MCDIARMID, C., AND HAYWARD, R. Large deviations for quicksort. J. Algorithms 21 (1996), 476--507.

....Discretes (MIMD) Paris, France probabilistic analysis to understand better the Quicksort behavior. In particular, one would like to know how likely (or rather unlikely) it is for such pathological behavior to occur. A large body of literature is devoted to analyzing the Quicksort algorithm [4, 5, 6, 8, 14, 15, 16, 17, 18, 19, 21]. However, many aspects of this problem are still largely unsolved. To review what is known and what is still unsolved, we introduce some notation. Let L n denote the number of comparisons needed to sort a random list of length n. It is known that after selecting randomly a key, the two sublists ....

....deviations of L n , i.e. Pr[jL n Gamma E[L n ]j eE[L n ] for e 0. Hennequin [8] used Chebyshev s inequality to show that the above probability is O(1= elog n) Recently, Rosler [19] showed that this probability is in fact Gammak ) for any fixed k, and soon after McDiarmid and Hayward [15] used the powerful method of bounded differences to obtain an even better estimate, namely that the tail is approximately equal to n Gamma2e log log n (see the comment after Theorem 1 of Section 2) In this paper, we obtain some new results for the tail probabilities. First of all, we ....

[Article contains additional citation context not shown here]

C.J. McDiarmid and R. Hayward, Large Deviations for Quicksort, J. Algorithms, 21, 476--507, 1996.

....even for the median of (2t 1) version of Quicksort. These include in particular asymptotic expressions for the means and variances, as well as limit laws for the scaled quantities, and large deviation inequalities, see Hennequin [22, 23] R egnier [42] R osler [43, 45] McDiarmid and Hayward [11], Bruhn [3] and for a detailed survey the book of Mahmoud [28] For the number of exchanges B n the mean and variance were for general t 2 N 0 studied in Hennequin [23] Chern and Hwang [5] re ned the analysis of the mean, and Hwang and Neininger [25] gave a limit law for the standard case t = 0. ....

McDiarmid, C. J. H. and R. B. Hayward (1996). Large deviations for Quicksort. J. Algorithms 21, 476-507.

....such that the performance deviates very much from the expected value, is very small. In the case of Quicksort, a recent result states that the probability that the real number of comparisons performed by the algorithm deviates more than ffl times their expectation is O(n Gammaffl ln ln n ) MH96] Another important technique in computer science and discrete mathematics is the probabilistic method. The basic idea behind the probabilistic method is to show that a certain property holds for some combinatorial structure. To do so, one constructs an appropriate probability space, and shows ....

C. McDiarmid and R. B. Hayward. Large deviations for quicksort. Journal of Algorithms, 21:476--507, 1996.

....Knessl and Wojciech Szpankowski probabilistic analysis to understand better the Quicksort behavior. In particular, one would like to know how likely (or rather unlikely) it is for such pathological behavior to occur. A large body of literature is devoted to analyzing the Quicksort algorithm [4, 5, 6, 8, 14, 15, 16, 17, 18, 19, 21]. However, many aspects of this problem are still largely unsolved. To review what is known and what is still unsolved, we introduce some notation. Let L n denote the number of comparisons needed to sort a random list of length n. It is known that after selecting randomly a key, the two sublists ....

....]j eE[L n ] for e 0. Hennequin [8] used Chebyshev s inequality to show Quicksort algorithm again revisited 45 that the above probability is O(1= elog 2 n) Recently, Rosler [19] showed that this probability is in fact O(n Gammak ) for any fixed k, and soon after McDiarmid and Hayward [15] used the powerful method of bounded differences to obtain an even better estimate, namely that the tail is approximately equal to n Gamma2e log log n (see the comment after Theorem 1 of Section 2) In this paper, we obtain some new results for the tail probabilities. First of all, we establish ....

[Article contains additional citation context not shown here]

C.J. McDiarmid and R. Hayward, Large Deviations for Quicksort, J. Algorithms, 21, 476--507, 1996.

....Knessl and Wojciech Szpankowski probabilistic analysis to understand better the Quicksort behavior. In particular, one would like to know how likely (or rather unlikely) it is for such pathological behavior to occur. A large body of literature is devoted to analyzing the Quicksort algorithm [4, 5, 6, 8, 14, 15, 16, 17, 18, 19, 21]. However, many aspects of this problem are still largely unsolved. To review what is known and what is still unsolved, we introduce some notation. Let n denote the number of comparisons needed to sort a random list of length n. It is known that after selecting randomly a key, the two sublists ....

....for e ) 0. Hennequin [8] used Chebyshev s inequality to show Quicksort algorithm again revisited 45 that the above probability is O 1 elog 2 n . Recently, Rosler [19] showed that this probability is in fact O n k for any fixed k, and soon after McDiarmid and Hayward [15] used the powerful method of bounded differences to obtain an even better estimate, namely that the tail is approximately equal to n 2eloglogn (see the comment after Theorem 1 of Section 2) In this paper, we obtain some new results for the tail probabilities. First of all, we establish an ....

[Article contains additional citation context not shown here]

C.J. McDiarmid and R. Hayward, Large Deviations for Quicksort, J. Algorithms, 21, 476--507, 1996.

....analysis to understand better the Quicksort behavior. In particular, one would like to know how likely (or rather unlikely) it is for such pathological behavior to occur. Our goal is to answer precisely this question. A large body of literature is devoted to analyzing the Quicksort algorithm [3, 4, 5, 7, 11, 12, 13, 14, 15, 17]. However, many aspects of this problem are still largely unsolved. To review what is known and what is still unsolved, we introduce some notation. Let L n denote the number of comparisons needed to sort a random list of length n. It is known that after selecting randomly a key, the two sublists ....

....of L n , i.e. Pr[jL n Gamma E[L n ]j E[L n ] for 0. Hennequin [7] used Chebyshev s inequality to show that the above probability is O(1= log 2 n) Recently, Rosler [15] showed that this probability is in fact O(n Gammak ) for any fixed k, and soon after McDiarmid and Hayward [12] used the powerful method of bounded differences to obtain an even better estimate, namely that the tail is asymptotically equal to n Gamma2 log log n (see the comment after Theorem 1 of Section 2) In this paper, we improve on some of the previous results. First of all, we establish an ....

[Article contains additional citation context not shown here]

C.J. McDiarmid and R. Hayward, Large Deviations for Quicksort, J. Algorithms, 21, 476--507, 1996.

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C.J.H. McDiarmid, R.B. Hayward, Large deviations for quicksort, J. Algorithms 21 (1996) 476--507.

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C. J. H. McDiarmid and R. B. Hayward. Large deviations for Quicksort. J. Algorithms, 21(3):476-507, 1996.

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C.J. McDiarmid and R. Hayward, Large deviation for quicksort, Journal Algorithms, 21 (1996), 476-507.

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