H.-H. Chern, H.-K. Hwang, Transitional behaviors of the average cost of quicksort with medianof 1), Algorithmica 29 (2001) 44--69.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... 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. Here we will give an asymptotic analysis of the joint distribution Y n : C n ; B n ) for general t 2 N 0 . A bivariate limit law is derived, which covers especially the missing ....

Chern, H.-H. and H.-K. Hwang (2001). Transitional behaviors of the average cost of Quicksort with median-of-(2t + 1). Algorithmica 29, 44-69.

.... ln(n) c wn 2 o(n 2 ) with c w = c p b E V 2 1 b E V 2 : Candidates could be random quadtrees, random m ary search trees or random median of (2k 1) search trees, where the techniques developed in Flajolet, Labelle, Laforest, and Salvy [10] Chern and Hwang [2] and Chern and Hwang [1] respectively could prove useful. Simply generated families of trees. As shown in Entringer, Meir, Moon, and Sz ekely [9] the mean of the Wiener index of some simply generated families of trees is of the order n 5=2 , where we obtained for the recursive and binary search trees n 2 ln(n) This ....

Chern, H.-H. and Hwang, H.-K. (2001) Transitional behaviors of the average cost of Quicksort with median-of-(2t + 1). Algorithmica 29 44-69.

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H.-H. Chern, H.-K. Hwang, Transitional behaviors of the average cost of quicksort with medianof 1), Algorithmica 29 (2001) 44--69.

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H.-H. Chern and H.-K. Hwang, Transitional behaviors of the average cost of quicksort with medianof -(2t+1), Algorithmica, 29 (2001), 44--69.

....such a situation (where explicit determinations of the dominant constants are dicult) include partial match queries in k d trees [45] partial match queries in quadtrees [41] and the number of consecutive records [21] Methodology. Restricting to CE equations, the method of linear operator (see [19, 20, 50, 54, 65, 102]) is especially useful for providing e ective forms of the coecients in the asymptotic expansion of the function (near its dominant singularity) it is also computationally simpler than other approaches such as the method of undetermined coecients and the method of variation of parameters. We ....

....jw 1j ; see Hille [55, x9.4] for the required theory. On the other hand, it is straightforward to prove that all zeros are simple. It follows, by Theorem 1, that ]M(w; z) K(w)n 1 (w) 1 (1 o(1) uniformly for jw 1j , where K(w) is analytic for jw 1j . Then the techniques used in [19] can be further applied to derive the asymptotics of E(X n;k ) w ]M(w; z) See also [19] for an analysis of the levelwise improvements of quicksort and other related problems, and [54, 77] for more examples of the same type. 4.11 Introspective quicksort and balanced BSTs Introspective ....

[Article contains additional citation context not shown here]

H.-H. Chern and H.-K. Hwang, Transitional behaviors of the average cost of quicksort with median-of-(2t + 1), Algorithmica, 29 (2001), 44-69.

....II: the limit law of Y n changes from normal to non normal if we fix I n (say I n = Uniform[0; 1; n Gamma 1] and vary T n . These two types of phase changes are similar to those in statistical physics where type I phase transition is discrete in nature and type II continuous; see also [7] for another type of phase transition for quicksort. 2.1 Type I phase change: from normal to non existence A simple instance of Type I phase change is the following quicksort using median of (2t 1) instead of choosing the pivot uniformly at random, choose the pivot as the median of a sample of ....

H.-H. Chern and H.-K. Hwang, Transitional behaviors of the average cost of quicksort with median-of-(2t + 1), Algorithmica, 29 (2001), 44--69.

....such a situation (where explicit determinations of the dominant constants are di#cult) include partial match queries in k d trees [45] partial match queries in quadtrees [41] and the number of consecutive records [21] Methodology. Restricting to CE equations, the method of linear operator (see [19, 20, 50, 54, 65, 102]) is especially useful for providing e#ective forms of the coe#cients in the asymptotic expansion of the function (near its dominant singularity) it is also computationally simpler than other approaches such as the method of undetermined coe#cients and the method of variation of parameters. We ....

....for #; see Hille [55, 9.4] for the required theory. On the other hand, it is straightforward to prove that all zeros are simple. It follows, by Theorem 1, that ]M(w, z) K(w)n # 1 (w) 1 (1 o(1) uniformly for #, where K(w) is analytic for #. Then the techniques used in [19] can be further applied to derive the asymptotics of E(X n,k ) w ]M(w, z) See also [19] for an analysis of the levelwise improvements of quicksort and other related problems, and [54, 77] for more examples of the same type. 4.11 Introspective quicksort and balanced BSTs Introspective ....

[Article contains additional citation context not shown here]

H.-H. Chern and H.-K. Hwang, Transitional behaviors of the average cost of quicksort with median-of-(2t + 1), Algorithmica, 29 (2001), 44-69.

....Type II: the limit law of Y n changes from normal to non normal if we fix I n (say I n = Uniform[0, 1, n 1] and vary T n . These two types of phase changes are similar to those in statistical physics where type I phase transition is discrete in nature and type II continuous; see also [7] for another type of phase transition for quicksort. 2.1 Type I phase change: from normal to non existence A simple instance of Type I phase change is the following quicksort using median of (2t 1) instead of choosing the pivot uniformly at random, choose the pivot as the median of a sample ....

H.-H. Chern and H.-K. Hwang, Transitional behaviors of the average cost of quicksort with median-of-(2t + 1), Algorithmica, 29 (2001), 44--69.

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