Optimal Sampling Strategies in Quicksort and Quickselect

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Optimal Sampling Strategies in Quicksort and Quickselect

It is well known that the performance of quicksort can be improved by selecting the median of a sample of elements as the pivot of each partitioning stage. For large samples the partitions are better, but the amount of additional comparisons and exchanges to find the median of the sample also increases. We show in this paper that the optimal sample size to minimize the average total cost of qui...

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(A) Let Q(A) be number of comparisons done on input array A: (A) R ij : event that rank i element is compared with rank j element, for 1 ≤ i < j ≤ n. (B) X ij is the indicator random variable for R ij. That is, X ij = 1 if rank i is compared with rank j element, otherwise 0. (B) Q(A) = ∑ 1≤i<j≤n X ij. (C) By linearity of expectation,

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Let Q(A) be number of comparisons done on input array A: (A) For 1 ≤ i < j < n let R ij be the event that rank i element is compared with rank j element. (B) X ij is the indicator random variable for R ij. That is, X ij = 1 if rank i is compared with rank j element, otherwise 0. Q(A) = ∑ 1≤i<j≤n X ij and hence by linearity of expectation,

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ژورنال

عنوان ژورنال: SIAM Journal on Computing

سال: 2001

ISSN: 0097-5397,1095-7111

DOI: 10.1137/s0097539700382108