Information Theory

Algorithms and Complexity (Internet edition, 1994) by Herbert S. Wilf

By Herbert S. Wilf

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Sample text

Whereas the performance of slowsort is pretty much always quadratic, no matter what the input is, Quicksort is usually a lot faster than its worst case discussed above. We want to show that on the average the running time of Quicksort is O(n log n). The first step is to get quite clear about what the word ‘average’ refers to. We suppose that the entries of the input array x are all distinct. Then the performance of Quicksort can depend only on the sequence of size relationships in the input array and the choices of the random splitting elements.

9) = (n + 1)yn . 10) as an equation for the yn ’s (y0 = 0). 10) is obviously n yn = 2 j=1 n =2 j −1 j(j + 1) { j=1 2 1 − } j +1 j n =2 1 − 4n/(n + 1). 11) j=1 is the average number of pairwise comparisons that we do if we Quicksort an array of length n. 2. 11), and is ∼ 2n log n (n → ∞). Quicksort is, on average, a very quick sorting method, even though its worst case requires a quadratic amount of labor. 2 1. Write out an array of 10 numbers that contains no splitter. Write out an array of 10 numbers that contains 10 splitters.

We will also average over all such random choices of the splitting elements. Therefore, when we speak of the function F (n), the average complexity of Quicksort, we are speaking of the average number of pairwise comparisons of array entries that are made by Quicksort, where the averaging 35 Chapter 2: Recursive Algorithms is done first of all over all n! of the possible input orderings of the array elements, and second, for each such input ordering, we average also over all sequences of choices of the splitting elements.

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