A guide to algorithm design paradigms, methods, and by Benoit A., Robert Y., Vivien F. PDF

By Benoit A., Robert Y., Vivien F.

ISBN-10: 1439898138

ISBN-13: 9781439898130

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Extra info for A guide to algorithm design paradigms, methods, and complexity analysis

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Each of three while loops executes at most n iterations, hence a complexity in O(n). 2: Algorithm identifying a star in a group. 3. First of all, we remark that the worst case is reached when the group of persons contains a star. To identify a star, it is necessary that each © 2014 by Taylor & Francis Group, LLC 16 Chapter 1. Introduction to complexity person, but the star, is involved in a question that identifies him or her as not being a star. This requires n 1 questions. It is also necessary that, if the group contains a star i, all n 1 questions: Does i know j?

All that is left is to insert g in the resulting set. In fact, because g h, we need only to insert g in a set of six elements if h i, or of seven elements if i < h. Such an insertion costs three additional comparisons. Overall, we sort 10 numbers in 5 1 + 7 + 2 + 2 + 3 + 3 = 22 comparisons. Sorting 11 numbers. We sort 10 of the numbers and then insert the 11th one using a binary search in 22 + 4 = 26 comparisons. 3. To sort 12 numbers, we first sort 11 of them and then insert the 12th number using a binary search, in 26 + 4 = 30 comparisons.

There are nevertheless two cases: • W < N : Then N becomes a winner and W an average element (it just lost a comparison and had previously won at least one as it was labeled a winner). The new situation is then (i 1, j, k, l + 1). • W > N : Then W remains a winner and N becomes a loser. The new situation is then (i 1, j, k + 1, l). The adversary chooses the latter case as it leads to a situation that is further from the completion of the algorithm than the former case. Indeed, in the former case the knowledge gain is that one element (W ) can be neither the minimum nor the maximum and thus can be safely ignored.

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A guide to algorithm design paradigms, methods, and complexity analysis by Benoit A., Robert Y., Vivien F.


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