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

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

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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. 1 is inspired from the book by Knuth [62] (another monument). Several exercises come from the wonderful book by Rawlins [92], but the solutions are ours. 8 (the toolbox).

A(n)) the number of multiplications (resp. number of additions) done by an algorithm to multiply two n-polynomials. 1. Compute M (n) and A(n) for the usual algorithm to multiply two npolynomials. 2. We assume that n is even, n = 2 m. We can then write P = P1 + X m P2 and Q = Q1 +X m Q2 . What is the degree of the polynomials P1 , P2 , Q1 , and Q2 ? 3. Let R1 = P1 Q1 , R2 = P2 Q2 , and R3 = (P1 + P2 ) (Q1 + Q2 ). Can you express R = P Q as a function of R1 , R2 , and R3 ? What is the degree of these three new polynomials?

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? and all n 1 questions: Does j know i?

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