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

Example text

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? are asked. However, these two sets of questions are not independent. As noted earlier, each question can, at most, invalidate one candidate as being a star. It is suboptimal for an algorithm to ask a question involving a person who is known not to be a star when there are still at least two star candidates. Indeed, in the worst case, this question will not provide any new knowledge.

2: Tree of comparisons to determine the maximum of a set of n = 2k values. 3: Tree of comparisons to determine the maximum of a set of n 6= 2k values. For the general case, we want to fall back to the case where n is a power of 2. Therefore, we take for k the value dlog(n)e. 3. As previously, we need n 1 comparisons to identify the maximum. Then, the longest branch in the tree has a length of dlog(n)e. Therefore, we need at most dlog(n)e 1 comparisons to identify the second maximum, for an overall complexity of n+dlog(n)e 2.

P1 , P2 , Q1 , and Q2 are m-polynomials and of degree m 1. 3. We have R = R1 + (R3 R2 R1 ) X m + R2 X 2m . R1 , R2 , and R3 are polynomials of degree 2m 2 = n 2, and thus (n 1)-polynomials. 2 Following this computation scheme, M (n) = 3M ( n2 ) = 3n4 , as the computation of R1 , R2 , and R3 each requires M (m) = M ( n2 ) multiplications. 5. Solutions to exercises 43 computations of R1 , R2 , and R3; (2) those involved in computing the polynomials P1 + P2 and Q1 + Q2 needed by R3 ; (3) those involved in computing R3 R2 R1 ; and (4) those needed to compute R from the partial results.

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