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

ISBN-10: 1439898138

ISBN-13: 9781439898130

**Read or Download A guide to algorithm design paradigms, methods, and complexity analysis PDF**

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**Example text**

B ! d with two comparisons. Finally, we are left only with inserting c in the sorted chain made of the three elements a, b, and e, which costs two additional comparisons (we already know that c d). Overall, we sort five numbers in 2 1 + 1 + 2 + 2 = 7 comparisons. Sorting 6 numbers. We sort five of the numbers and then insert the sixth one using a binary search in 7 + 3 = 10 comparisons. Sorting 7 numbers. We sort six of the numbers and then insert the seventh one using a binary search in 10 + 3 = 13 comparisons.

We start with homogeneous recurrences and then consider the most general case of recurrences with a second member. 1 Solving homogeneous recurrences A homogeneous linear recurrence with constant coefficients has the form p0 sn +p1 sn−1 + +pk sn−k = 0, where each pi is a constant and (si )i 0 is an unknown sequence. , the linear combination is set equal to zero. Solving such recurrences k requires finding all the roots of the polynomial P = i=0 pi X k−i , together with their multiplicity order. However, we see that P is a polynomial of degree k, and no algebraic method can find the roots of arbitrary polynomials © 2014 by Taylor & Francis Group, LLC 38 Chapter 2.

1. Can we easily apply Strassen’s algorithm to compute the product A B? 2. Apart from the classical multiplication algorithm, another way to view the product consists of multiplying columns of A with rows of B. Give an expression of A B, using Ac [ ], the -th column of A, and Br [ ], the -th row of B. 3. To optimize the matrix product, the idea is to partition the columns of A and the rows of B into n/k equal-sized groups of size k (we can assume, for simplicity, that k divides n; otherwise, the last group is smaller).

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