By Benoit A., Robert Y., Vivien F.
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Additional resources for A guide to algorithm design paradigms, methods, and complexity analysis
Example text
The problem of establishing the complexity of matrix product is still open. The only known lower bound is a disappointing O(n2 ); we need to touch each coefficient at least once. Strassen’s algorithm provides, however, an excellent illustration of the divideand-conquer paradigm, which we formalize in the next section through the master theorem. © 2014 by Taylor & Francis Group, LLC 36 Chapter 2. 2 Master theorem Before formulating the master theorem, we need to formalize the divide-andconquer paradigm that was illustrated in the previous section through the Strassen’s algorithm.
4. Exercises Now by induction for k (E c)k+1 fcn 39 1, writing Qk (n) = a0 nk + Qk−1 (n): c)k+1 fcn (a0 nk + Qk−1 (n))g c)k [(E c)fcn (a0 nk + Qk−1 (n))g] c)k fcn+1 (a0 (n + 1)k + Qk−1 (n + 1)) cn+1 (a0 nk + Qk−1 (n))g c)k [cn+1 Rk−1 (n)], Qk (n)g = (E = (E = (E = (E where Rk−1 (n) is a polynomial in n of degree k 1, because both (n+1)k nk and Qk−1 (n+1) Qk−1 (n) are polynomials of degree k 1. With the induction hypothesis, we obtain the result: (E c)k+1 fcn Qk (n)g = f0g. 1)): A(n) = 7 A n 18 + 2 4 n2 .
In both cases, c is inserted with, at most, two comparisons. 5. Bibliographical notes 31 have sorted the set fa, b, c, d, e, f, h, jg. We then insert i in the set of seven elements fa, b, c, d, e, f, h, g using three comparisons (we already know that i 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.
