April 20, 2017

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

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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? Compute M (n) and A(n), assuming that we use the classical multiplication algorithm to compute R1 , R2 , and R3 .

Overall, we sort nine numbers in 4 1 + 5 + 2 + 2 + 3 + 3 = 19 comparisons. Sorting 10 numbers. To sort 10 numbers, we proceed as previously. We start by creating five sorted pairs with five comparisons. Then we sort the greatest elements of the pairs in seven comparisons. We then obtain the following configuration: j f b d h a c e g i Next we insert e in a ! b ! d with two comparisons. Then, if e d, we insert c in a ! b. Otherwise, we insert c in fa, b, eg. In both cases, c is inserted with, at most, two comparisons.

Each leave corresponds to a possible output (several leaves can correspond to the same output). From what precedes, any decision tree must have at least m+n leaves, n and its height must be at least log m+n . n (2n)! 2. Here we consider the case m = n. )2 . We use Stirling’s approximation of the factorial function: n! p p then n 2πn ne . We have 2 2π 3. Therefore, for n sufficiently large, p p p n n n we have 2 n ne 2πn ne 3 n ne , and p 2 2n p 3 n (2n)! )2 2n 2n e n n 2 e = p 2 2 1 2n p 2 . 9 n Finally, for n sufficiently large, 2n log n log p 2 2 9 1 log(n) + 2n 2 2n 1 log(n) 2 2.

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