02 - Spring 2010 CS530 Analysis of Algorithms Homework 2...

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Spring 2010 CS530 – Analysis of Algorithms Homework 2 Homework 2, due Feb 3 You must prove your answer to every question. Problems with a ( * ) in place of a score may be a little too advanced, or too challenging to most students, so I do not assign a score to them. But I will still note if you solve them. Problem 1 (28.1-6) . (10pts) Let A , B be n × n matrices such that AB = I . Prove that if A 0 is obtained from A by adding row j into row i 6= j then the inverse B 0 of A 0 is obtained by subtracting column i from column j of B . Solution. Let T = ( t kl ) be the matrix obtained from the unit matrix I after we add row j into row i . Thus, t kl = 1 if k = l or if k = i , l = j . It is clearly the same matrix as the one we obtain from I by adding column j to column i . It is easy to check that the inverse T - 1 is obtained if we subtract row j of I from row i , and that this is the same as when we work on the columns of I instead of rows. It is easy to check that A 0 = TA and hence B 0 = BT - 1 . But BT - 1 is obtained by subtract- ing column j of B from column i . Problem 2. (10pts) In class, I have shown how to compute the Fourier transform fast, writiting up a recurrence F ( n ) ± 2 · F ( n /2) + c · n for the cost of transforming a vector of length n , (here c is a constant). I indicated that this leads to the inequality F ( n ) ± d · n log n for some constant d . Derive this inequality now. Solution.
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02 - Spring 2010 CS530 Analysis of Algorithms Homework 2...

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