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Computer Science 170 - Fall 1999 - Demmel - Midterm 2

# Computer Science 170 - Fall 1999 - Demmel - Midterm 2 - CS...

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CS 170, Midterm #2, Fall 1999, SOLUTIONS CS 170, Fall 1999 Midterm 2 with Solutions Professor Demmel Problem #1 1) (15 points) The following is a forest formed after some number of UNIONs and FINDs, starting with the disjoint sets A,B,C,D, E, F, G, H, and I. Both union-by-rank and path compression were used. (a) Starting with the forest above, we now call the following routines in order: FIND(B), UNION (G,H), UNION (A,G), UNION (E,I) Draw the resulting forest, using both union-by-rank and path compression. In case of tie during UNION, assume that UNION will put the lexicographically first letter as root: Answer: (b) Starting with the disjoint sets A, B, C, D, E, F, G, H, and I, give a sequence of UNIONs and FINDs that results in the forest shown at the top of the page. In case of a tie during union, assume that UNION will put the lexicographically first letter as a root. Answer: One solution is UNION (F,G), UNION (A,C), UNION (B,E), UNION (B,D), UNION (D,A) file:///C|/Documents%20and%20Settings/Jason%20Raft...-%20Fall%201999%20-%20Demmel%20-%20Midterm%202.htm (1 of 4)1/27/2007 5:30:44 PM

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CS 170, Midterm #2, Fall 1999, SOLUTIONS Problem #2 2) (25 points) Let p(x) = SUM_FROM_i=0_to_n (p_sub_i*x^i) and q(x) = SUM_FROM_i=0_to_m (q_sub_i*x^i) be polynomials of degrees n and m, respectively, where n and m can be any integers such that n>=m. (a) Give an algorithm using the FFT that computes the coefficients of r(x) = p(x)_DOT_q(x). How many arithmetic operations does it perform, as a function of m and n? Your answer can use O() notation. Answer: (1) Round up n+m+1 to the nearest power of 2, ie find the smallest k such that 2^k>=n+m+1: k = CEILING_OF(LOGbase2(n + m + 1)). (2) Pad the vectors [p_sub_0,...,p_sub_n] and [q_sub_0,..., q_sub_n] with enough zeroes to make vectors p_prime and q_prime of length 2^k. (3) Compute p_hat
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