CS 170, Fall 1999
Midterm II
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.
(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 tie during union, assume that UNION will put the
lexicographically first letter as root.
2. (25 points) Let
be polynomials of degrees n and m, respectively, where n and m can be integers such that n> m
(a) Give an algorithm using the FFT that computes the coefficients of r(x) = p(x)· q(x). How many arithmetic
operations does it perform, as a function of m and n? Your answer can use O() notation.
(b) Give an algorithm not using the FFT that computes the coefficients of r(x) = p(x)· q(x). How many
arithmetic operations does it perform, as a function of m and n?
(c) Combine the above algorithms to give the fastest possible algorithm depending on m and n. How many
arithmatic operations does it perform? Roughly how small (in an O() sense) does m have to be for the
non-FFT algorithm to be at least as fast as the FFT algroithm?
3. (25 points) Giaven a set S = {s1, s2, ..., sn} of n nonnegative integers, and a positive integer T, find a subset
of S that adds up to T. Use dynamic programming; your solution should not have a cost growing like 2n.
You should (1) Formulate your algorithm recursively, (2) describe how it would be implemented in a
bottom-up iterative manner, (3) give a bound on its running time in terms of n and T, and (4) give a short
justification of both the correctness of the algorithm and its running time.
CS170 Fall 1999 Midterm II
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