COT 5405: Algorithms
Spring 2003
Due Date: 06/02/2003, 5pm
Homework Assignment 1.
Please use DivideandConquer approach to solve each of these problems (unless indicated otherwise).
Also, you are expected to derive the best possible algorithm to solve the problem (unless indicated
otherwise). Poor performance solutions (even if they are correct) may not get any credit. Divide your
solution into four parts:
1.
High Level Algorithm
2.
Pseudo Code (or code in any high level language such as Java, C, C++)
3.
Proof of Correctness
4.
Complexity Analysis
Please write legibly.
1. (10 points)
Solve the recurrence relation iteratively (not using the Master’s theorem):
T (N) = 2T ( N /2 ) + log N
Make any reasonable assumptions. Find an exact formula for T (N), and then give
a Theta expression.
2. (10 points)
Suppose T(N) = T(N1) + T(N2) + 1 and T(1) = T(0) = 1. Prove that T(N) =
(F
N+1
) where F
N+1
represents Fibonacci sequence.
3. (10 points)
Show that F
n
>= (3/2)
n2
for all n >= 1 where F
n
denotes the nth Fibonacci
number. In other words, prove that F
n
=
( (3/2)
n2
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 Fall '08
 UNGOR
 Algorithms

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