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# HW1 - COT 5405 Algorithms Spring 2003 Due Date 5pm Homework...

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COT 5405: Algorithms Spring 2003 Due Date: 06/02/2003, 5pm Homework Assignment 1. Please use Divide-and-Conquer 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(N-1) + T(N-2) + 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) n-2 for all n >= 1 where F n denotes the n-th Fibonacci number. In other words, prove that F n = ( (3/2) n-2

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