Assignment 3 -- due Monday, September 27 at the beginning of lab
CS 4521 Fall Semester, 2012
20 Points

Topics: Growth of Functions, Master Theorem

The assignment:
Do the following Exercises from the text, which ask you to prove properties of n!, the Fibonacci numbers, and polynomials.

• (7 points) Exercise 3.2-3, page 60. First show that lg(n!) = O(n lg n), which follows from lg(n!) ≤ n lg(n) (easy to show).
Second show lg(n!) = Ω(n lg n), which follows from lg(n!) ≥ 1/4 n lg(n) (for n ≥ 4).
Then lg(n!) = Θ(n lg n) follows from "First" and "Second" and Theorem 3.1, page 48).
Hints for "Second": first show that lg(n!) ≥ ceiling((n+1)/2) lg(ceiling(n/2)) which is ≥
(n/2) lg(n/2) (show this for both odd and even n),
then show that (n/2) lg(n/2) ≥ 1/4 n lg(n) for n ≥ 4.
To show n! = ω(2n) and n! = o(nn), you can show that the limits as n goes to infinity of n!/2n and n!/nn are infinity and 0 respectively, since n!, 2n, and nn are all (asymptotically) nonnegative.
• (3 points) Exercise 3.2-7, page 60. Hint: use the fact that φ satisfies the equation
x2 = x + 1 -- you need to prove this (and use a "strong" form of induction).
• (5 points) Problem 3-1, parts a., b., and c., page 61. Hint: for part c., use parts a., b., and Theorem 3.1.
• (4 points) Exercise 4.5-1, page 96.
• (1 point) Exercise 4.5-3, page 97.

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