Assignment 3 -- due Monday, September 27
at the beginning of lab
CS 4521 Fall Semester, 2010
Topics: Growth of Functions, Master Theorem
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 ≥
(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 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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