UMBC CMSC441, Design & Analysis of Algorithms, Fall 1999, Section 0101

Homework Assignments 1-4

Homework 1, Due Thursday September 9

1. For each of the following pairs of functions f(n) and g(n), give the values of n₀, f(n₀) and g(n₀) where n₀ is the smallest positive integer such that f(n₀) >= g(n₀). Hint: you may need to use a calculator. Use the log identity

   log₂ x = (log₁₀ x)/(log₁₀ 2)

   if your calculator does not take logs in base 2.

   1. f(n) = n and g(n) = 50 lg² n

   2. f(n) = 2^n/2 and g(n) = n¹⁸

   3. f(n) = n^1/5 and g(n) = lg¹² n
    

2. Prove that the following statements are correct:

   1. 5n² - 6n = Theta(n²)

   2. n! = O(nⁿ)

   3. n^1.001 + n lg n = Theta(n^1.001)
    

3. Show that the following statements are incorrect:

   1. 10 n² + 9 = O(n)

   2. n²/lg n = Theta(n²)
    

Homework 2, Due Thursday September 16

1. Prove by induction that for x != 1 and n >= 0 that

   (Summation of xⁱ for i=0 to n) = (xⁿ⁺¹ - 1)/(x-1).

2. Show that (summation of i² for i=0 to n) = Theta(n³).

3. Exercise 4.1-5, page 57.
   Hint: use the fact that log(0.75 n) > log (n/2 + 17) for large enough n.

4. Exercise 4.2-1, page 60.

Homework 3, Due Thursday September 23

1. Problem 4-1, page 72.
   In the cases where you use the Master Theorem, clearly state which case you are using and the asymptotic bounds on T(n).

2. Exercise 7.5-4, page 151.

3. Exercise 7.5-5, page 151.

4. Problem 7-1, page 152.

Homework 4, Due Thursday September 30

1. Problem 8-3, page 169.

2. Problem 8-4, page 169-170.

3. Exercise 9.3-4, page 180.