Lecture 7

Learning objectives

After this class, you should be able to:

1. Define the big-O, big-Omega, and big-Theta notations.
2. Given an algorithm, derive its asymptotic time complexity, and express it using the big-O notation.
3. Prove simple properties of the big-o notation, such as those given in section 2.3.
4. Given functions f and g, prove or disprove that f = O(g). You should be able to do the proofs by directly using the definition, and also by using properties of big-O.

Reading assignment

1. Read the note on complexity in www.cs.fsu.edu/~asriniva/courses/DS04/lectures/Lec6/complexity.html
2. Section 2.2 - 2.7.
3. Class notes.

Exercises and review questions

• Exercises and review questions on current lecture's material
  1. Prove, using induction, that 2n2 + 3n + 1 <4n2, n > 2.
  2. Prove that hi-lo+1 decreases by a factor of at least 2 in each iteration of the binary search algorithm presented in class.
  3. Chapter 2, exercise 2 a, d, and e.
  4. Chapter 2, exercise 3f.
  5. Chapter 2, exercise 4a.
  6. Prove that n3 + n is O(n3) directly from the definition of big-O. Show constants c and N that satisfy the definition.
  7. Prove that n3 + n is O(n3) using properties of big-O.
  8. Prove that n - 1 is big-Omega(n), directly from the definition of big-Omega.
  9. Prove that n - 1 is big-Theta(n), directly from the definition of big-Theta.

• Questions on next lecture's material
  1. None.