Lecture 6

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. Section 2.3 - 2.5.
2. Class notes.
3. Page 76 and 77.

Exercises and review questions

• Exercises and review questions on current lecture's material
  1. Prove, using induction, that 2n2 + 3n + 1 <4n2, n > 2. (Please don't post the solution on blackboard. You may give us a hardcopy of your solution, if you wish to get feedback from us.)
  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. (Skip this question, for now.)
  3. Chapter 2, exercise 2 a, d, and e. (Please don't post the solution on blackboard. You may give us a hardcopy of your solution, if you wish to get feedback from us.)
  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. Consider a variable q that is declared as: IntNode *q;, where IntNode is defined on pages 75 and 76 of the text. Write code that will do the following. It will make q point to an IntNode object that has info=7. This IntNode object, in turn, should point to another IntNode object that has info = 3. Then print out the info field of the second IntNode object (the one with info = 3).