Lecture Notes: Solving Problems with Big Omega and Big Theta Notations

Introduction

• Topics Covered: Solved problems for Big Omega and Big Theta notations
• Number of Problems: Two problems; one for each notation

Big Omega Notation Solved Problem

Problem Statement

• Objective: Find the lower bound for the function F(N) = 10N² + 5

Key Concepts

• Lower Bound: A function that grows asymptotically lesser than F(N)
• Upper Bound: A function that grows asymptotically greater than F(N)
• Dominant Term: The term in a function that grows fastest as N increases

Steps to Find the Lower Bound

1. Find the Dominant Term: For F(N) = 10N² + 5, the dominant term is 10N²
2. Assume a Function G(N): Based on the dominant term, assume G(N) = N²
3. Apply Big Omega Definition: F(N) is Big Omega of G(N) if for constants C > 0 and N > N_0, F(N) >= C * G(N)

Solution

• Substitute Values:
  • F(N) = 10N² + 5
  • Assume C = 10
  • G(N) = N²
• Inequality Check:
  • 10N² + 5 >= 10N² for N >= 1
• Result: 10N² + 5 is Big Omega of N² which proves that N² is the lower bound of F(N)

Big Theta Notation Solved Problem

Problem Statement

• Objective: Show that F(N) = N³ + 3N² is Theta of N³

Key Concepts

• Theta Notation: F(N) is Big Theta of G(N) if both C1 * G(N) <= F(N) <= C2 * G(N) for constants C1, C2 > 0 and N >= N_0

Steps to Prove Big Theta Notation

1. Prove Upper Bound: Show F(N) <= C2 * G(N)
2. Prove Lower Bound: Show C1 * G(N) <= F(N)

Solution

Upper Bound

• Assume C2 = 2
• Inequality Check:
  • N³ + 3N² <= 2N³ for N >= 3
• Result: Proves F(N) <= C2 * G(N) for N >= 3

Lower Bound

• Assume C1 = 1
• Inequality Check:
  • N³ + 3N² >= N³ for N >= 1
• Result: Proves C1 * G(N) <= F(N) for N >= 1

Combining Results

• N Condition: Both inequalities satisfied for N >= 3
• Final Result: F(N) = N³ + 3N² is Theta of N³ for C1 = 1, C2 = 2, and N_0 = 3

Conclusion

• Solved problems for both Big Omega and Big Theta notations
• Successfully demonstrated the methodology to find lower and tight bounds
• End of lecture