DIVIDE and CONQUER technique, Exercises of Algorithms and Programming

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Module 2 Divide and Conquer Prof. Hemachander N, Dept. of Computer Science & Engineering Cauvery Institute of Technology, Mandya

Contents

1. General method
2. Recurrence equation
3. Algorithm: Binary search
4. Algorithm: Finding the maximum and minimum
5. Algorithm: Merge sort
6. Algorithm: Quick sort
7. Algorithm: Strassen’s matrix multiplication
8. Advantages and Disadvantages
9. Decrease and Conquer Approach
10. Algorithm: Topological Sort

Divide and Conquer

Control Abstraction for Divide &Conquer

General Time functionm of DandC

Technique:

where f(n) is a function that accounts for the time spent on dividing the problem into smaller ones and on combining their solutions.

• (^) T(n)=

Finding Time Complexity of DandC

Algorithms

• (^) Substitution Method
• (^) Masters Theorem

Master Theorem

• (^) It states that, in recurrence equation T(n)=aT(n/b) + f(n) Time complexity can be computed based on how the value of a & bd^ relates to each other. Here d is power of n in f(n)

Example

Find the time complexity of the following equation using substitution method and master theorm [assume T(1)= 0 ]

Excercise

1. T(n)=T(n-1)+n T(1)=
2. Solve Recurrence Relation a=2, b=2, f(n)=cn
3. Solve T(n)=T(n/2)+n, T(1)=
4. T(n)=2 T(n/2)+1, T(1)=