1. Growth Functions Algorithms Lecture 8

2. Many algorithms are recursive in structure to solve a given problem
Recursive functions
Many algorithms are recursive in structure to solve a given problem
That means they call themselves recursively one or more times
Algorithms lecture Dr. Arwa Zabian

3. Divide and Conquer
It breaks the problem into several sub problems that are similar to the original problem but smaller in size, solve the sub problems recursively, and then combine these solutions to create a solution to the original problem.
Divide and conquer paradigm involves three steps at each level of recursion:
Divide : the problem into a number of sub problems
Conquer : solve the sub problems recursively
Combine : the solution to the sub problems into the solution for the original problem
Algorithms lecture Dr. Arwa Zabian

4. Example: Merge Sort Algorithm
Is an algorithm that uses divide and conquer approach to sort an array of size n
It works as follow:
Divide A into two sub arrays A1 , A2 of size n/2 ( this operation is done recursively until arriving at one element)
Sort each sub array separately with Merge Sort
Merge the two sorted sub arrays into the final array.
Algorithms lecture Dr. Arwa Zabian

5. Exercise Apply Merge sort on the following array
The size of the array n=10
The solution :
Divide a into two sub arrays of size 5 , repeat this operation until arriving at two sub arrays of size 2 each one.
Sort these two sub arrays separately and then combine the two sorted arrays
The solution is presented in the following figure:.
Algorithms lecture Dr. Arwa Zabian

6. Algorithms lecture 8 Dr. Arwa Zabian

7. Merge Sort Pseudo code MERGE-SORT (A, p,r) If p < r
Then q  (p + r)/2
Merge-Sort (A, p, q)
Merge-Sort (A, q+1, r)
Merge (A, p, q,r)
Algorithms lecture Dr. Arwa Zabian

8. Exercise Consider the following sequence of integers:
Sort this sequence using Insertion sort and Merge sort
Calculate the number of comparisons and the running time
Algorithms lecture Dr. Arwa Zabian

9. Analyze of Merge Sort
When the sequence to be sorted has length 1, so it is sorted .
The key operation in Merge sort is the combine or merge operations. For that, we use an auxiliary procedure MERGE (A, p, q, r)
Where A is an array , and p, q, r are indices numbering elements of the array such that the subarrays A[p….q], and A[q+1,…..r] are sorted. It merges them to form a single sorted array.
Merge sort take time  (n), where n is the number of elements to be merged.
If p  r , the sub array has at most one element and is already sorted
Algorithms lecture Dr. Arwa Zabian

10. Analyzing Divide and Conquer Algorithms:
The running time T(n) of divide and conquer is based on three steps:
T(n)  (1) if n  c n too small
a T(n/b) + D(n) + C(n) otherwise
Number of
Sub problems
Time needed
to solve each
Sub problem
Time of
division
Time of
Merging
So, we have a problem of size n
We divide n into a sub problem, each one of size 1/b
Algorithms lecture Dr. Arwa Zabian

11. Analysis of Merge Sort Dividing steps D(n) =  (1)
Conquer solving the problem recursively 2T(n/2) of size n/2
Combine for n elements C(n)=  (n)
 (1) if n=1
T(n) =
2 T(n/2)+  (n) if n > 1
T(n) =  (n log n) where n is the size of the input
Algorithms lecture Dr. Arwa Zabian

12. Exercise.1
Suppose we are comparing implementation of Insertion sort and Merge sort on the same machine. For input of size n, insertion sort runs in 8n2 steps , while Merge sort runs in 64 n logn steps. For which values of n does insertion sort beat Merge sort?
Algorithms lecture Dr. Arwa Zabian

13. Exercise 2
Consider A is sorted array, how you can find the location and the value of the Mid point. And what characteristics does satisfy all the elements on the left and on the right of the Mid point.
Write a pseudo code for finding the Mid point
Algorithms lecture Dr. Arwa Zabian

14. Exercise 3
What is the smallest value of n such that an algorithm whose running time is 100 n2 run faster than an algorithm whose running time is 2n on the same machine.
Algorithms lecture Dr. Arwa Zabian