Understand mergesort and the working of recursion. The time and space complexity are also dealt with.

1. MERGE SORT -
HOW RECURSION WORKS
10 4 126152
2 4 6 10 12 15
Dr.S.Lovelyn Rose
PSGCollege ofTechnology
Coimbatore, India

2. Splitting
Solve subproblems
Combine solution
ALGORITHM
MergeSort(low,high)
{
if(low<high)
{
mid=(low+high)/2
MergeSort(low,mid)
MergeSort(mid+1,high)
Merge(low,mid,high)
}
}

3. Merge(low,mid,high)
{
h=low; i=low; j=mid+1
while((h<=mid) and (j<=high))
{
if(a[h]<=a[j])
{
b[i]=a[h]
h=h+1
}
else
{
b[i]=a[j]
j=j+1
}
i=i+1
}//End of while
temp array 2nd sub array1st sub array

4. temp to original array
for k=low to high
a[k]=b[k]
}
If(h>mid)
{
for k=j to high
{
b[i]=a[k]
i=i+1
}
}
else
{
for k=h to mid
{
b[i]=a[k]
i=i+1
}
}

5. Recursion Order
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
8 3 2 9 7 1 5 4
3 8 4 51 72 9
2 3 8 9 1 4 5 7
1 2 3 4 5 7 8 9
1
2
3 4
5
6
7 8
9
10
11
12
13 14
15
16
17 18
19
20
21
Note :The numbers in red help to understand the order in which recursion takes place

6. Formulating the Recurrence Relation
T(n)=2T(n/2)+(n-1)
Size of a sublist
Number of sublists
(n-1)
 Number of times basic operation is performed when a function is called
 Inside Merge() a minimum of (n-1) comparisons is performed.
Eg.
 If 1 element no comparison
 If 2 elements->max 1 comparison
 If 4 elements->3 comparisons

7. Solving Recurrence
MastersTheorem
a=2,b=2,f(n)=n-1
log b a=1
By case 2,f(n)=Ѳ(n1 )=O(n)
T(n)= Ѳ(n log n)

8. By Substitution Method
T(n)=2T(n/2)+(n-1)
=2(2T(n/22 )+((n/2)-1))+(n-1)
=22 T(n/22 )+(n-2)+(n-1)
=22 (2T(n/23 )+((n/4)-1))+(2n-(1+2))
=23T (n/23 )+3n-(1+2+22 )
.
.
=2k T(n/2k )+kn-(1+2+…+2k-1 )
Let n=2k
=2k T(1 )+n log n-(2k -1 )
=n log n-n+1
= Ѳ(n log n)

9. Space Complexity
= n+n = 2n = Ѳ(n)
Additional array ‘b’ of size ‘n’
Input Array of size ‘n’

10. My Blogs
http://datastructuresinterview.blogspot.in/
http://talkcoimbatore.blogspot.in/
http://simpletechnical.blogspot.in/