# Python Program - Merge Sort

Merge sort is a divide and conquer algorithm. It is based on the idea of dividing the unsorted array into several sub-array until each sub-array consists of a single element and merging those sub-array in such a way that results into a sorted array. The process step of merge sort can be summarized as follows:

• Divide: Divide the unsorted array into several sub-array until each sub-array contains only single element.
• Merge: Merge the sub-arrays in such way that results into sorted array and merge until achieves the original array.
• Merging technique: the first element of the two sub-arrays is considered and compared. For ascending order sorting, the element with smaller value is taken from the sub-array and becomes a new element of the sorted array. This process is repeated until both sub-array are emptied and the merged array becomes sorted array.

### Example:

To understand the merge sort, lets consider an unsorted array [4, 9, -4] (right side array created after 11th process in the below diagram) and discuss each step taken to sort the array in ascending order.

At the first step, the array [4, 9, -4] is divided into two sub-array. The first sub-array contains [4, 9] and second sub-array contains [-4]. As the number of element in the first sub-array is greater than one, it is further divided into sub-arrays consisting of elements [4] and [9] respectively. As the number of elements in all sub-arrays is one, hence the further dividing of the array can not be done.

In the merging process, The sub-arrays formed in the last step are combined together using the process mentioned above to form a sorted array. First, [4] and [9] sub-arrays are merged together to form a sorted sub-array [4, 9]. Then [4, 9] and [-4] sub-arrays are merged together to form final sorted array [-4, 4, 9]

## Implementation of Merge Sort

```# function for merge sort - splits the MyList
# and call merge function to sort and merge the MyList
# mergesort is a recursive function
def mergesort(MyList, left, right):
if left < right:
mid = left + (right - left)//2
mergesort(MyList, left, mid)
mergesort(MyList, mid+1, right)
merge(MyList, left, mid, right)

# merge function performs sort and merge operations
# for mergesort function
def merge(MyList, left, mid, right):
#  Create two temporary List to hold split
#  elements of main MyList
n1 = mid - left + 1  # no of elements in LeftList
n2 = right - mid     # no of elements in RightList
LeftList = MyList[left:mid+1]
RightList = MyList[mid+1:right+1]

#  In below section x, y and z represents index number
#  of LeftList, RightList and MyList respectively
x, y, z = 0, 0, left
while x < n1 and y < n2:
if LeftList[x] < RightList[y]:
MyList[z] = LeftList[x]
x+=1
else:
MyList[z] = RightList[y]
y+=1
z+=1

#  Copying the remaining elements of LeftList
while x < n1:
MyList[z] = LeftList[x]
x+=1
z+=1
#  Copying the remaining elements of RightList
while y < n2:
MyList[z] = RightList[y]
y+=1
z+=1

# function to print list
def PrintList(MyList):
for i in MyList:
print(i, end=" ")
print("\n")

# test the code
MyList = [10, 1, 23, 50, 4, 9, -4]
n = len(MyList)
print("Original List")
PrintList(MyList)

mergesort(MyList, 0, n-1)
print("Sorted List")
PrintList(MyList)
```

The above code will give the following output:

```Original List
10 1 23 50 4 9 -4

Sorted List
-4 1 4 9 10 23 50
```

### Time Complexity:

In all cases (worst, average and best), merge sort always divides the array until all sub-arrays contains single element and takes linear time to merge those sub-arrays. Dividing process has time complexity Θ(logN) and merging process has time complexity Θ(N). Therefore, in all cases, the time complexity of merge sort is Θ(NlogN).

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