# Python Program to Implement Binary Insertion Sort

This is a Python program to implement binary insertion sort.

Problem Description

The program sorts a list by binary insertion sort.

Problem Solution

1. Create a function binary insertion_sort that takes a list as argument.
2. Inside the function create a loop with a loop variable i that counts from 1 to the length of the list – 1.
3. Set temp equal to the element at index i.
4. Call binary_search with key = temp, start = 0 and end = i.
5. Set pos equal to one plus the return value of the above call.
6. Shift all elements one position right from indexes pos + 1 to i inclusive and set alist[pos] to temp.
7. Create a function binary_search that takes a list, a key and two indexes start and end as arguments.
8. The function performs binary search to find key in the list between indexes start and end – 1 inclusive. If the key is not found, it returns the index of the largest element smaller than key. If key is smaller than the first element, it returns -1.

Program/Source Code

Here is the source code of a Python program to implement binary insertion sort. The program output is shown below.

```def binary_insertion_sort(alist):
for i in range(1, len(alist)):
temp = alist[i]
pos = binary_search(alist, temp, 0, i) + 1

for k in range(i, pos, -1):
alist[k] = alist[k - 1]

alist[pos] = temp

def binary_search(alist, key, start, end):
'''If key is in the list at index p, then return p.
If there are multiple such keys in the list, then return the index of any one.
If key is not in the list and a < key < b where a and b are elements in the list, then return the index of a.
If key is not in the list and key < a where a is the first element in the list, then return -1.
Only elements with indexes start to end - 1 inclusive are considered.
'''
if end - start <= 1:
if key < alist[start]:
return start - 1
else:
return start

mid = (start + end)//2
if alist[mid] < key:
return binary_search(alist, key, mid, end)
elif alist[mid] > key:
return binary_search(alist, key, start, mid)
else:
return mid

alist = input('Enter the list of numbers: ').split()
alist = [int(x) for x in alist]
binary_insertion_sort(alist)
print('Sorted list: ', end='')
print(alist)```
Program Explanation

1. The user is prompted to enter a list of numbers.
2. The list is passed to the binary insertion_sort function.
3. The sorted list is displayed.

Runtime Test Cases
```Case 1:
Enter the list of numbers: 5 2 7 10 3 5 2 1 8 9
Sorted list: [1, 2, 2, 3, 5, 5, 7, 8, 9, 10]

Case 2:
Enter the list of numbers: 7 5 5 4 3 2 1
Sorted list: [1, 2, 3, 4, 5, 5, 7]

Case 3:
Enter the list of numbers: 2
Sorted list: ```

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