Insertion Sort by Swapping Elements

Insertion Sort is suitable for arrays of small size. It also achieves best-case complexity of O(n) if the arrays are already sorted. We have discussed at both Iterative Insertion Sort and Recursive Insertion Sort. In this article slightly different implementations for both iterative and recursive versions are discussed.

Iterative Insertion Sort

Let us look at the algorithm for the iterative insertion sort

```function insertionSort(V)
i, j, k
for i from  1..length(V)
k = V[i]
j = i-1
while j > 0 and  k < V[j]
V[j+1] = V[j]
j -= 1
V[j] = k
return V
```

Inside the while loop, we shift all values larger than k by one position and then insert k into the first position where k is larger than the array value. The same effect is obtained if we swap consecutive array elements. By repeated swapping k will travel to its correct position.

Let’s take an example to illustrate this

```Insert 3 in A = {1, 2, 4, 5, 6}
Put 3 at the end of list.
A = {1, 2, 4, 5, 6, 3}
3 < 6, swap 3 and 6
A = {1, 2, 4, 5, 3, 6}
3 < 5 swap 3 and 5
A = {1, 2, 4, 3, 5, 6}
3 < 4 swap 3 and 4
A = {1, 2, 3, 4, 5, 6}
3 > 2 so stop

By repeatedly swapping 3 travels to its proper position in the list
```

Therefore the above algorithm can be modified as

```function insertionSort(V)
for i in 1...length(V)
j = i
while ( j > 0 and V[j] < V[j-1])
Swap V[j] and V[j-1]
j -= 1
return V
```

The CPP code for this algorithm is given below

C++

 `// Iterative CPP program to sort  ` `// an array by swapping elements ` `#include ` `#include ` `using` `namespace` `std; ` `using` `Vector = vector<``int``>; ` ` `  `// Utility function to print a Vector ` `void` `printVector(``const` `Vector& V) ` `{ ` `    ``for` `(``auto` `e : V) { ` `        ``cout << e << ``" "``; ` `    ``} ` `    ``cout << endl; ` `} ` ` `  `// Function performs insertion sort on ` `// vector V ` `void` `insertionSort(Vector& V) ` `{ ` `    ``int` `N = V.size(); ` `    ``int` `i, j, key; ` ` `  `    ``for` `(i = 1; i < N; i++) { ` `        ``j = i; ` ` `  `        ``// Insert V[i] into list 0..i-1 ` `        ``while` `(j > 0 and V[j] < V[j - 1]) { ` ` `  `            ``// Swap V[j] and V[j-1] ` `            ``swap(V[j], V[j - 1]); ` ` `  `            ``// Decrement j by 1 ` `            ``j -= 1; ` `        ``} ` `    ``} ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``Vector A = { 9, 8, 7, 5, 2, 1, 2, 3 }; ` ` `  `    ``cout << ``"Array: "` `<< endl; ` `    ``printVector(A); ` ` `  `    ``cout << ``"After Sorting :"` `<< endl; ` `    ``insertionSort(A); ` `    ``printVector(A); ` ` `  `    ``return` `0; ` `} `

Java

 `// Iterative Java program to sort  ` `// an array by swapping elements ` `import` `java.io.*; ` `import` `java.util.*; ` ` `  `class` `GFG  ` `{ ` `    ``// Utility function to print a Vector ` `    ``static` `void` `printVector( Vector V) ` `    ``{ ` `    ``for` `(``int` `i = ``0``; i < V.size(); i++) { ` `            ``System.out.print(V.get(i)+``" "``); ` `                     `  `            ``} ` `            ``System.out.println(); ` `    ``} ` `     `  `    ``// Function performs insertion sort on ` `    ``// vector V ` `    ``static` `void` `insertionSort(Vector V) ` `    ``{ ` `        ``int` `N = V.size(); ` `        ``int` `i, j, key; ` `     `  `        ``for` `(i = ``1``; i < N; i++) { ` `            ``j = i; ` `     `  `            ``// Insert V[i] into list 0..i-1 ` `            ``while` `(j > ``0` `&& V.get(j) < V.get(j - ``1``)) { ` `     `  `                ``// Swap V[j] and V[j-1] ` `                ``int` `temp= V.get(j); ` `                ``V.set(j, V.get(j - ``1``)); ` `                ``V.set(j - ``1``, temp); ` `     `  `                ``// Decrement j by 1 ` `                ``j -= ``1``; ` `            ``} ` `        ``} ` `    ``} ` `     `  `    ``public` `static` `void` `main (String[] args)  ` `    ``{ ` `        ``Vector A = ``new` `Vector (); ` `        ``A.add(``0``, ``9``); ` `        ``A.add(``1``, ``8``); ` `        ``A.add(``2``, ``7``); ` `        ``A.add(``3``, ``5``); ` `        ``A.add(``4``, ``2``); ` `        ``A.add(``5``, ``1``); ` `        ``A.add(``6``, ``2``); ` `        ``A.add(``7``, ``3``); ` `        ``System.out.print(``"Array: "``); ` `        ``printVector(A); ` `        ``System.out.print(``"After Sorting :"``); ` `        ``insertionSort(A); ` `        ``printVector(A); ` `    ``} ` `} ` ` `  `//This code is contributed by Gitanjali. `

Python3

 `# Iterative python program to sort  ` `# an array by swapping elements ` `import` `math ` ` `  `# Utility function to print a Vector ` `def` `printVector( V): ` ` `  `    ``for` `i ``in` `V: ` `        ``print``(i ,end``=` `" "``) ` `    ``print` `(``" "``) ` ` `  `def` `insertionSort( V): ` ` `  `    ``N ``=` `len``(V) ` `     `  `    ``for` `i ``in` `range``(``1``,N): ` `    ``j ``=` `i ` ` `  `        ``# Insert V[i] into list 0..i-1 ` `    ``while` `(j > ``0` `and` `V[j] < V[j ``-` `1``]) : ` ` `  `        ``# Swap V[j] and V[j-1] ` `        ``temp ``=` `V[j]; ` `        ``V[j] ``=` `V[j ``-` `1``]; ` `        ``V[j``-``1``] ``=` `temp; ` ` `  `        ``# Decrement j ` `        ``j ``-``=` `1` `     `  ` `  `# Driver method ` `A ``=` `[ ``9``, ``8``, ``7``, ``5``, ``2``, ``1``, ``2``, ``3` `] ` `n ``=` `len``(A) ` `print``(``"Array"``) ` `printVector(A) ` `print``( ``"After Sorting :"``) ` `insertionSort(A) ` `printVector(A) ` ` `  `# This code is contributed by Gitanjali. `

Output:

```Array:
9 8 7 5 2 1 2 3
After Sorting :
1 2 2 3 5 7 8 9
```

Recursive Insertion Sort

Consider an Array A of size N

• First recursively sort the sublist of A which is of size N-1
• Insert the last element of A into the sorted sublist.

To perform the insertion step use repeated swapping as discussed above.

Algorithm

```function insertionSortRecursive(A, N)
if N >= 1
insertionSortRecursive(A, N-1)
j = N-1
while j > 0 and A[j] < A[j-1]
Swap A[j] and A[j-1]
j = j-1
[end of while]
[end of if]
```

The CPP implementation is given below

CPP

 `// Recursive CPP program to sort an array ` `// by swapping elements ` `#include ` `#include ` ` `  `using` `namespace` `std; ` `using` `Vector = vector<``int``>; ` ` `  `// Utility function to print a Vector ` `void` `printVector(``const` `Vector& V) ` `{ ` `    ``for` `(``auto` `e : V) { ` `        ``cout << e << ``" "``; ` `    ``} ` `    ``cout << endl; ` `} ` ` `  `// Function to perform Insertion Sort recursively ` `void` `insertionSortRecursive(Vector& V, ``int` `N) ` `{ ` `    ``if` `(N <= 1) ` `        ``return``; ` ` `  `    ``// General Case ` `    ``// Sort V till second last element and ` `    ``// then insert last element into V ` `    ``insertionSortRecursive(V, N - 1); ` ` `  `    ``// Insertion step ` `    ``int` `j = N - 1; ` `    ``while` `(j > 0 and V[j] < V[j - 1]) { ` ` `  `        ``// Swap V[j] and V[j-1] ` `        ``swap(V[j], V[j - 1]); ` ` `  `        ``// Decrement j ` `        ``j -= 1; ` `    ``} ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` ` `  `    ``// Declare a vector of size 10 ` `    ``Vector A = { 9, 8, 7, 5, 2, 1, 2, 3 }; ` ` `  `    ``cout << ``"Array: "` `<< endl; ` `    ``printVector(A); ` ` `  `    ``cout << ``"After Sorting :"` `<< endl; ` `    ``insertionSortRecursive(A, A.size()); ` `    ``printVector(A); ` `    ``return` `0; ` `} `

Java

 `// Recursive Java program to sort  ` `// an array by swapping elements ` `import` `java.io.*; ` `import` `java.util.*; ` ` `  `class` `GFG  ` `{ ` `    ``// Utility function to print a Vector ` `    ``static` `void` `printVector( Vector V) ` `    ``{ ` `    ``for` `(``int` `i = ``0``; i < V.size(); i++) { ` `            ``System.out.print(V.get(i) + ``" "``); ` `                     `  `            ``} ` `            ``System.out.println(); ` `    ``} ` `     `  `    ``// Function performs insertion sort on ` `    ``// vector V ` `    ``static` `void` `insertionSortRecursive(Vector V,``int` `N) ` `    ``{ ` `        ``if` `(N <= ``1``) ` `            ``return``; ` `     `  `        ``// General Case ` `        ``// Sort V till second last element and ` `        ``// then insert last element into V ` `        ``insertionSortRecursive(V, N - ``1``); ` `     `  `     `  `        ``// Insertion step ` `        ``int` `j = N - ``1``; ` `         `  `            ``// Insert V[i] into list 0..i-1 ` `            ``while` `(j > ``0` `&& V.get(j) < V.get(j - ``1``)) ` `            ``{ ` `     `  `                ``// Swap V[j] and V[j-1] ` `                ``int` `temp= V.get(j); ` `                ``V.set(j, V.get(j - ``1``)); ` `                ``V.set(j - ``1``, temp); ` `     `  `                ``// Decrement j by 1 ` `                ``j -= ``1``; ` `            ``} ` `         `  `    ``} ` `     `  `    ``// Driver code  ` `    ``public` `static` `void` `main (String[] args)  ` `    ``{ ` `        ``Vector A = ``new` `Vector (); ` `        ``A.add(``0``, ``9``); ` `        ``A.add(``1``, ``8``); ` `        ``A.add(``2``, ``7``); ` `        ``A.add(``3``, ``5``); ` `        ``A.add(``4``, ``2``); ` `        ``A.add(``5``, ``1``); ` `        ``A.add(``6``, ``2``); ` `        ``A.add(``7``, ``3``); ` `        ``System.out.print(``"Array: "``); ` `        ``printVector(A); ` `        ``System.out.print(``"After Sorting :"``); ` `        ``insertionSortRecursive(A,A.size()); ` `        ``printVector(A); ` `    ``} ` `} ` ` `  `// This code is contributed by Gitanjali. `

Python3

 `# Recursive python program  ` `# to sort an array ` `# by swapping elements ` `import` `math  ` ` `  `# Utility function to print ` `# a Vector ` `def` `printVector( V): ` ` `  `    ``for` `i ``in` `V: ` `        ``print``(i, end ``=` `" "``) ` `    ``print` `(``" "``) ` ` `  `# Function to perform Insertion ` `# Sort recursively ` `def` `insertionSortRecursive(V, N): ` ` `  `    ``if` `(N <``=` `1``): ` `        ``return` `0` `  `  `    ``# General Case ` `    ``# Sort V till second ` `    ``# last element and ` `    ``# then insert last element ` `    ``# into V ` `    ``insertionSortRecursive(V, N ``-` `1``) ` `  `  `    ``# Insertion step ` `    ``j ``=` `N ``-` `1` `    ``while` `(j > ``0` `and` `V[j] < V[j ``-` `1``]) : ` `  `  `        ``# Swap V[j] and V[j-1] ` `        ``temp ``=` `V[j]; ` `        ``V[j] ``=` `V[j ``-` `1``]; ` `        ``V[j``-``1``] ``=` `temp; ` `  `  `        ``# Decrement j ` `        ``j ``-``=` `1` `     `  ` `  `# Driver method ` `A ``=` `[ ``9``, ``8``, ``7``, ``5``, ``2``, ``1``, ``2``, ``3` `] ` `n``=``len``(A) ` `print``(``"Array"``) ` `printVector(A) ` `print``( ``"After Sorting :"``) ` ` `  `insertionSortRecursive(A,n) ` `printVector(A) ` ` `  `# This code is contributed ` `# by Gitanjali. `

Output

```Array:
9 8 7 5 2 1 2 3
After Sorting :
1 2 2 3 5 7 8 9
```

Note
The Time Complexity of the algorithm is still O(N^2) in the worst case. Moreover, these versions are potentially slower since repeated swapping requires more operations. However, these versions are discussed because of their implementation simplicity and ease of understanding.

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