Step-By-Step Insertion Sort Algorithm With Example

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In this tutorial, you will learn how the insertion sort algorithm works. Also, you will find working examples of insertion sort in Python. Before we get started, if you want to about the bubble sort algorithm, please go through the article.

Introduction to Insertion Sort Algorithm

The insertion sort algorithm works similarly as we sort cards in our hand in a card game. We assume that the first card is already sorted then, we select an unsorted card. If the unsorted card is greater than the card in hand, it is placed on the right otherwise, to the left. In the same way, other unsorted cards are taken and put in the right place. A similar approach is used by insertion sort. Insertion sort is a sorting algorithm that places an unsorted element at its suitable place in each iteration.

How does the Insertion Sort Algorithm work?

Suppose we need to sort the following array.

The first element in the array is assumed to be sorted. Take the second element and store it separately in key. Compare the key with the first element. If the first element is greater than, then the key is placed in front of the first element. If the first element is greater than the key, then the key is placed in front of the first element.
Now, the first two elements are sorted. Take the third element and compare it with the elements on the left of it. Placed it just behind the element smaller than it. If there is no element smaller than it, then place it at the beginning of the array.
Similarly, place every unsorted element at its correct position.

a) Place 12 behind 85

b) Place 59 behind 85 and the array is sorted

Insertion Sort Algorithm


insertionSort(array)
  mark first element as sorted
  for each unsorted element X
    'extract' the element X
    for j <- lastSortedIndex down to 0
      if current element j > X
        move sorted element to the right by 1
    break loop and insert X here
end insertionSort

insertionSort(array)

mark first element as sorted

for each unsorted element X

'extract' the element X

for j <- lastSortedIndex down to 0

if current element j > X

move sorted element to the right by 1

break loop and insert X here

end insertionSort

Python Examples


# Insertion sort in Python


def insertionSort(array):

    for step in range(1, len(array)):
        key = array[step]
        j = step - 1
        
        # Compare key with each element on the left of it until an element smaller than it is found
        # For descending order, change key<array[j] to key>array[j].        
        while j >= 0 and key < array[j]:
            array[j + 1] = array[j]
            j = j - 1
        
        # Place key at after the element just smaller than it.
        array[j + 1] = key


data = [9, 5, 1, 4, 3]
insertionSort(data)
print('Sorted Array in Ascending Order:')
print(data)

# Insertion sort in Python

def insertionSort(array):

for step in range(1, len(array)):

key = array[step]

j = step - 1

# Compare key with each element on the left of it until an element smaller than it is found

# For descending order, change key<array[j] to key>array[j].

while j >= 0 and key < array[j]:

array[j + 1] = array[j]

j = j - 1

# Place key at after the element just smaller than it.

array[j + 1] = key

data = [9, 5, 1, 4, 3]

insertionSort(data)

print('Sorted Array in Ascending Order:')

print(data)

Complexity

Time Complexities

Worst Case Complexity: O(n²)
Suppose, an array is in ascending order, and you want to sort it in descending order. In this case, the worst-case complexity occurs. Each element has to be compared with each of the other elements so, for every nth element, (n-1) a number of comparisons are made. Thus, the total number of comparisons = n*(n-1) ~ n²
Best Case Complexity: O(n)
When the array is already sorted, the outer loop runs for n number of times whereas the inner loop does not run at all. So, there is only n number of comparisons. Thus, complexity is linear.
Average Case Complexity: O(n²)
It occurs when the elements of an array are in jumbled order (neither ascending nor descending).