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"Insertion sort algorithm" is one of the simplest and most commonly used sorting algorithms when it comes to the ordering of a deck of cards in everyday life. You insert each element into its proper place in the sorted array using this algorithm. Despite its simplicity, Insertion sort is quite inefficient compared to its allies such as quicksort, and merge sort to name a few.

By the end of this tutorial, you will have a better understanding of the fundamental technicalities of the Insertion sort with all the necessary details along with practical implementations.

What Is the Insertion Sort Algorithm?

Insertion sort algorithm is a basic sorting algorithm that sequentially sorts each item in the final sorted array or list. It is significantly low on efficiency while working on comparatively larger data sets. While other algorithms such as quicksort, heapsort, or merge sort have time and again proven to be far more effective and efficient.

Now, have a look at the working of the Insertion sort algorithm to get a better understanding of the insertion sort.

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How Does Insertion Sort Work?

To sort an array in ascending order, you will follow the steps below:

1. Iterate through the array from arr[1] to arr[n].
2. Compare the current element (key) to one that came before it.
3. If the data at the current index is less than the data at the previous index, you will compare it to the element before it.
4. You will shift bigger elements to the next index to make space for swapped elements, and then you will iterate the same steps again to sort the complete array.

And this is how the Insertion sort algorithm works. Now implement this algorithm through a simple C++ code.

How to Implement the Insertion Sort Algorithm?

You will be provided with a one-dimensional array of elements {6, 5, 4, 2, 3}. You have to write a code to sort this array using the insertion sort algorithm. The final array should come out to be as {2, 3, 4, 5, 6}.

Code:

//A C++ program to sort an array using insertion sort

#include <bits/stdc++.h>

using namespace std;

//A Function to sort an array using insertion sort

void insertion_sort(int array[], int size)

{

int i, key, k;

for (i = 1; i < size; i++)

{

key = array[i];

k = i - 1;

// Move Ar[0..i-1] elements, 

//which are larger than the key, 

//to one place over their present location

while (k >= 0 && array[k] > key)

{

array[k + 1] = array[k];

k = k - 1;

}

array[k + 1] = key;

}

}

// A function to print the array of size n

void print_array(int array[], int size)

{

int i;

for (i = 0; i < size; i++)

cout << array[i] << " ";

cout << endl;

}

/* Driver code */

int main()

{

int arr[] = { 6, 5, 4, 2, 3 };

int size = sizeof(arr) / sizeof(arr[0]);

cout<<"Array before sorting:\n";

print_array(arr, size);

insertion_sort(arr, size);

cout<<"\nArray after sorting:\n";

print_array(arr, size);

return 0;

}

You have now explored the working of insertion sort with a code. Now, look at some of the advantages of the Insertion sort.

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What Are the Advantages of the Insertion Sort?

You will now look at a few major benefits of using insertion sort and a few scenarios where insertion sort is proven to be delivering the best performance.

• It, like other quadratic sorting algorithms, is efficient for small data sets.
• It just necessitates a constant amount of O(1) extra memory space.
• It works well with data sets that have been sorted in a significant way.
• It does not affect the relative order of elements with the same key.

What Are the Disadvantages of the Insertion Sort?

Despite its simplicity and effectiveness over smaller data sets, Insertion sort does a few downfalls. This tutorial will now address a few major drawbacks which you should consider before you implement insertion sort in real-time

• Insertion sort is inefficient against more extensive data sets
• The insertion sort exhibits the worst-case time complexity of O(n2)
• It does not perform well than other, more advanced sorting algorithms

With this, you have come to an end of this tutorial. You will now look at what could be your next steps to master sorting algorithms.

Next Steps

Your next stop in mastering data structures should be the selection Sort Algorithm. The selection sort algorithm divides the list into two parts, with the sorted half on the left and the unsorted half on the right, using in-place comparisons.

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What are Sorting Algorithms?

Sorting algorithms are ways to organize an array of items from smallest to largest. These algorithms can be used to organize messy data and make it easier to use. Furthermore, having an understanding of these algorithms and how they work is fundamental for a strong understanding of Computer Science which is becoming more and more critical in a world of premade packages. This blog focuses on the speed, uses, advantages, and disadvantages of specific Sorting Algorithms.

Although there is a wide variety of sorting algorithms, this blog explains Straight Insertion, Shell Sort, Bubble Sort, Quick Sort, Selection Sort, and Heap Sort. The first two algorithms (Straight Insertion and Shell Sort) sort arrays with insertion, which is when elements get inserted into the right place. The next 2 (Bubble Sort and Quick Sort) sort arrays with exchanging which is when elements move around the array. The last one is heap sort which sorts through selection where the right elements are selected as the algorithm runs down the array.

Big-O Notation

Before this blog goes any further, it is essential to explain the methods that professionals use to analyze and assess algorithm complexity and performance. The current standard is called “Big O notation” named according to its notation which is an “O” followed by a function such as “O(n).”

Formal definition

Suppose f(x) and g(x) are two functions defined on some subset of the real numbers. We write f(x) = O(g(x)) (or f(x) = O(g(x)) for x -> ∞ to be more precise) if and only if there exist constants N and C such that |f(x)| <= C |g(x)| for all x>N. Intuitively, this means that f does not grow faster than g.

Big O is used to denote either the time complexity of an algorithm or how much space it takes up. This blog focuses mainly on the time complexity part of this notation. The way people can calculate this is by identifying the worst case for the targeted algorithm and formulating a function of its performance given an n amount of elements. For example, if there were an algorithm that searched for the number 2 in an array, then the worst case would be if the 2 was at the very end of the array. Therefore, the Big O notation would be O(n) since it would have to run through the entire n-element array before finding the number 2.

To help you, find below a table with algorithms and its complexity.

Straight Insertion Sort

Straight insertion sort is one of the most basic sorting algorithms that essentially inserts an element into the right position of an already sorted list. It is usually added at the end of a new array and moves down until it finds an element smaller thank itself (the desired position). The process repeats for all the elements in the unsorted array. Consider the array {3,1,2,5,4}, we begin at 3, and since there are no other elements in the sorted array, the sorted array becomes just {3}. Afterward, we insert 1 which is smaller than 3, so it would move in front of 3 making the array {1,3}. This same process is repeated down the line until we get the array {1,2,3,4,5}.

The advantages of this process are that it is straightforward and easy to implement. Also, it is relatively quick when there are small amounts of elements to sort. It can also turn into binary insertion which is when you compare over longer distances and narrow it down to the right spot instead of comparing against every single element before the right place. However, a straight insertion sort is usually slow whenever the list becomes large.

Main Characteristics:

• Insertion sort family
• Straightforward and simple
• Worst case = O(n^2)

Python implementation:

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Sun Mar 10 17:13:56 2019

@note: Insertion sort algorithm
@source: http://interactivepython.org/courselib/static/pythonds/SortSearch/TheInsertionSort.html

"""

def insertionSort(alist):
   for index in range(1,len(alist)):

     currentvalue = alist[index]
     position = index

     while position>0 and alist[position-1]>currentvalue:
         alist[position]=alist[position-1]
         position = position-1

     alist[position]=currentvalue

Shell Sort

Shell sort is an insertion sort that first partially sorts its data and then finishes the sort by running an insertion sort algorithm on the entire array. It generally starts by choosing small subsets of the array and sorting those arrays. Afterward, it repeats the same process with larger subsets until it reaches a point where the subset is the array, and the entire thing becomes sorted. The advantage of doing this is that having the array almost entirely sorted helps the final insertion sort achieve or be close to its most efficient scenario.

Furthermore, increasing the size of the subsets is achieved through a decreasing increment term. The increment term essentially chooses every kth element to put into the subset. It starts large, leading to smaller (more spread out) groups, and it becomes smaller until it becomes 1 (all of the array).

The main advantage of this sorting algorithm is that it is more efficient than a regular insertion sort. Also, there is a variety of different algorithms that seek to optimize shell sort by changing the way the increment decreases since the only restriction is that the last term in the sequence of increments is 1. The most popular is usually Knuth’s method which uses the formula h=((3^k)-1)/2 giving us a sequence of intervals of 1 (k=1),4 (k=2),13 (k=3), and so on. On the other hand, shell sort is not as efficient as other sorting algorithms such as quicksort and merge sort.

Main Characteristics:

• Sorting by insertion
• Can optimize algorithm by changing increments
• Using Knuth’s method, the worst case is O(n^(3/2))

Python implementation:

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Sun Mar 10 17:13:56 2019

@note: Insertion sort - Shell Method algorithm
@source: https://www.w3resource.com/python-exercises/data-structures-and-algorithms/python-search-and-sorting-exercise-7.php

"""

def shellSort(alist):
    sublistcount = len(alist)//2
    while sublistcount > 0:
      for start_position in range(sublistcount):
        gap_InsertionSort(alist, start_position, sublistcount)

      sublistcount = sublistcount // 2

def gap_InsertionSort(nlist,start,gap):
    for i in range(start+gap,len(nlist),gap):

        current_value = nlist[i]
        position = i

        while position>=gap and nlist[position-gap]>current_value:
            nlist[position]=nlist[position-gap]
            position = position-gap

        nlist[position]=current_value

Bubble Sort

Bubble sort compares adjacent elements of an array and organizes those elements. Its name comes from the fact that large numbers tend to “float” (bubble) to the top. It loops through an array and sees if the number at one position is greater than the number in the following position which would result in the number moving up. This cycle repeats until the algorithm has gone through the array without having to change the order. This method is advantageous because it is simple and works very well for mostly sorted lists. As a result, programmers can quickly and easily implement this sorting algorithm. However, the tradeoff is that this is one of the slower sorting algorithms.

Main Characteristics: Exchange sorting Easy to implement Worst Case = O(n^2)

Python implementation:

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Sun Mar 10 18:05:51 2019

@note: Exchanging sort - Bubble Sort algorithm
@source: http://interactivepython.org/runestone/static/pythonds/SortSearch/TheBubbleSort.html

"""

def bubbleSort(alist):
    for passnum in range(len(alist)-1,0,-1):
        for i in range(passnum):
            if alist[i]>alist[i+1]:
                temp = alist[i]
                alist[i] = alist[i+1]
                alist[i+1] = temp

Quicksort

Quicksort is one of the most efficient sorting algorithms, and this makes of it one of the most used as well. The first thing to do is to select a pivot number, this number will separate the data, on its left are the numbers smaller than it and the greater numbers on the right. With this, we got the whole sequence partitioned. After the data is partitioned, we can assure that the partitions are oriented, we know that we have bigger values on the right and smaller values on the left. The quicksort uses this divide and conquer algorithm with recursion. So, now that we have the data divided we use recursion to call the same method and pass the left half of the data, and after the right half to keep separating and ordinating the data. At the end of the execution, we will have the data all sorted.

Main characteristics:

• From the family of Exchange Sort Algorithms
• Divide and conquer paradigm
• Worst case complexity O(n²)

Python implementation:

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Sun Mar 10 18:09:35 2019

@note: Exchanging sort - Bubble Sort algorithm
@source: http://interactivepython.org/courselib/static/pythonds/SortSearch/TheQuickSort.html

"""

def quickSort(alist):
   quickSortHelper(alist,0,len(alist)-1)

def quickSortHelper(alist,first,last):
   if first<last:

       splitpoint = partition(alist,first,last)

       quickSortHelper(alist,first,splitpoint-1)
       quickSortHelper(alist,splitpoint+1,last)


def partition(alist,first,last):
   pivotvalue = alist[first]

   leftmark = first+1
   rightmark = last

   done = False
   while not done:

       while leftmark <= rightmark and alist[leftmark] <= pivotvalue:
           leftmark = leftmark + 1

       while alist[rightmark] >= pivotvalue and rightmark >= leftmark:
           rightmark = rightmark -1

       if rightmark < leftmark:
           done = True
       else:
           temp = alist[leftmark]
           alist[leftmark] = alist[rightmark]
           alist[rightmark] = temp

   temp = alist[first]
   alist[first] = alist[rightmark]
   alist[rightmark] = temp


   return rightmark

Heapsort

Heapsort is a sorting algorithm based in the structure of a heap. The heap is a specialized data structure found in a tree or a vector. In the first stage of the algorithm, a tree is created with the values to be sorted, starting from the left, we create the root node, with the first value. Now we create a left child node and insert the next value, at this moment we evaluate if the value set to the child node is bigger than the value at the root node, if yes, we change the values. We do this to all the tree. The initial idea is that the parent nodes always have bigger values than the child nodes.

At the end of the first step, we create a vector starting with the root value and walking from left to right filling the vector.

Now we start to compare parent and child nodes values looking for the biggest value between them, and when we find it, we change places reordering the values. In the first step, we compare the root node with the last leaf in the tree. If the root node is bigger, then we change the values and continue to repeat the process until the last leaf is the larger value. When there are no more values to rearrange, we add the last leaf to the vector and restart the process. We can see this in the image below.

The main characteristics of the algorithm are:

• From the family of sorting by selection
• Comparisons in the worst case = O(n log n)
• Not stable

Python implementation:

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Sun Mar 10 18:18:25 2019

@source: https://www.geeksforgeeks.org/heap-sort/

"""
# Python program for implementation of heap Sort 

# To heapify subtree rooted at index i. 
# n is size of heap 
def heapify(arr, n, i): 
	largest = i # Initialize largest as root 
	l = 2 * i + 1	 # left = 2*i + 1 
	r = 2 * i + 2	 # right = 2*i + 2 

	# See if left child of root exists and is 
	# greater than root 
	if l < n and arr[i] < arr[l]: 
		largest = l 

	# See if right child of root exists and is 
	# greater than root 
	if r < n and arr[largest] < arr[r]: 
		largest = r 

	# Change root, if needed 
	if largest != i: 
		arr[i],arr[largest] = arr[largest],arr[i] # swap 

		# Heapify the root. 
		heapify(arr, n, largest) 

# The main function to sort an array of given size 
def heapSort(arr): 
	n = len(arr) 

	# Build a maxheap. 
	for i in range(n, -1, -1): 
		heapify(arr, n, i) 

	# One by one extract elements 
	for i in range(n-1, 0, -1): 
		arr[i], arr[0] = arr[0], arr[i] # swap 
		heapify(arr, i, 0) 

heapSort(arr) 
 

Benchmark: How fast they are?

After development of the algorithms it is good for us to test how fast they can be. In this part we developed a simple program using the code above to generate a basic benchmark, just to see how much time they can use to sort a list of integers. Important observations about the code:

• Python default recursion call limit is 1,000, in this test we are using big numbers, so we needed to improve that number to run the benchmark without errors. The limit was set to 10,000.
• This code just measure the running time of each algorithm.
• It was made 20 tests with different size of lists ranging from 2500 to 50000.
• The numbers were generate randonly ranging from 1 to 10000. The results are the following: Shell sort and Heap Sort algorithms performed well despite the length of the lists, in the other side we found that Insertion sort and Bubble sort algorithms were far the worse, increasing largely the computing time. See the results in the chart below.

Conclusion

In this post, we showed 5 of the most common sorting algorithms used today. Before using any of them is extremely important to know how fast it runs and how much space is going to use. So it’s the tradeoff between complexity, speed, and volume. Another critical characteristic of the sorting algorithms that are important to know is its stability. The stability means that the algorithm keeps the order of elements with equal key values. The best algorithm changes for each different set of data and as a result, understanding our data plays a significant role in the process of choosing the right algorithm.

As we can see, understanding our data plays a very important role in the process of choosing the right algorithm.

If this post got your attention, take a look at video below, it will give you a concise explanation about 15 sorting algorithms.

References:

Big O Notation - https://en.wikipedia.org/wiki/Big_O_notation

Knuth, Donald Ervin, 1938 - The art of computer programming / Donald Ervin Knuth. xiv,782 p. 24 cm.

15 Sorting Algorithms in 6 minutes - https://www.youtube.com/watch?v=kPRA0W1kECg

Insertion sort - https://pt.wikipedia.org/wiki/Insertion_sort

Insertion sort - http://interactivepython.org/courselib/static/pythonds/SortSearch/TheInsertionSort.html

Shell sort - https://en.wikipedia.org/wiki/Shellsort

Shell sort - http://www.bebetterdeveloper.com/algorithms/sorting/sorting-shell-sort.html

Shell sort - https://www.researchgate.net/publication/234791427_ENHANCED_SHELL_SORT_ALGORITHM

Shell sort - https://www.w3resource.com/python-exercises/data-structures-and-algorithms/python-search-and-sorting-exercise-7.php

Bubble sort - https://en.wikibooks.org/wiki/A-level_Computing/AQA/Paper_1/Fundamentals_of_algorithms/Sorting_algorithms#Bubble_Sort

Quicksort - https://commons.wikimedia.org/wiki/File:Quicksort.gif

Quicksort - http://interactivepython.org/courselib/static/pythonds/SortSearch/TheQuickSort.html

Heapsort Algorithm - https://en.wikipedia.org/wiki/Heapsort

Heapsort Algorithm - https://www.geeksforgeeks.org/heap-sort/

Bubble sort - http://interactivepython.org/runestone/static/pythonds/SortSearch/TheBubbleSort.html

The Advantages & Disadvantages of Sorting Algorithms - Joe Andy - https://sciencing.com/the-advantages-disadvantages-of-sorting-algorithms-12749529.html

Sedgewick, R., & Wayne, K. (2011). Algorithms, 4th Edition. (p. I–XII, 1-955). Addison-Wesley. - https://algs4.cs.princeton.edu/20sorting/

Sorting algorithms - https://brilliant.org/wiki/sorting-algorithms/

A tour of the top 5 sorting algorithms with Python code - https://medium.com/@george.seif94/a-tour-of-the-top-5-sorting-algorithms-with-python-code-43ea9aa02889

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