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Reviewing Sorting Algorithms: Merge Sort

Let’s sort it out!

In a series of posts, we will be discussing some of the sorting algorithms listed in the below order:

  1. Bubble Sort

  2. Selection Sort

  3. Insertion Sort

  4. Merge Sort

  5. Quick Sort

  6. Heap Sort

As promised, we are back to explore the next in a series of sorting algorithms, the Merge Sort. Till now, we have reviewed three sorting algorithms as listed above, to conclude that the common traits they possess are inefficiency and slowness. Bubble Sort, Selection Sort, and Insertion Sort algorithms have minor differences among themselves with the same quadratic running time; meaning the time complexity of these three algorithms is O(n^2). In this post, we shall determine if Merge Sort is any better than its already discussed peers, and does it fit the bill to be chosen as "The One"? Let’s find out.

Why is it called Merge Sort?

Merge Sort is a divide and conquer algorithm that was invented in 1945 by John von Neumann, who was a founding figure in computing. The Merge Sort algorithm sorts an input collection by breaking it into two halves. It then sorts those two halves and merges them together as one sorted array. Most of the merge sort implementations use divide and conquer, a common algorithm design paradigm based on recursion. The idea behind the merge sort is, it’s easier to sort sublists or smaller sets and combine them rather than sorting the long list of an array’s items. In our previous post for Insertion Sort, we have clearly identified the common issue in all the discussed sorting techniques as sorting a long list of values that had slower and inefficient runtimes, with a quadratic time complexity O(n^2).

Let’s first understand the divide and conquer algorithm.

A divide-and-conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same or related type until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.” [Ref. 1]

A divide-and-conquer algorithm divides the problem into the simplest form. The smaller problems are the ones most easier to solve. The solution is then applied to the bigger chunks of the complex problem. Hence, the larger problem is conquered by using the same solution recursively.

The three parts of the divide and conquer algorithm are as below:

Divide: The problem is divided into the smallest possible number of the same problem; meaning the problem is divided into a subproblem, which indeed is divided into another subproblem and so forth until the smallest set is achieved.

Conquer: The smallest subproblem is conquered first by finding the solution as a base case. Once the solution is achieved, the same technique can be used to tackle bigger subproblems recursively.

Combine: The solution to the small problem is combined and build-up to solve the bigger problem.

The below pictures demonstrate the divide-and-conquer algorithm, wherein a problem is divided into two subproblems and the algorithm makes multiple recursive calls.

Further simplification of the recursive steps is demonstrated below:

At this point, we have better clarity on how the Divide-and-conquer algorithm works in theory, it’s time to illustrate this design paradigm implementation in the Merge Sort. The idea is to discuss the Merge Sort in a very simple way so that it sticks well in our memory, very much like solving small problems to achieve the original big problem solution. We have now cemented in theory how the Merge Sort will split the collections in a number of sublists to find the basic solution. Let’s break it down with our post series sample array [10,8,4,6,1,3,2,5]. Additional element number [3,2,5] is added to the array to make it an even array for easier demonstration purposes.

Divide: The array is divided into two halves initially and each half is again divided into halves until the smallest subset is attained. Below is the illustration for the divide step and we achieve a sorted smallest sub list at the end. You may ask, how is it sorted? Well, the final eight sublists are considered sorted as there is only one value and nothing else to compare. In short, the smallest sublist is always a sorted list by itself and considered as a base case. Now that we have solved the smallest problem, the next step would be to solve the next biggest subproblem. How do we do that? This is the time we introduce the “conquer” part of the algorithm.

Conquer: The sorted sublists are merged together maintaining the sorted list. Two sorted lists are required to merge together and create one single sorted list. The sorted lists are merged together as a part of the “combine” step recursively building towards the final single sorted list. The base case is the first single element sorted list. If you observe closely, the sorting has taken place recursively for each of the below sublists.