Introduction to Merge Sort: A Fast, Stable Sorting Algorithm

Have you ever struggled to put a shuffled deck of cards back in order? It might seem like an impossible task if there are enough cards mixed up randomly. Lucky for you, merge sort is an ingeniously simple sorting algorithm that can order elements efficiently, even if they start completely out of order!

Read on as I explain exactly what merge sort is and break down step-by-step how it works with easy to follow examples. I‘ll also analyze key performance metrics like speed and memory usage so you know when to use it. Let‘s get started!

What Is Merge Sort?

Merge sort falls into a class of sorting algorithms called "divide and conquer"…

Now that you have the big picture for what merge sort is, let‘s look at exactly how it works.

How Merge Sort Works – Conceptual Overview

On a high level, merge sort works by breaking an array down into…

That probably still seems a bit magical – so let‘s step through a full example to make the divide-and-conquer process completely concrete!

Walkthrough: Sorting An Array with Merge Sort

Imagine we had the following randomly ordered list of numbers to sort:

[29, 72, 98, 13, 87, 54, 68, 42]

Let me show you how merge sort can efficiently order this array from low to high through its elegant divide and conquer approach…

Step 1) Divide Array In Half

We first break down the problem by dividing the starting array in half:

Left: [29, 72, 98, 13] 
Right: [87, 54, 68, 42]

Step 2) Break Down Halves

We divide the halves again, getting:

Left: [29, 72] [98, 13] 
Right: [87, 54] [68, 42]

Step 3) Recurse Until Base Case

We keep dividing the subarrays in half recursively until we reach singleton arrays, which are already "sorted":

Left: [29] [72] [13] [98] 
Right: [87] [54] [68] [42]  

Step 4) Start Merging Back Up

Next we compare and merge pairs while keeping elements ordered:

Left: [29, 72] [13, 98]  
Right: [54, 87] [42, 68]   

Step 5) Repeat Merge Process

We repeat step 4 on the merged sublists, and keep merging sorted runs back together:

Left: [13, 29, 72, 98]
Right: [42, 54, 68, 87] 

Result: [13, 29, 42, 54, 68, 72, 87, 98]

And there we have the final sorted array in only 5 basic steps! Pretty slick right?

Now that you have an idea for how merge sort works, let‘s analyze its performance and efficiency…

Analyzing Merge Sort‘s Speed and Memory Usage

Merge sort owes its popularity to great time performance and stability – it sustains O(n log n) run time even as data grows huge!

Time Complexity – Fast Even For Large Data

You can see how even with 1 million elements, merge sort quickly orders data in 20 million simple comparisons, despite items starting completely out of order.

Its speed comes from each recursive call working on smaller inputs – divide inputs in half til sorting is trivial.

Space Complexity – Managing The Tradeoff

Merge sort does require copying data with its divide strategy, costing additional memory. Each divided subarray has to be handled separately.

So there is a memory vs speed tradeoff…

Luckily with modern systems having plenty of cheap memory, merge sort‘s linear O(n) space costs become acceptable when you consider the speed and stability gained.

Now that you grasp the performance better, let‘s dive deeper into implementation…

Implementing Merge Sort In Python

Let‘s walk through a Python implementation to connect the conceptual dots…

Step 1) Define The Merge Sort Function

def mergeSort(arr):

    # Base case 
    if len(arr) <= 1:
        return

    # Otherwise split input array    
    mid = len(arr) // 2
    left = arr[:mid] 
    right = arr[mid:]

Start by defining the mergeSort function which will orchestrate sorting.

It first checks if the input array is trivially small. If so, return it unchanged since by definition a single element array is "sorted".

Otherwise, it splits the input array in half to create two subarrays – a left and right half.

Step 2) Recurse On The Halves

    # Recurse on left & right subarrays
    mergeSort(left)  
    mergeSort(right)

Next we apply mergeSort recursively on the left and right halves created from the split.

This recursion continues splitting halves down further and further til we reach sorted subarrays of just 1 element each.

Step 3) Merge Halves Back Together

    # Merge sorted halves 
    i, j = 0, 0

    # Final index position tracker 
    k = 0  

    while i < len(left) and j < len(right):
        if left[i] < right[j]:
          arr[k] = left[i]
          i += 1
        else:
          arr[k] = right[j]
          j += 1
        k += 1

    # Append any extras
    while i < len(left):
       arr[k] = left[i]
       i += 1
       k += 1

    while j < len(right):
        arr[k] = right[j]
        j += 1
        k += 1

The last phase handles merging the now sorted subarrays back together into properly ordered full arrays via comparisons.

It does this iteratively via indices i, j, and tracker k while moving up the recursion tree.

Once all levels are merged back together, we‘ve produced the final sorted array!

And there you have it – a clean implementation of merge sort in Python. Feel free to tweak it on some test cases!

FAQs About Merge Sort Algorithms

Q: Is merge sort stable?

Yes! Merge sort is stable since it does item-by-item comparisons…

Q: When should I avoid merge sort?

For nearly sorted data, insertion sort may outperform…

Conclusion & Next Steps

Thanks for following along on this deep dive into merge sort! I explained what merge sort is, how it cleverly divides inputs for fast sorting, demonstrated it visually on a full example, analyzed key performance metrics, and showed a Python implementation…

I hope you now feel confident to utilize merge sort when writing performant programs. If you have any other questions feel free to reach out!