# Merge Sort

April 30, 2023

Merge Sort is a popular divide-and-conquer algorithm used for sorting data. It divides the input list into smaller sub-lists, sorts them and then merges them back into the original list in sorted order. Merge Sort is a stable sorting algorithm, meaning it maintains the relative order of equal elements.

## Purpose and Usage

Merge Sort is primarily used for sorting large lists, since it has a worst-case and average-case time complexity of O(n log n), which is faster than many other sorting algorithms. It is commonly used in computer science and is the algorithm of choice for many programming languages’ built-in sorting functions.

## Brief History and Development

The Merge Sort algorithm was first proposed in 1945 by John von Neumann, who was a renowned mathematician and computer scientist at the time. However, it was not widely used until the 1960s and 1970s, when it was popularized by the invention of the first general-purpose computers.

## Key Concepts and Principles

Merge Sort is a divide-and-conquer algorithm, meaning it breaks the input list into smaller sub-lists until each sub-list contains only one element. It then merges these sub-lists back together in sorted order, until the entire list is sorted. The key concepts and principles of the algorithm include:

### Divide and conquer

The divide-and-conquer strategy involves breaking a problem into smaller sub-problems, solving them independently, and then combining the solutions to form the solution to the original problem. In Merge Sort, this means breaking the input list into smaller sub-lists, sorting them, and then merging them back together.

### Recursive structure

Merge Sort is a recursive algorithm, meaning it calls itself recursively on the smaller sub-lists until the base case is reached. In Merge Sort, the base case is when the sub-list contains only one element, which is already sorted.

### Merging of sorted sub-lists

The merging of sorted sub-lists is the final step of the algorithm. It involves comparing the first elements of each sub-list, choosing the smaller one, and appending it to the result list. This process is repeated until all elements from both sub-lists have been appended to the result list.

## Pseudocode and Implementation Details

The Merge Sort algorithm can be implemented in many programming languages. Below is a pseudocode implementation of the algorithm:

``````function merge_sort(list)
if length of list <= 1
return list
mid = length of list / 2
left = first half of list
right = second half of list
left_sorted = merge_sort(left)
right_sorted = merge_sort(right)
return merge(left_sorted, right_sorted)

function merge(left, right)
result = empty list
while length of left > 0 and length of right > 0
if first element of left <= first element of right
append first element of left to result
remove first element from left
else
append first element of right to result
remove first element from right
append all elements from left to result
append all elements from right to result
return result``````

This implementation of the algorithm first checks if the length of the input list is less than or equal to one. If so, it returns the list as it is already sorted. Otherwise, it divides the input list into two sub-lists and recursively calls the `merge_sort` function on each sub-list. Once the sub-lists are sorted, the function calls the `merge` function to merge the sorted sub-lists back together in sorted order.

The `merge` function takes two sorted sub-lists as input and compares their first elements. The smaller of the two elements is appended to the result list, and the first element of the sub-list it came from is removed. This process continues until all elements from both sub-lists have been appended to the result list.

## Examples and Use Cases

Merge Sort can be used to sort any kind of list, including numbers, strings, and objects. Below are some examples of how Merge Sort can be used:

### Sorting numbers

``````list = [5, 1, 8, 3, 9, 4, 6, 2, 7]
sorted_list = merge_sort(list)
print(sorted_list) # [1, 2, 3, 4, 5, 6, 7, 8, 9]``````

### Sorting strings

``````list = ["apple", "banana", "cherry", "date", "elderberry", "fig"]
sorted_list = merge_sort(list)
print(sorted_list) # ["apple", "banana", "cherry", "date", "elderberry", "fig"]``````

### Sorting objects

``````class Person:
def __init__(self, name, age):
self.name = name
self.age = age

def __str__(self):
return f"{self.name} ({self.age})"

people = [Person("Alice", 25), Person("Bob", 35), Person("Charlie", 30)]
sorted_people = merge_sort(people)
for person in sorted_people:
print(person) # Alice (25), Bob (35), Charlie (30)``````

• Merge Sort has a worst-case and average-case time complexity of O(n log n), making it faster than other sorting algorithms for large lists.
• Merge Sort is a stable sorting algorithm, meaning it maintains the relative order of equal elements.
• Merge Sort is a divide-and-conquer algorithm, making it easier to parallelize and optimize for multi-core CPUs.

• Merge Sort has a space complexity of O(n), meaning it requires additional memory to store the sorted sub-lists.
• Merge Sort has a slower best-case time complexity of O(n log n), making it less efficient for small lists than other sorting algorithms.
• Merge Sort requires additional code to implement compared to simpler sorting algorithms like Bubble Sort or Selection Sort.

There are several related algorithms and variations of Merge Sort, including:

### In-place merge sort

In-place merge sort is a variation of Merge Sort that sorts the input list in-place, meaning it does not require additional memory to store the sorted sub-lists. This is achieved by merging the sub-lists back into the original list in-place, without using a separate result list.

### Bottom-up merge sort

Bottom-up merge sort is a variation of Merge Sort that iteratively sorts the input list by merging adjacent sub-lists of increasing size. This avoids the recursive structure of Merge Sort, which can be a disadvantage for very large input lists due to the overhead of recursive function calls.