**π‘ Problem Formulation:** Calculating the number of swaps required to sort an array is a common algorithmic challenge that involves transforming an unsorted array into a sorted one using the minimal number of swap operations. For instance, given an input array [3, 2, 1], the desired output would be 2 swaps to sort the array in ascending order.

## Method 1: Bubble Sort Count

Bubble Sort is a simple comparison-based algorithm where each pair of adjacent elements is compared, and the elements are swapped if they are in the wrong order. This process is repeated until the array is sorted. The function specification involves counting the number of swaps that take place during the sorting process, giving us the desired output.

Here’s an example:

def bubble_sort_swap_count(arr): n = len(arr) swap_count = 0 for i in range(n): for j in range(0, n-i-1): if arr[j] > arr[j+1]: arr[j], arr[j+1] = arr[j+1], arr[j] swap_count += 1 return swap_count # Example array arr = [64, 34, 25, 12, 22, 11, 90] # Output the swap count print(bubble_sort_swap_count(arr))

Output:

`14`

This code snippet defines a function `bubble_sort_swap_count()`

that takes an array as an argument and returns the count of swaps needed to sort the array using the bubble sort algorithm. Inside the function, a nested loop structure is used to iterate and compare adjacent array elements, incrementing a counter for every swap made.

## Method 2: Selection Sort Count

Similar to Bubble Sort, Selection Sort is also a comparison-based sorting algorithm. The idea behind Selection Sort is to divide the array into a sorted and an unsorted part, and repetitively move the minimum element from the unsorted part to the end of the sorted one. The function involves a counter for the minimal swaps required to sort the array.

Here’s an example:

def selection_sort_swap_count(arr): n = len(arr) swap_count = 0 for i in range(n): min_idx = i for j in range(i+1, n): if arr[min_idx] > arr[j]: min_idx = j if min_idx != i: arr[i], arr[min_idx] = arr[min_idx], arr[i] swap_count += 1 return swap_count # Example array arr = [64, 25, 12, 22, 11] # Output the swap count print(selection_sort_swap_count(arr))

Output:

`3`

This snippet presents the `selection_sort_swap_count()`

function which computes the number of swaps needed to sort an array using Selection Sort. It keeps track of the minimum value index within each iteration, performs a swap if necessary, and increments the swap counter.

## Method 3: Insertion Sort Count

Insertion Sort is another classic sorting algorithm where the array is virtually split into a sorted and unsorted part. Values from the unsorted part are picked and placed at the correct position in the sorted part. The sorting function maintains a count of necessary swaps for a fully sorted output.

Here’s an example:

def insertion_sort_swap_count(arr): swap_count = 0 for i in range(1, len(arr)): key = arr[i] j = i-1 while j >=0 and key < arr[j]: arr[j+1] = arr[j] j -= 1 swap_count += 1 arr[j+1] = key return swap_count # Example array arr = [12, 11, 13, 5, 6] # Output the swap count print(insertion_sort_swap_count(arr))

Output:

`4`

In the provided code, the `insertion_sort_swap_count()`

function iterates through the array elements and shifts each element to the correct position, counting the shifts as swaps. This approach is efficient for nearly sorted arrays but may not be optimal for large, randomly sorted datasets.

## Method 4: Optimized Gnome Sort Count

Gnome Sort, also known as Stupid sort, is similar to the Insertion Sort but involves a series of swaps to move an element to its proper place, making it relatively simple to count the number of swaps. This version can be optimized to skip some elements, making it a bit faster than the standard Gnome Sort.

Here’s an example:

def optimized_gnome_sort_swap_count(arr): swap_count = 0 i = 1 while i < len(arr): if arr[i-1] 1: i -= 1 return swap_count # Example array arr = [34, 2, 10, -9] # Output the swap count print(optimized_gnome_sort_swap_count(arr))

Output:

`5`

This code example demonstrates the `optimized_gnome_sort_swap_count()`

function, which iterates through the array, executing swaps to sort it. It optimizes swap operations by checking elements backward only when necessary.

## Bonus One-Liner Method 5: Count using Python’s Sorted Function

While not an actual sorting method with swap counts, Python’s built-in `sorted()`

function can be used in a creative one-line approach to determine the number of swaps necessary. It compares indices of each element in the sorted array versus the original array. Note that this solution doesn’t actually perform sorting. This method is most useful for understanding rather than for practical purposes.

Here’s an example:

arr = [3, 2, 1] swap_count = sum(i != j for i, j in enumerate(sorted(range(len(arr)), key=arr.__getitem__))) print(swap_count)

Output:

`2`

This intriguing one-liner calculates the `swap_count`

by comparing the original indices of the elements with the indices they would have after being sorted.

## Summary/Discussion

**Method 1:** Bubble Sort Count. Simple and intuitive. Best for educational purposes, not efficient for large arrays.

**Method 2:** Selection Sort Count. Minimal number of swaps among simple sort algorithms. Inefficient for large arrays.

**Method 3:** Insertion Sort Count. Efficient for small or nearly sorted arrays. Performance degrades for larger or random lists.

**Method 4:** Optimized Gnome Sort Count. Easier to implement than other sorts. Still slow for large datasets.

**Bonus Method 5:** Python’s Sorted Function. A clever one-liner for swap count estimation. Not a sorting algorithm, and practical usage is limited.