How does the permutation sort algorithm function?

The permutation sort algorithm functions by generating all permutations of the input list and checking each for being sorted.

Permutation sort, also known as exhaustive sort, is a simple but highly inefficient sorting algorithm. It works by generating all possible permutations of the input list and checking each one to see if it is sorted. If a permutation is sorted, the algorithm stops and returns that permutation as the sorted list.

The algorithm starts by generating the first permutation of the input list. This is typically the list in its original, unsorted order. It then generates the next permutation and checks if it is sorted. This process is repeated until a sorted permutation is found or all permutations have been generated and checked.

Generating all permutations of a list can be done in several ways. One common method is to use recursion. Starting with the first element of the list, the algorithm swaps it with each of the other elements in turn and recursively generates the permutations of the remaining elements. This process is repeated for each element in the list.

Checking if a list is sorted is straightforward. The algorithm simply iterates through the list from start to end, comparing each element with the next. If it finds an element that is greater than the next, it knows the list is not sorted. If it reaches the end of the list without finding any such elements, it knows the list is sorted.

The main drawback of permutation sort is its inefficiency. The number of permutations of a list of n elements is n factorial (n!), which grows very rapidly as n increases. This means the algorithm has a worst-case and average time complexity of O(n!). For this reason, permutation sort is not used in practice for large lists. However, it can be useful for teaching purposes, as it provides a clear illustration of the concept of permutations and how they can be generated and checked.