Can you find all the permutations of a given string?

Finding all the permutations of a given string is a common problem in computer programming and can be solved using recursion and backtracking.

Here is a Python code snippet that accomplishes this:

def permute(data, i, length): 
    if i==length: 
        print(''.join(data))
    else: 
        for j in range(i, length): 
            data[i], data[j] = data[j], data[i] 
            permute(data, i+1, length) 
            data[i], data[j] = data[j], data[i]
            
string = 'ABC'
n = len(string)
data = list(string)
permute(data, 0, n)

The function ‘permute‘ generates all the permutations of a string. This is done by swapping each character in the string with every other character (including itself). For each swap, ‘permute‘ is called recursively with the next character, until all characters have been used. The swapped characters are then swapped back to their original positions before the next iteration.

This process can be better understood with the following tree diagram:

ABC
/ | \
A BC ACB
/ \ / \
AB C AC B
/ \ / \
ABC ACB ABC ACB

In the diagram above, each level of the tree corresponds to a character position in the string. Each node represents a string permutation. When the tree is fully expanded, all possible permutations are printed.

There are n! permutations in a string of length n. This is because there are n choices for the first character, n-1 choices for the second character, n-2 choices for the third character, and so on, until only 1 choice is left for the last character. This can be represented mathematically as:

n! = n * (n − 1) * (n − 2) * ... * 2 * 1.

Also, for each permutation, there are n*(n-1) swaps made, so the time complexity of this algorithm is O(n² * n!).