Algorithm to generate all possible permutations of a list

algorithmlistpermutation

Say I have a list of n elements, I know there are n! possible ways to order these elements. What is an algorithm to generate all possible orderings of this list? Example, I have list [a, b, c]. The algorithm would return [[a, b, c], [a, c, b,], [b, a, c], [b, c, a], [c, a, b], [c, b, a]].

I'm reading this here
http://en.wikipedia.org/wiki/Permutation#Algorithms_to_generate_permutations

But Wikipedia has never been good at explaining. I don't understand much of it.

Best Answer

Basically, for each item from left to right, all the permutations of the remaining items are generated (and each one is added with the current elements). This can be done recursively (or iteratively if you like pain) until the last item is reached at which point there is only one possible order.

So with the list [1,2,3,4] all the permutations that start with 1 are generated, then all the permutations that start with 2, then 3 then 4.

This effectively reduces the problem from one of finding permutations of a list of four items to a list of three items. After reducing to 2 and then 1 item lists, all of them will be found.
Example showing process permutations using 3 coloured balls:
(from https://en.wikipedia.org/wiki/Permutation#/media/File:Permutations_RGB.svg - https://commons.wikimedia.org/wiki/File:Permutations_RGB.svg)

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Python – How to check if a list is empty

if not a:
  print("List is empty")

Using the implicit booleanness of the empty list is quite pythonic.

Python – How to generate all permutations of a list

There's a function in the standard-library for this: itertools.permutations.

import itertools
list(itertools.permutations([1, 2, 3]))

If for some reason you want to implement it yourself or are just curious to know how it works, here's one nice approach, taken from http://code.activestate.com/recipes/252178/:

def all_perms(elements):
    if len(elements) <=1:
        yield elements
    else:
        for perm in all_perms(elements[1:]):
            for i in range(len(elements)):
                # nb elements[0:1] works in both string and list contexts
                yield perm[:i] + elements[0:1] + perm[i:]

A couple of alternative approaches are listed in the documentation of itertools.permutations. Here's one:

def permutations(iterable, r=None):
    # permutations('ABCD', 2) --> AB AC AD BA BC BD CA CB CD DA DB DC
    # permutations(range(3)) --> 012 021 102 120 201 210
    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    if r > n:
        return
    indices = range(n)
    cycles = range(n, n-r, -1)
    yield tuple(pool[i] for i in indices[:r])
    while n:
        for i in reversed(range(r)):
            cycles[i] -= 1
            if cycles[i] == 0:
                indices[i:] = indices[i+1:] + indices[i:i+1]
                cycles[i] = n - i
            else:
                j = cycles[i]
                indices[i], indices[-j] = indices[-j], indices[i]
                yield tuple(pool[i] for i in indices[:r])
                break
        else:
            return

And another, based on itertools.product:

def permutations(iterable, r=None):
    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    for indices in product(range(n), repeat=r):
        if len(set(indices)) == r:
            yield tuple(pool[i] for i in indices)

Best Answer

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