# Algorithmic complexity of this algorithm to find all ordered permutations of length X

I have written a recursive function to generate the list of all ordered permutations of length X for a list of chars.

For instance: ['a', 'b', 'c', 'd'] with X=2 will give [['a', 'a'], ['a', 'b'], ['a', 'c'], ['a', 'd'], ['b', 'a'], ['b', 'b'], ..., ['d', 'd']]

I'm not sure about its algorithmic complexity though (at least I know it's pretty horrible). I would say it's something around:

O(X * N^(L + X))

(where L is the number of different chars, 4 here because we have 'A', 'B', 'C', 'D', and X the length of the permutations we want to generate). Because I have 2 nested loops, which will be run X times (well, X-1 because of the special case when X = 1). Is it correct?

def generate_permutations(symbols, permutations_length):
if permutations_length == 1:
return [[symbol] for symbol in symbols]

tails = generate_permutations(symbols, permutations_length-1)
permutations = []

for symbol in symbols:
for tail in tails:
permutation = [symbol] + tail

permutations.append(permutation)

return permutations

print(generate_permutations(['a', 'b', 'c', 'd'], 2))


By the way: I know this is not idiomatic Python and I apologize if it's ugly but it's just some prototyping I'm doing before writing this code in a different, less expressive language. And I also know that I could use itertools.permutations to do this task. By the way, I'd be interested if someone happens to know the algorithmic complexity of itertool's permutations function.

• "And I also know that I could use itertools.permutations to do this task." So why didn't you? – Mast Apr 27 '18 at 11:25
• For the sake of the exercise and to learn how to do things by myself. – user168305 Apr 27 '18 at 11:53

def generate_permutations(symbols, permutations_length):


These are not permutations. They are Cartesian products. In addition, calling the function generate_something suggests that what's returned might be a generator, which is probably what you want for a function which returns a stupendously large data structure, but is not what you get from this function.

    for symbol in symbols:
for tail in tails:
permutation = [symbol] + tail

permutations.append(permutation)


If you use extend instead of append then you give better hints as to how much underlying buffers might need to be increased:

    for tail in tails:
result.extend([symbol] + tail for symbol in symbols)


Performance: with an alphabet of size $L$ let $T_L(X)$ be the cost of calling this function for products of length $X$. Then $T_L(1) = L$ and $T_L(X+1) = T_L(X) + L L^X (X+2)$ assuming that constructing the list [symbol] + tail takes time proportional to its length and extending the list of results take amortised constant time per element added. This gives $\Theta(L^{X}X)$ complexity.

The same complexity but much better memory efficiency can be achieved with a true generator approach which is based on the fact that all you really need to do is count to $L^X$ and convert into base $L$.

• Thanks a lot for this explanation! And I agree that generate_something was not the best way to name it. May I ask how you would do the same thing with a generator instead? – user168305 Apr 27 '18 at 11:55
• @Matthieu: suppose that instead of 'a', 'b', 'c', 'd' your elements are 0, 1, 2, 3 and the Cartesian products are 00, 01, 02, 03, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33. Does that make it clear what I mean by counting to 16 and converting into base 4? – Peter Taylor Apr 27 '18 at 12:37