Example of a factorial time algorithm O( n! )

Question

I\'m studying time complexity in school and our main focus seems to be on polynomial time O(n^c) algorithms and quasi-linear time O(nlog

Answer 1

List all permutations of an array is O(n!). Below is a recursive implementation using the swap method. The recursion is inside the for loop and the elements in the array are swapped in position until no more elements remain. As you can see from the result count, the number of elements in the array is n!. Each permutation is an operation and there are n! operations.

def permutation(array, start, result)
    if (start == array.length) then
        result << array.dup  
    end
    for i in start..array.length-1 do
        array[start], array[i] = array[i], array[start]
        permutation(array, start+1,result)
        array[start], array[i] = array[i], array[start]
    end 
    result   
end        


p permutation([1,2,3], 0, []).count  #> 6 = 3!
p permutation([1,2,3,4], 0, []).count #> 24 = 4!
p permutation([1,2,3,4,5], 0, []).count #> 120 = 5!