Monday, September 24, 2012

Permutations in Python and Scala

A permutations of a collection of objects describes all possible, ordered arrangements of those objects. For instance, all permutations of (1,2,3) are (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). The following Python code generates all permutation of the elements in the given list:
def permutations(ls):
    if len(ls) == 1: yield ls
    for i in xrange(len(ls)):
        for p in permutations(ls[:i]+ls[i+1:]):
            yield [ls[i]]+p
Here a usage example:
ls = [1,2,3]            
for p in permutations(ls):
    print p
[1, 2, 3]
[1, 3, 2]
[2, 1, 3]
[2, 3, 1]
[3, 1, 2]
[3, 2, 1]
Now, let's compute permutations in Scala (2.9):
def permutations[T](ls: List[T]): List[List[T]] = ls match {
  case List() => List(List())
  case _ => for(e <- ls; r <- permutations(ls filterNot(_==e))) yield e::r
In contrast to the Python version the Scala version above is not a generator and creates all permutations in memory. While the "yield" keyword in Python defines a generator in Scala it is a way to describe a list comprehension. To create a generator version Iterators can be used:
def permutations[T](ls: List[T]): Iterator[List[T]] = ls match {
  case List() => Iterator(List())
  case _ => for(e <- ls.iterator; r <- permutations(ls.filterNot(_==e))) yield e::r
And a more compact but slightly less readable version:
def perms[T](ls:List[T]): Iterator[List[T]] =
    if(ls.isEmpty) Iterator(sl) else for (h <- ls.iterator; t <- perms(ls.filterNot(_==h))) yield h::t
The time complexity is n factorial (O(n!)), which is easy to see. Let's say there is one element, then there is only one permutations. If there are two elements, the there is two possible permutations. With three elements let's hold one, that there is two left, which gives us two permutations times three. Obviously the sequence is 1*2*3*...*n, which is n!. BTW, this can be beautifully written in Scala:
def fac(n:Int) = (1 to n) reduceLeft (_ * _)
or more cryptically-compact using left fold instead of reduce:
def fac(n:Int) = (1/:(2 to n))(_*_)