Question

I need to implement quicksort in SML for a homework assignment, and I’m lost. I was previously unfamiliar with how quicksort was implemented, so I read up on that, but every implementation I read about was in an imperative one. These don’t look too difficult, but I had no idea how to implement quicksort functionally.

Wikipedia happens to have quicksort code in Standard ML (which is the language required for my assignment), but I don’t understand how it’s working.

Wikipedia code:

val filt = List.filter
fun quicksort << xs = let
fun qs [] = []
 | qs [x] = [x]
 | qs (p::xs) = let
     val lessThanP = (fn x => << (x, p))
     in
       qs (filt lessThanP xs) @ p :: (qs (filt (not o lessThanP) xs))
     end
in
  qs xs
end

In particular, I don’t understand this line: qs (filt lessThanP xs) @ p :: (qs (filt (not o lessThanP) xs)). filt will return a list of everything in xs less than p*, which is concatenated with p, which is cons-ed onto everything >= p.*

*assuming the << (x, p) function returns true when x < p. Of course it doesn’t have to be that.

Actually, typing this out is helping me understand what’s going on a bit. Anyways, I’m trying to compare that SML function to wiki’s quicksort pseudocode, which follows.

function quicksort(array, ‘left’, ‘right’)

  // If the list has 2 or more items
  if 'left' < 'right'

      // See "Choice of pivot" section below for possible choices
      choose any 'pivotIndex' such that 'left' ≤ 'pivotIndex' ≤ 'right'

      // Get lists of bigger and smaller items and final position of pivot
      'pivotNewIndex' := partition(array, 'left', 'right', 'pivotIndex')

      // Recursively sort elements smaller than the pivot
      quicksort(array, 'left', 'pivotNewIndex' - 1)

      // Recursively sort elements at least as big as the pivot
      quicksort(array, 'pivotNewIndex' + 1, 'right')

Where partition is defined as

// left is the index of the leftmost element of the array
// right is the index of the rightmost element of the array (inclusive)
//   number of elements in subarray = right-left+1
function partition(array, 'left', 'right', 'pivotIndex')
  'pivotValue' := array['pivotIndex']
  swap array['pivotIndex'] and array['right']  // Move pivot to end
  'storeIndex' := 'left'
  for 'i' from 'left' to 'right' - 1  // left ≤ i < right
      if array['i'] < 'pivotValue'
          swap array['i'] and array['storeIndex']
          'storeIndex' := 'storeIndex' + 1
  swap array['storeIndex'] and array['right']  // Move pivot to its final place
  return 'storeIndex'

So, where exactly is the partitioning happening? Or am I thinking about SMLs quicksort wrongly?

Answer 1

So, where exactly is the partitioning happening? Or am I thinking about SMLs quicksort wrongly?

A purely functional implementation of quicksort works by structural recursion on the input list (IMO, this is worth mentioning). Moreover, as you see, the two calls to “filt” allow you to partition the input list into two sublists (say A and B), which then can be treated individually. What is important here is that:

• all elements of A are less than or equal to the pivot element (“p” in the code)
• all elements of B are greater than the pivot element

An imperative implementation works in-place, by swapping elements in the same array. In the pseudocode you’ve provided, the post-invariant of the “partition” function is that you have two subarrays, one starting at ‘left’ of input array (and ending at ‘pivotIndex’), and another starting right after ‘pivotIndex’ and ending at ‘right’. What is important here is that the two subarrays can be seen as representations of the sublists A and B.

I think that by now, you have the idea where the partitioning step is happening (or conversely, how the imperative and functional are related).