What does this list permutations implementation in Haskell exactly do?

Question

I am studying the code in the Data.List module and can't exactly wrap my head around this implementation of permutations:

permutations            :: [a] -> [[a]]
permutations xs0        =  xs0 : perms xs0 []
  where
    perms []     _  = []
    perms (t:ts) is = foldr interleave (perms ts (t:is)) (permutations is)
      where interleave    xs     r = let (_,zs) = interleave' id xs r in zs
            interleave' _ []     r = (ts, r)
            interleave' f (y:ys) r = let (us,zs) = interleave' (f . (y:)) ys r
                                     in  (y:us, f (t:y:us) : zs)

Can somebody explain in detail how these nested functions connect/work with each other?

Answer 1

Sorry about the late answer, it took a bit longer to write down than expected.

---

So, first of all to maximize lazyness in a list function like this there are two goals:

• Produce as many answers as possible before inspecting the next element of the input list
• The answers themselves must be lazy, and so there the same must hold.

Now consider the permutation function. Here maximal lazyness means:

• We should determine that there are at least n! permutations after inspecting just n elements of input
• For each of these n! permutations, the first n elements should depend only on the first n elements of the input.

The first condition could be formalized as

length (take (factorial n) $ permutations ([1..n] ++ undefined))) `seq` () == ()

David Benbennick formalized the second condition as

map (take n) (take (factorial n) $ permutations [1..]) == permutations [1..n] 

Combined, we have

map (take n) (take (factorial n) $ permutations ([1..n] ++ undefined)) == permutations [1..n] 

Let's start with some simple cases. First permutation [1..]. We must have

permutations [1..] = [1,???] : ???

And with two elements we must have

permutations [1..] = [1,2,???] : [2,1,???] : ???

Note that there is no choice about the order of the first two elements, we can't put [2,1,...] first, since we already decided that the first permutation must start with 1. It should be clear by now that the first element of permutations xs must be equal to xs itself.

---

Now on to the implementation.

First of all, there are two different ways to make all permutations of a list:

1. Selection style: keep picking elements from the list until there are none left

   permutations []  = [[]]
   permutations xxs = [(y:ys) | (y,xs) <- picks xxs, ys <- permutations xs]
     where
       picks (x:xs) = (x,xs) : [(y,x:ys) | (y,ys) <- picks xs]
   

2. Insertion style: insert or interleave each element in all possible places

   permutations []     = [[]]
   permutations (x:xs) = [y | p <- permutations xs, y <- interleave p]
     where
       interleave []     = [[x]]
       interleave (y:ys) = (x:y:ys) : map (y:) (interleave ys)
   

Note that neither of these is maximally lazy. The first case, the first thing this function does is pick the first element from the entire list, which is not lazy at all. In the second case we need the permutations of the tail before we can make any permutation.

To start, note that interleave can be made more lazy. The first element of interleave yss list is [x] if yss=[] or (x:y:ys) if yss=y:ys. But both of these are the same as x:yss, so we can write

interleave yss = (x:yss) : interleave' yss
interleave' [] = []
interleave' (y:ys) = map (y:) (interleave ys)

The implementation in Data.List continues on this idea, but uses a few more tricks.

It is perhaps easiest to go through the mailing list discussion. We start with David Benbennick's version, which is the same as the one I wrote above (without the lazy interleave). We already know that the first elment of permutations xs should be xs itself. So, let's put that in

permutations xxs     = xxs : permutations' xxs
permutations' []     = []
permutations' (x:xs) = tail $ concatMap interleave $ permutations xs
  where interleave = ..

The call to tail is of course not very nice. But if we inline the definitions of permutations and interleave we get

permutations' (x:xs)
  = tail $ concatMap interleave $ permutations xs
  = tail $ interleave xs ++ concatMap interleave (permutations' xs)
  = tail $ (x:xs) : interleave' xs ++ concatMap interleave (permutations' xs)
  = interleave' xs ++ concatMap interleave (permutations' xs)

Now we have

permutations xxs     = xxs : permutations' xxs
permutations' []     = []
permutations' (x:xs) = interleave' xs ++ concatMap interleave (permutations' xs)
  where
   interleave yss = (x:yss) : interleave' yss
   interleave' [] = []
   interleave' (y:ys) = map (y:) (interleave ys)

The next step is optimization. An important target would be to eliminate the (++) calls in interleave. This is not so easy, because of the last line, map (y:) (interleave ys). We can't immediately use the foldr/ShowS trick of passing the tail as a parameter. The way out is to get rid of the map. If we pass a parameter f as the function that has to be mapped over the result at the end, we get

permutations' (x:xs) = interleave' id xs ++ concatMap (interleave id) (permutations' xs)
  where
   interleave f yss = f (x:yss) : interleave' f yss
   interleave' f [] = []
   interleave' f (y:ys) = interleave (f . (y:)) ys

Now we can pass in the tail,

permutations' (x:xs) = interleave' id xs $ foldr (interleave id) [] (permutations' xs)
  where
   interleave  f yss    r = f (x:yss) : interleave' f yss r
   interleave' f []     r = r
   interleave' f (y:ys) r = interleave (f . (y:)) ys r

This is starting to look like the one in Data.List, but it is not the same yet. In particular, it is not as lazy as it could be. Let's try it out:

*Main> let n = 4
*Main> map (take n) (take (factorial n) $ permutations ([1..n] ++ undefined))
[[1,2,3,4],[2,1,3,4],[2,3,1,4],[2,3,4,1]*** Exception: Prelude.undefined

Uh oh, only the first n elements are correct, not the first factorial n. The reason is that we still try to place the first element (the 1 in the above example) in all possible locations before trying anything else.

---

Yitzchak Gale came up with a solution. Considered all ways to split the input into an initial part, a middle element, and a tail:

[1..n] == []    ++ 1 : [2..n]
       == [1]   ++ 2 : [3..n]
       == [1,2] ++ 3 : [4..n]

If you haven't seen the trick to generate these before before, you can do this with zip (inits xs) (tails xs). Now the permutations of [1..n] will be

• [] ++ 1 : [2..n] aka. [1..n], or
• 2 inserted (interleaved) somewhere into a permutation of [1], followed by [3..n]. But not 2 inserted at the end of [1], since we already go that result in the previous bullet point.
• 3 interleaved into a permutation of [1,2] (not at the end), followed by [4..n].
• etc.

You can see that this is maximally lazy, since before we even consider doing something with 3, we have given all permutations that start with some permutation of [1,2]. The code that Yitzchak gave was

permutations xs = xs : concat (zipWith newPerms (init $ tail $ tails xs)
                                                (init $ tail $ inits xs))
  where
    newPerms (t:ts) = map (++ts) . concatMap (interleave t) . permutations3
    interleave t [y]        = [[t, y]]
    interleave t ys@(y:ys') = (t:ys) : map (y:) (interleave t ys') 

Note the recursive call to permutations3, which can be a variant that doesn't have to be maximally lazy.

As you can see this is a bit less optimized than what we had before. But we can apply some of the same tricks.

The first step is to get rid of init and tail. Let's look at what zip (init $ tail $ tails xs) (init $ tail $ inits xs) actually is

*Main> let xs = [1..5] in zip (init $ tail $ tails xs) (init $ tail $ inits xs)
[([2,3,4,5],[1]),([3,4,5],[1,2]),([4,5],[1,2,3]),([5],[1,2,3,4])]

The init gets rid of the combination ([],[1..n]), while the tail gets rid of the combination ([1..n],[]). We don't want the former, because that would fail the pattern match in newPerms. The latter would fail interleave. Both are easy to fix: just add a case for newPerms [] and for interleave t [].

permutations xs = xs : concat (zipWith newPerms (tails xs) (inits xs))
  where
    newPerms [] is = []
    newPerms (t:ts) is = map (++ts) (concatMap (interleave t) (permutations is))
    interleave t []         = []
    interleave t ys@(y:ys') = (t:ys) : map (y:) (interleave t ys') 

Now we can try to inline tails and inits. Their definition is

tails xxs = xxs : case xxs of
  []     -> []
  (_:xs) -> tails xs

inits xxs = [] : case xxs of
  []     -> []
  (x:xs) -> map (x:) (inits xs)

The problem is that inits is not tail recursive. But since we are going to take a permutation of the inits anyway, we don't care about the order of the elements. So we can use an accumulating parameter,

inits' = inits'' []
  where
  inits'' is xxs = is : case xxs of
    []     -> []
    (x:xs) -> inits'' (x:is) xs

Now we make newPerms a function of xxs and this accumulating parameter, instead of tails xxs and inits xxs.

permutations xs = xs : concat (newPerms' xs [])
  where
    newPerms' xxs is =
      newPerms xxs is :
      case xxs of
        []     -> []
        (x:xs) -> newPerms' xs (x:is)
    newPerms [] is = []
    newPerms (t:ts) is = map (++ts) (concatMap (interleave t) (permutations3 is))

inlining newPerms into newPerms' then gives

permutations xs = xs : concat (newPerms' xs [])
  where
    newPerms' []     is = [] : []
    newPerms' (t:ts) is =
      map (++ts) (concatMap (interleave t) (permutations is)) :
      newPerms' ts (t:is)

inlining and unfolding concat, and moving the final map (++ts) into interleave,

permutations xs = xs : newPerms' xs []
  where
    newPerms' []     is = []
    newPerms' (t:ts) is =
        concatMap interleave (permutations is) ++
        newPerms' ts (t:is)
      where
      interleave []     = []
      interleave (y:ys) = (t:y:ys++ts) : map (y:) (interleave ys) 

Then finally, we can reapply the foldr trick to get rid of the (++):

permutations xs = xs : newPerms' xs []
  where
    newPerms' []     is = []
    newPerms' (t:ts) is =
        foldr (interleave id) (newPerms' ts (t:is)) (permutations is)
      where
      interleave f []     r = r
      interleave f (y:ys) r = f (t:y:ys++ts) : interleave (f . (y:)) ys r

Wait, I said get rid of the (++). We got rid of one of them, but not the one in interleave. For that, we can see that we are always concatenating some tail of yys to ts. So, we can unfold the calculating (ys++ts) along with the recursion of interleave, and have the function interleave' f ys r return the tuple (ys++ts, interleave f ys r). This gives

permutations xs = xs : newPerms' xs []
  where
    newPerms' []     is = []
    newPerms' (t:ts) is =
        foldr interleave (newPerms' ts (t:is)) (permutations is)
      where
      interleave ys r = let (_,zs) = interleave' id ys r in zs
      interleave' f []     r = (ts,r)
      interleave' f (y:ys) r = 
        let (us,zs) = interleave' (f . (y:)) ys r
        in  (y:us, f (t:y:us) : zs)

And there you have it, Data.List.permutations in all its maximally lazy optimized glory.

---

Great write-up by Twan! I (@Yitz) will just add a few references:

• The original email thread where Twan developed this algorithm, linked above by Twan, is fascinating reading.

• Knuth classifies all possible algorithms that satisfy these criteria in Vol. 4 Fasc. 2 Sec. 7.2.1.2.

• Twan's permutations3 is essentially the same as Knuth's "Algorithm P". As far as Knuth knows, that algorithm was first published by English church bell ringers in the 1600's.

Answer 2

The basic algorithm is based on the idea of taking one item from the list at a time, finding every permutation of items including that new one, and then repeating.

To explain what this looks like, [1..] will mean a list from one up, where no values (no even the first) have been examined yet. It is the parameter to the function. The resulting list is something like:

[[1..]] ++
[[2,1,3..]] ++
[[3,2,1,4..], [2,3,1,4..]] ++ [[3,1,2,4..], [1,3,2,4..]]
[[4,3,2,1,5..], etc

The clustering above reflects the core idea of the algorithm... each row represents a new item taken from the input list, and added to the set of items that are being permuted. Furthermore, it is recursive... on each new row, it takes all the existing permutations, and places the item in each place it hasn't been yet (all the places other then the last one). So, on the third row, we have the two permutations [2,1] and [1,2], and then we take place 3 in both available slots, so [[3,2,1], [2,3,1]] and [[3,1,2], [1,3,2]] respectively, and then append whatever the unobserved part is.

Hopefully, this at least clarifies the algorithm a little. However, there are some optimizations and implementation details to explain.

(Side note: There are two central performance optimizations that are used: first, if you want to repeatedly prepend some items to multiple lists, map (x:y:z:) list is a lot faster then matching some conditional or pattern matching, because it has not branch, just a calculated jump. Second, and this one is used a lot, it is cheap (and handy) to build lists from the back to the front, by repeatedly prepending items; this is used in a few places.

The first thing the function does is establish a two bases cases: first, every list has one permutation at least: itself. This can be returned with no evaluation whatsoever. This could be thought of as the "take 0" case.

The outer loop is the part that looks like the following:

perms (t:ts) is = <prepend_stuff_to> (perms ts (t:is))

ts is the "untouched" part of the list, that we are not yet permuting and haven't even examined yet, and is initially the entire input sequence.

t is the new item we will be sticking in between the permutations.

is is the list of items that we will permute, and then place t in between, and is initially empty.

Each time we calculate one of the above rows, we reach the end of the items we have prepended to the thunk containing (perms ts (t:is)) and will recurse.

---

The second loop in is a foldr. It for each permutation of is (the stuff before the current item in the original list), it interleaves the item into that list, and prepends it to the thunk.

foldr interleave <thunk> (permutations is)

---

The third loop is one of the most complex. We know that it prepends each possible interspersing of our target item t in a permutation, followed by the unobserved tail onto the result sequence. It does this with a recursive call, where it folds the permutation into a stack of functions as it recurses, and then as it returns, it executes what amounts to a two little state machines to build the results.

Lets look at an example: interleave [<thunk>] [1,2,3] where t = 4 and is = [5..]

First, as interleave' is called recursively, it builds up ys and fs on the stack, like this:

y = 1, f = id
y = 2, f = (id . (1:))
y = 3, f = ((id . (1:)) . (2:))
(the functions are conceptually the same as ([]++), ([1]++), and ([1,2]++) respectively)

Then, as we go back up, we return and evalute a tuple containing two values, (us, zs).

us is the list to which we prepend the ys after our target t.

zs is the result accumulator, where each time we get a new permutation, we prepend it to the results lists.

Thus, to finish the example, f (t:y:us) gets evaluated and returned as a result for each level of the stack above.

([1,2]++) (4:3:[5..]) === [1,2,4,3,5..]
([1]++) (4:2[3,5..])  === [1,4,2,3,5..]
([]++) (4:1[2,3,5..]) === [4,1,2,3,5..]

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Hopefully that helps, or at least supplements the material linked in the author's comment above.

(Thanks to dfeuer for bringing this up on IRC and discussing it for a few hours)