Common/Lib/Set.hs

	Set.hs revision b10d6cef708b7a659f2d3b367e8e0db0d03ae3f5
--------------------------------------------------------------------------------
{-| Module      :  Common.Lib.Set
    Copyright   :  (c) Daan Leijen 2002
    License     :  BSD-style

    Maintainer  :  daan@cs.uu.nl
    Stability   :  provisional
    Portability :  portable

  An efficient implementation of sets.

  1) The 'filter' function clashes with the "Prelude".
      If you want to use "Set" unqualified, this function should be hidden.

      > import Prelude hiding (filter)
      > import Set

      Another solution is to use qualified names. This is also the only way how
      a "Map", "Set", and "MultiSet" can be used within one module.

      > import qualified Set
      >
      > ... Set.single "Paris"

      Or, if you prefer a terse coding style:

      > import qualified Set as S
      >
      > ... S.single "Berlin"

  2) The implementation of "Set" is based on /size balanced/ binary trees (or
     trees of /bounded balance/) as described by:

     * Stephen Adams, \"/Efficient sets: a balancing act/\", Journal of Functional
       Programming 3(4):553-562, October 1993, <http://www.swiss.ai.mit.edu/~adams/BB>.

     * J. Nievergelt and E.M. Reingold, \"/Binary search trees of bounded balance/\",
       SIAM journal of computing 2(1), March 1973.

  3) Note that the implementation /left-biased/ -- the elements of a first argument
      are always perferred to the second, for example in 'union' or 'insert'.
      Off course, left-biasing can only be observed when equality an equivalence relation
      instead of structural equality.

  4) Another implementation of sets based on size balanced trees
      exists as "Data.Set" in the Ghc libraries. The good part about this library
      is that it is highly tuned and thorougly tested. However, it is also fairly old,
      it is implemented indirectly on top of "Data.FiniteMap" and only supports
      the basic set operations.
      The "Set" module overcomes some of these issues:

      * It tries to export a more complete and consistent set of operations, like
        'partition', 'subset' etc.

      * It uses the efficient /hedge/ algorithm for both 'union' and 'difference'
        (a /hedge/ algorithm is not applicable to 'intersection').

      * It converts ordered lists in linear time ('fromAscList').

      * It takes advantage of the module system with names like 'empty' instead of 'Data.Set.emptySet'.

      * It is implemented directly, instead of using a seperate finite map implementation.
-}
---------------------------------------------------------------------------------
module Common.Lib.Set  (
            -- * Set type
              Set          -- instance Eq, Ord, Show

            -- * Operators
            , (\\)

            -- * Query
            , isEmpty
            , size
            , member
            , subset
            , properSubset

            -- * Construction
            , empty
            , single
            , insert
            , delete

            -- * Combine
            , union, unions
            , difference
            , intersection

            -- * Filter
            , filter
            , partition
            , split
            , splitMember

            -- * Fold
            , fold
            , image

            -- * Min\/Max
            , findMin
            , findMax
            , deleteMin
            , deleteMax
            , deleteFindMin
            , deleteFindMax

            -- * Conversion

            -- ** List
            , elems
            , toList
            , fromList

            -- ** Ordered list
            , toAscList
            , fromAscList
            , fromDistinctAscList

            -- * Debugging
            , showTree
            , showTreeWith
            , valid
            ) where

import Prelude hiding (filter)

{-
-- just for testing
import QuickCheck
import List (nub,sort)
import qualified List
-}

{--------------------------------------------------------------------
  Operators
--------------------------------------------------------------------}
infixl 9 \\

-- | /O(n+m)/. See 'difference'.
(\\) :: Ord a => Set a -> Set a -> Set a
m1 \\ m2 = difference m1 m2

{--------------------------------------------------------------------
  Sets are size balanced trees
--------------------------------------------------------------------}
-- | A set of values @a@.
data Set a    = Tip
              | Bin !Size a !(Set a) !(Set a)

type Size     = Int

{--------------------------------------------------------------------
  Query
--------------------------------------------------------------------}
-- | /O(1)/. Is this the empty set?
isEmpty :: Set a -> Bool
isEmpty t
  = case t of
      Tip           -> True
      Bin sz x l r  -> False

-- | /O(1)/. The number of elements in the set.
size :: Set a -> Int
size t
  = case t of
      Tip           -> 0
      Bin sz x l r  -> sz

-- | /O(log n)/. Is the element in the set?
member :: Ord a => a -> Set a -> Bool
member x t
  = case t of
      Tip -> False
      Bin sz y l r
          -> case compare x y of
               LT -> member x l
               GT -> member x r
               EQ -> True

{--------------------------------------------------------------------
  Construction
--------------------------------------------------------------------}
-- | /O(1)/. The empty set.
empty  :: Set a
empty
  = Tip

-- | /O(1)/. Create a singleton set.
single :: a -> Set a
single x
  = Bin 1 x Tip Tip

{--------------------------------------------------------------------
  Insertion, Deletion
--------------------------------------------------------------------}
-- | /O(log n)/. Insert an element in a set.
insert :: Ord a => a -> Set a -> Set a
insert x t
  = case t of
      Tip -> single x
      Bin sz y l r
          -> case compare x y of
               LT -> balance y (insert x l) r
               GT -> balance y l (insert x r)
               EQ -> Bin sz x l r


-- | /O(log n)/. Delete an element from a set.
delete :: Ord a => a -> Set a -> Set a
delete x t
  = case t of
      Tip -> Tip
      Bin sz y l r
          -> case compare x y of
               LT -> balance y (delete x l) r
               GT -> balance y l (delete x r)
               EQ -> glue l r

{--------------------------------------------------------------------
  Subset
--------------------------------------------------------------------}
-- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
properSubset :: Ord a => Set a -> Set a -> Bool
properSubset s1 s2
  = (size s1 < size s2) && (subset s1 s2)


-- | /O(n+m)/. Is this a subset?
subset :: Ord a => Set a -> Set a -> Bool
subset t1 t2
  = (size t1 <= size t2) && (subsetX t1 t2)

subsetX Tip t = True
subsetX t Tip = False
subsetX (Bin _ x l r) t
  = found && subsetX l lt && subsetX r gt
  where
    (found,lt,gt) = splitMember x t


{--------------------------------------------------------------------
  Minimal, Maximal
--------------------------------------------------------------------}
-- | /O(log n)/. The minimal element of a set.
findMin :: Set a -> a
findMin (Bin _ x Tip r) = x
findMin (Bin _ x l r)   = findMin l
findMin Tip             = error "Set.findMin: empty set has no minimal element"

-- | /O(log n)/. The maximal element of a set.
findMax :: Set a -> a
findMax (Bin _ x l Tip)  = x
findMax (Bin _ x l r)    = findMax r
findMax Tip              = error "Set.findMax: empty set has no maximal element"

-- | /O(log n)/. Delete the minimal element.
deleteMin :: Set a -> Set a
deleteMin (Bin _ x Tip r) = r
deleteMin (Bin _ x l r)   = balance x (deleteMin l) r
deleteMin Tip             = Tip

-- | /O(log n)/. Delete the maximal element.
deleteMax :: Set a -> Set a
deleteMax (Bin _ x l Tip) = l
deleteMax (Bin _ x l r)   = balance x l (deleteMax r)
deleteMax Tip             = Tip


{--------------------------------------------------------------------
  Union.
--------------------------------------------------------------------}
-- | The union of a list of sets: (@unions == foldl union empty@).
unions :: Ord a => [Set a] -> Set a
unions ts
  = foldlStrict union empty ts


-- | /O(n+m)/. The union of two sets. Uses the efficient /hedge-union/ algorithm.
union :: Ord a => Set a -> Set a -> Set a
union Tip t2  = t2
union t1 Tip  = t1
union t1 t2  -- hedge-union is more efficient on (bigset `union` smallset)
  | size t1 >= size t2  = hedgeUnion (const LT) (const GT) t1 t2
  | otherwise           = hedgeUnion (const LT) (const GT) t2 t1

hedgeUnion cmplo cmphi t1 Tip
  = t1
hedgeUnion cmplo cmphi Tip (Bin _ x l r)
  = join x (filterGt cmplo l) (filterLt cmphi r)
hedgeUnion cmplo cmphi (Bin _ x l r) t2
  = join x (hedgeUnion cmplo cmpx l (trim cmplo cmpx t2))
           (hedgeUnion cmpx cmphi r (trim cmpx cmphi t2))
  where
    cmpx y  = compare x y

{--------------------------------------------------------------------
  Difference
--------------------------------------------------------------------}
-- | /O(n+m)/. Difference of two sets.
-- The implementation uses an efficient /hedge/ algorithm comparable with /hedge-union/.
difference :: Ord a => Set a -> Set a -> Set a
difference Tip t2  = Tip
difference t1 Tip  = t1
difference t1 t2   = hedgeDiff (const LT) (const GT) t1 t2

hedgeDiff cmplo cmphi Tip t
  = Tip
hedgeDiff cmplo cmphi (Bin _ x l r) Tip
  = join x (filterGt cmplo l) (filterLt cmphi r)
hedgeDiff cmplo cmphi t (Bin _ x l r)
  = merge (hedgeDiff cmplo cmpx (trim cmplo cmpx t) l)
          (hedgeDiff cmpx cmphi (trim cmpx cmphi t) r)
  where
    cmpx y = compare x y

{--------------------------------------------------------------------
  Intersection
--------------------------------------------------------------------}
-- | /O(n+m)/. The intersection of two sets.
intersection :: Ord a => Set a -> Set a -> Set a
intersection Tip t = Tip
intersection t Tip = Tip
intersection t1 t2  -- intersection is more efficient on (bigset `intersection` smallset)
  | size t1 >= size t2  = intersect t1 t2
  | otherwise           = intersect t2 t1

intersect Tip t = Tip
intersect t Tip = Tip
intersect t (Bin _ x l r)
  | found     = join x tl tr
  | otherwise = merge tl tr
  where
    (found,lt,gt) = splitMember x t
    tl            = intersect lt l
    tr            = intersect gt r


{--------------------------------------------------------------------
  Filter and partition
--------------------------------------------------------------------}
-- | /O(n)/. Filter all elements that satisfy the predicate.
filter :: Ord a => (a -> Bool) -> Set a -> Set a
filter p Tip = Tip
filter p (Bin _ x l r)
  | p x       = join x (filter p l) (filter p r)
  | otherwise = merge (filter p l) (filter p r)

-- | /O(n)/. Partition the set into two sets, one with all elements that satisfy
-- the predicate and one with all elements that don't satisfy the predicate.
-- See also 'split'.
partition :: Ord a => (a -> Bool) -> Set a -> (Set a,Set a)
partition p Tip = (Tip,Tip)
partition p (Bin _ x l r)
  | p x       = (join x l1 r1,merge l2 r2)
  | otherwise = (merge l1 r1,join x l2 r2)
  where
    (l1,l2) = partition p l
    (r1,r2) = partition p r

{--------------------------------------------------------------------
  Fold
--------------------------------------------------------------------}
-- | /O(n)/. Fold the elements of a set.
fold :: (a -> b -> b) -> b -> Set a -> b
fold f z s
  = foldR f z s

-- | /O(n)/. Post-order fold.
foldR :: (a -> b -> b) -> b -> Set a -> b
foldR f z Tip           = z
foldR f z (Bin _ x l r) = foldR f (f x (foldR f z r)) l

{--------------------------------------------------------------------
  Image (Added by T.M.)
--------------------------------------------------------------------}
-- | /n/. Image of a set under a function
image :: Ord b => (a -> b) -> Set a -> Set b
image f t = foldR (insert . f) empty t

{--------------------------------------------------------------------
  List variations
--------------------------------------------------------------------}
-- | /O(n)/. The elements of a set.
elems :: Set a -> [a]
elems s
  = toList s

{--------------------------------------------------------------------
  Lists
--------------------------------------------------------------------}
-- | /O(n)/. Convert the set to a list of elements.
toList :: Set a -> [a]
toList s
  = toAscList s

-- | /O(n)/. Convert the set to an ascending list of elements.
toAscList :: Set a -> [a]
toAscList t
  = foldR (:) [] t


-- | /O(n*log n)/. Create a set from a list of elements.
fromList :: Ord a => [a] -> Set a
fromList xs
  = foldlStrict ins empty xs
  where
    ins t x = insert x t

{--------------------------------------------------------------------
  Building trees from ascending/descending lists can be done in linear time.

  Note that if [xs] is ascending that:
    fromAscList xs == fromList xs
--------------------------------------------------------------------}
-- | /O(n)/. Build a map from an ascending list in linear time.
fromAscList :: Eq a => [a] -> Set a
fromAscList xs
  = fromDistinctAscList (combineEq xs)
  where
  -- [combineEq xs] combines equal elements with [const] in an ordered list [xs]
  combineEq xs
    = case xs of
        []     -> []
        [x]    -> [x]
        (x:xx) -> combineEq' x xx

  combineEq' z [] = [z]
  combineEq' z (x:xs)
    | z==x      = combineEq' z xs
    | otherwise = z:combineEq' x xs


-- | /O(n)/. Build a set from an ascending list of distinct elements in linear time.
fromDistinctAscList :: [a] -> Set a
fromDistinctAscList xs
  = build const (length xs) xs
  where
    -- 1) use continutations so that we use heap space instead of stack space.
    -- 2) special case for n==5 to build bushier trees.
    build c 0 xs   = c Tip xs
    build c 5 xs   = case xs of
                       (x1:x2:x3:x4:x5:xx)
                            -> c (bin x4 (bin x2 (single x1) (single x3)) (single x5)) xx
    build c n xs   = seq nr $ build (buildR nr c) nl xs
                   where
                     nl = n `div` 2
                     nr = n - nl - 1

    buildR n c l (x:ys) = build (buildB l x c) n ys
    buildB l x c r zs   = c (bin x l r) zs

{--------------------------------------------------------------------
  Eq converts the set to a list. In a lazy setting, this
  actually seems one of the faster methods to compare two trees
  and it is certainly the simplest :-)
--------------------------------------------------------------------}
instance Eq a => Eq (Set a) where
  t1 == t2  = (size t1 == size t2) && (toAscList t1 == toAscList t2)

instance Ord a => Ord (Set a) where
  t1 <= t2  = toAscList t1 <= toAscList t2

{--------------------------------------------------------------------
  Show
--------------------------------------------------------------------}
instance Show a => Show (Set a) where
  showsPrec d s  = showSet (toAscList s)

showSet :: (Show a) => [a] -> ShowS
showSet []
  = showString "{}"
showSet (x:xs)
  = showChar '{' . shows x . showTail xs
  where
    showTail []     = showChar '}'
    showTail (x:xs) = showChar ',' . shows x . showTail xs


{--------------------------------------------------------------------
  Utility functions that return sub-ranges of the original
  tree. Some functions take a comparison function as argument to
  allow comparisons against infinite values. A function [cmplo x]
  should be read as [compare lo x].

  [trim cmplo cmphi t]  A tree that is either empty or where [cmplo x == LT]
                        and [cmphi x == GT] for the value [x] of the root.
  [filterGt cmp t]      A tree where for all values [k]. [cmp k == LT]
  [filterLt cmp t]      A tree where for all values [k]. [cmp k == GT]

  [split k t]           Returns two trees [l] and [r] where all values
                        in [l] are <[k] and all keys in [r] are >[k].
  [splitMember k t]     Just like [split] but also returns whether [k]
                        was found in the tree.
--------------------------------------------------------------------}

{--------------------------------------------------------------------
  [trim lo hi t] trims away all subtrees that surely contain no
  values between the range [lo] to [hi]. The returned tree is either
  empty or the key of the root is between @lo@ and @hi@.
--------------------------------------------------------------------}
trim :: (a -> Ordering) -> (a -> Ordering) -> Set a -> Set a
trim cmplo cmphi Tip = Tip
trim cmplo cmphi t@(Bin sx x l r)
  = case cmplo x of
      LT -> case cmphi x of
              GT -> t
              le -> trim cmplo cmphi l
      ge -> trim cmplo cmphi r

trimMemberLo :: Ord a => a -> (a -> Ordering) -> Set a -> (Bool, Set a)
trimMemberLo lo cmphi Tip = (False,Tip)
trimMemberLo lo cmphi t@(Bin sx x l r)
  = case compare lo x of
      LT -> case cmphi x of
              GT -> (member lo t, t)
              le -> trimMemberLo lo cmphi l
      GT -> trimMemberLo lo cmphi r
      EQ -> (True,trim (compare lo) cmphi r)


{--------------------------------------------------------------------
  [filterGt x t] filter all values >[x] from tree [t]
  [filterLt x t] filter all values <[x] from tree [t]
--------------------------------------------------------------------}
filterGt :: (a -> Ordering) -> Set a -> Set a
filterGt cmp Tip = Tip
filterGt cmp (Bin sx x l r)
  = case cmp x of
      LT -> join x (filterGt cmp l) r
      GT -> filterGt cmp r
      EQ -> r

filterLt :: (a -> Ordering) -> Set a -> Set a
filterLt cmp Tip = Tip
filterLt cmp (Bin sx x l r)
  = case cmp x of
      LT -> filterLt cmp l
      GT -> join x l (filterLt cmp r)
      EQ -> l


{--------------------------------------------------------------------
  Split
--------------------------------------------------------------------}
-- | /O(log n)/. The expression (@split x set@) is a pair @(set1,set2)@
-- where all elements in @set1@ are lower than @x@ and all elements in
-- @set2@ larger than @x@.
split :: Ord a => a -> Set a -> (Set a,Set a)
split x Tip = (Tip,Tip)
split x (Bin sy y l r)
  = case compare x y of
      LT -> let (lt,gt) = split x l in (lt,join y gt r)
      GT -> let (lt,gt) = split x r in (join y l lt,gt)
      EQ -> (l,r)

-- | /O(log n)/. Performs a 'split' but also returns whether the pivot
-- element was found in the original set.
splitMember :: Ord a => a -> Set a -> (Bool,Set a,Set a)
splitMember x Tip = (False,Tip,Tip)
splitMember x (Bin sy y l r)
  = case compare x y of
      LT -> let (found,lt,gt) = splitMember x l in (found,lt,join y gt r)
      GT -> let (found,lt,gt) = splitMember x r in (found,join y l lt,gt)
      EQ -> (True,l,r)

{--------------------------------------------------------------------
  Utility functions that maintain the balance properties of the tree.
  All constructors assume that all values in [l] < [x] and all values
  in [r] > [x], and that [l] and [r] are valid trees.

  In order of sophistication:
    [Bin sz x l r]    The type constructor.
    [bin x l r]       Maintains the correct size, assumes that both [l]
                      and [r] are balanced with respect to each other.
    [balance x l r]   Restores the balance and size.
                      Assumes that the original tree was balanced and
                      that [l] or [r] has changed by at most one element.
    [join x l r]      Restores balance and size.

  Furthermore, we can construct a new tree from two trees. Both operations
  assume that all values in [l] < all values in [r] and that [l] and [r]
  are valid:
    [glue l r]        Glues [l] and [r] together. Assumes that [l] and
                      [r] are already balanced with respect to each other.
    [merge l r]       Merges two trees and restores balance.

  Note: in contrast to Adam's paper, we use (<=) comparisons instead
  of (<) comparisons in [join], [merge] and [balance].
  Quickcheck (on [difference]) showed that this was necessary in order
  to maintain the invariants. It is quite unsatisfactory that I haven't
  been able to find out why this is actually the case! Fortunately, it
  doesn't hurt to be a bit more conservative.
--------------------------------------------------------------------}

{--------------------------------------------------------------------
  Join
--------------------------------------------------------------------}
join :: a -> Set a -> Set a -> Set a
join x Tip r  = insertMin x r
join x l Tip  = insertMax x l
join x l@(Bin sizeL y ly ry) r@(Bin sizeR z lz rz)
  | delta*sizeL <= sizeR  = balance z (join x l lz) rz
  | delta*sizeR <= sizeL  = balance y ly (join x ry r)
  | otherwise             = bin x l r


-- insertMin and insertMax don't perform potentially expensive comparisons.
insertMax,insertMin :: a -> Set a -> Set a
insertMax x t
  = case t of
      Tip -> single x
      Bin sz y l r
          -> balance y l (insertMax x r)

insertMin x t
  = case t of
      Tip -> single x
      Bin sz y l r
          -> balance y (insertMin x l) r

{--------------------------------------------------------------------
  [merge l r]: merges two trees.
--------------------------------------------------------------------}
merge :: Set a -> Set a -> Set a
merge Tip r   = r
merge l Tip   = l
merge l@(Bin sizeL x lx rx) r@(Bin sizeR y ly ry)
  | delta*sizeL <= sizeR = balance y (merge l ly) ry
  | delta*sizeR <= sizeL = balance x lx (merge rx r)
  | otherwise            = glue l r

{--------------------------------------------------------------------
  [glue l r]: glues two trees together.
  Assumes that [l] and [r] are already balanced with respect to each other.
--------------------------------------------------------------------}
glue :: Set a -> Set a -> Set a
glue Tip r = r
glue l Tip = l
glue l r
  | size l > size r = let (m,l') = deleteFindMax l in balance m l' r
  | otherwise       = let (m,r') = deleteFindMin r in balance m l r'


-- | /O(log n)/. Delete and find the minimal element.
deleteFindMin :: Set a -> (a,Set a)
deleteFindMin t
  = case t of
      Bin _ x Tip r -> (x,r)
      Bin _ x l r   -> let (xm,l') = deleteFindMin l in (xm,balance x l' r)
      Tip           -> (error "Set.deleteFindMin: can not return the minimal element of an empty set", Tip)

-- | /O(log n)/. Delete and find the maximal element.
deleteFindMax :: Set a -> (a,Set a)
deleteFindMax t
  = case t of
      Bin _ x l Tip -> (x,l)
      Bin _ x l r   -> let (xm,r') = deleteFindMax r in (xm,balance x l r')
      Tip           -> (error "Set.deleteFindMax: can not return the maximal element of an empty set", Tip)


{--------------------------------------------------------------------
  [balance x l r] balances two trees with value x.
  The sizes of the trees should balance after decreasing the
  size of one of them. (a rotation).

  [delta] is the maximal relative difference between the sizes of
          two trees, it corresponds with the [w] in Adams' paper,
          or equivalently, [1/delta] corresponds with the $\alpha$
          in Nievergelt's paper. Adams shows that [delta] should
          be larger than 3.745 in order to garantee that the
          rotations can always restore balance.

  [ratio] is the ratio between an outer and inner sibling of the
          heavier subtree in an unbalanced setting. It determines
          whether a double or single rotation should be performed
          to restore balance. It is correspondes with the inverse
          of $\alpha$ in Adam's article.

  Note that:
  - [delta] should be larger than 4.646 with a [ratio] of 2.
  - [delta] should be larger than 3.745 with a [ratio] of 1.534.

  - A lower [delta] leads to a more 'perfectly' balanced tree.
  - A higher [delta] performs less rebalancing.

  - Balancing is automatic for random data and a balancing
    scheme is only necessary to avoid pathological worst cases.
    Almost any choice will do in practice

  - Allthough it seems that a rather large [delta] may perform better
    than smaller one, measurements have shown that the smallest [delta]
    of 4 is actually the fastest on a wide range of operations. It
    especially improves performance on worst-case scenarios like
    a sequence of ordered insertions.

  Note: in contrast to Adams' paper, we use a ratio of (at least) 2
  to decide whether a single or double rotation is needed. Allthough
  he actually proves that this ratio is needed to maintain the
  invariants, his implementation uses a (invalid) ratio of 1.
  He is aware of the problem though since he has put a comment in his
  original source code that he doesn't care about generating a
  slightly inbalanced tree since it doesn't seem to matter in practice.
  However (since we use quickcheck :-) we will stick to strictly balanced
  trees.
--------------------------------------------------------------------}
delta,ratio :: Int
delta = 4
ratio = 2

balance :: a -> Set a -> Set a -> Set a
balance x l r
  | sizeL + sizeR <= 1    = Bin sizeX x l r
  | sizeR >= delta*sizeL  = rotateL x l r
  | sizeL >= delta*sizeR  = rotateR x l r
  | otherwise             = Bin sizeX x l r
  where
    sizeL = size l
    sizeR = size r
    sizeX = sizeL + sizeR + 1

-- rotate
rotateL x l r@(Bin _ _ ly ry)
  | size ly < ratio*size ry = singleL x l r
  | otherwise               = doubleL x l r

rotateR x l@(Bin _ _ ly ry) r
  | size ry < ratio*size ly = singleR x l r
  | otherwise               = doubleR x l r

-- basic rotations
singleL x1 t1 (Bin _ x2 t2 t3)  = bin x2 (bin x1 t1 t2) t3
singleR x1 (Bin _ x2 t1 t2) t3  = bin x2 t1 (bin x1 t2 t3)

doubleL x1 t1 (Bin _ x2 (Bin _ x3 t2 t3) t4) = bin x3 (bin x1 t1 t2) (bin x2 t3 t4)
doubleR x1 (Bin _ x2 t1 (Bin _ x3 t2 t3)) t4 = bin x3 (bin x2 t1 t2) (bin x1 t3 t4)


{--------------------------------------------------------------------
  The bin constructor maintains the size of the tree
--------------------------------------------------------------------}
bin :: a -> Set a -> Set a -> Set a
bin x l r
  = Bin (size l + size r + 1) x l r


{--------------------------------------------------------------------
  Utilities
--------------------------------------------------------------------}
foldlStrict f z xs
  = case xs of
      []     -> z
      (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)


{--------------------------------------------------------------------
  Debugging
--------------------------------------------------------------------}
-- | /O(n)/. Show the tree that implements the set. The tree is shown
-- in a compressed, hanging format.
showTree :: Show a => Set a -> String
showTree s
  = showTreeWith True False s


{- | /O(n)/. The expression (@showTreeWith hang wide map@) shows
 the tree that implements the set. If @hang@ is
 @True@, a /hanging/ tree is shown otherwise a rotated tree is shown. If
 @wide@ is true, an extra wide version is shown.

> Set> putStrLn $ showTreeWith True False $ fromDistinctAscList [1..5]
> 4
> +--2
> |  +--1
> |  +--3
> +--5
>
> Set> putStrLn $ showTreeWith True True $ fromDistinctAscList [1..5]
> 4
> |
> +--2
> |  |
> |  +--1
> |  |
> |  +--3
> |
> +--5
>
> Set> putStrLn $ showTreeWith False True $ fromDistinctAscList [1..5]
> +--5
> |
> 4
> |
> |  +--3
> |  |
> +--2
>    |
>    +--1

-}
showTreeWith :: Show a => Bool -> Bool -> Set a -> String
showTreeWith hang wide t
  | hang      = (showsTreeHang wide [] t) ""
  | otherwise = (showsTree wide [] [] t) ""

showsTree :: Show a => Bool -> [String] -> [String] -> Set a -> ShowS
showsTree wide lbars rbars t
  = case t of
      Tip -> showsBars lbars . showString "|\n"
      Bin sz x Tip Tip
          -> showsBars lbars . shows x . showString "\n"
      Bin sz x l r
          -> showsTree wide (withBar rbars) (withEmpty rbars) r .
             showWide wide rbars .
             showsBars lbars . shows x . showString "\n" .
             showWide wide lbars .
             showsTree wide (withEmpty lbars) (withBar lbars) l

showsTreeHang :: Show a => Bool -> [String] -> Set a -> ShowS
showsTreeHang wide bars t
  = case t of
      Tip -> showsBars bars . showString "|\n"
      Bin sz x Tip Tip
          -> showsBars bars . shows x . showString "\n"
      Bin sz x l r
          -> showsBars bars . shows x . showString "\n" .
             showWide wide bars .
             showsTreeHang wide (withBar bars) l .
             showWide wide bars .
             showsTreeHang wide (withEmpty bars) r


showWide wide bars
  | wide      = showString (concat (reverse bars)) . showString "|\n"
  | otherwise = id

showsBars :: [String] -> ShowS
showsBars bars
  = case bars of
      [] -> id
      _  -> showString (concat (reverse (tail bars))) . showString node

node           = "+--"
withBar bars   = "|  ":bars
withEmpty bars = "   ":bars

{--------------------------------------------------------------------
  Assertions
--------------------------------------------------------------------}
-- | /O(n)/. Test if the internal set structure is valid.
valid :: Ord a => Set a -> Bool
valid t
  = balanced t && ordered t && validsize t

ordered t
  = bounded (const True) (const True) t
  where
    bounded lo hi t
      = case t of
          Tip           -> True
          Bin sz x l r  -> (lo x) && (hi x) && bounded lo (<x) l && bounded (>x) hi r

balanced :: Set a -> Bool
balanced t
  = case t of
      Tip           -> True
      Bin sz x l r  -> (size l + size r <= 1 || (size l <= delta*size r && size r <= delta*size l)) &&
                       balanced l && balanced r


validsize t
  = (realsize t == Just (size t))
  where
    realsize t
      = case t of
          Tip          -> Just 0
          Bin sz x l r -> case (realsize l,realsize r) of
                            (Just n,Just m)  | n+m+1 == sz  -> Just sz
                            other            -> Nothing

{-
{--------------------------------------------------------------------
  Testing
--------------------------------------------------------------------}
testTree :: [Int] -> Set Int
testTree xs   = fromList xs
test1 = testTree [1..20]
test2 = testTree [30,29..10]
test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]

{--------------------------------------------------------------------
  QuickCheck
--------------------------------------------------------------------}
qcheck prop
  = check config prop
  where
    config = Config
      { configMaxTest = 500
      , configMaxFail = 5000
      , configSize    = \n -> (div n 2 + 3)
      , configEvery   = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
      }


{--------------------------------------------------------------------
  Arbitrary, reasonably balanced trees
--------------------------------------------------------------------}
instance (Enum a) => Arbitrary (Set a) where
  arbitrary = sized (arbtree 0 maxkey)
            where maxkey  = 10000

arbtree :: (Enum a) => Int -> Int -> Int -> Gen (Set a)
arbtree lo hi n
  | n <= 0        = return Tip
  | lo >= hi      = return Tip
  | otherwise     = do{ i  <- choose (lo,hi)
                      ; m  <- choose (1,30)
                      ; let (ml,mr)  | m==(1::Int)= (1,2)
                                     | m==2       = (2,1)
                                     | m==3       = (1,1)
                                     | otherwise  = (2,2)
                      ; l  <- arbtree lo (i-1) (n `div` ml)
                      ; r  <- arbtree (i+1) hi (n `div` mr)
                      ; return (bin (toEnum i) l r)
                      }


{--------------------------------------------------------------------
  Valid tree's
--------------------------------------------------------------------}
forValid :: (Enum a,Show a,Testable b) => (Set a -> b) -> Property
forValid f
  = forAll arbitrary $ \t ->
--    classify (balanced t) "balanced" $
    classify (size t == 0) "empty" $
    classify (size t > 0  && size t <= 10) "small" $
    classify (size t > 10 && size t <= 64) "medium" $
    classify (size t > 64) "large" $
    balanced t ==> f t

forValidIntTree :: Testable a => (Set Int -> a) -> Property
forValidIntTree f
  = forValid f

forValidUnitTree :: Testable a => (Set Int -> a) -> Property
forValidUnitTree f
  = forValid f


prop_Valid
  = forValidUnitTree $ \t -> valid t

{--------------------------------------------------------------------
  Single, Insert, Delete
--------------------------------------------------------------------}
prop_Single :: Int -> Bool
prop_Single x
  = (insert x empty == single x)

prop_InsertValid :: Int -> Property
prop_InsertValid k
  = forValidUnitTree $ \t -> valid (insert k t)

prop_InsertDelete :: Int -> Set Int -> Property
prop_InsertDelete k t
  = not (member k t) ==> delete k (insert k t) == t

prop_DeleteValid :: Int -> Property
prop_DeleteValid k
  = forValidUnitTree $ \t ->
    valid (delete k (insert k t))

{--------------------------------------------------------------------
  Balance
--------------------------------------------------------------------}
prop_Join :: Int -> Property
prop_Join x
  = forValidUnitTree $ \t ->
    let (l,r) = split x t
    in valid (join x l r)

prop_Merge :: Int -> Property
prop_Merge x
  = forValidUnitTree $ \t ->
    let (l,r) = split x t
    in valid (merge l r)


{--------------------------------------------------------------------
  Union
--------------------------------------------------------------------}
prop_UnionValid :: Property
prop_UnionValid
  = forValidUnitTree $ \t1 ->
    forValidUnitTree $ \t2 ->
    valid (union t1 t2)

prop_UnionInsert :: Int -> Set Int -> Bool
prop_UnionInsert x t
  = union t (single x) == insert x t

prop_UnionAssoc :: Set Int -> Set Int -> Set Int -> Bool
prop_UnionAssoc t1 t2 t3
  = union t1 (union t2 t3) == union (union t1 t2) t3

prop_UnionComm :: Set Int -> Set Int -> Bool
prop_UnionComm t1 t2
  = (union t1 t2 == union t2 t1)


prop_DiffValid
  = forValidUnitTree $ \t1 ->
    forValidUnitTree $ \t2 ->
    valid (difference t1 t2)

prop_Diff :: [Int] -> [Int] -> Bool
prop_Diff xs ys
  =  toAscList (difference (fromList xs) (fromList ys))
    == List.sort ((List.\\) (nub xs)  (nub ys))

prop_IntValid
  = forValidUnitTree $ \t1 ->
    forValidUnitTree $ \t2 ->
    valid (intersection t1 t2)

prop_Int :: [Int] -> [Int] -> Bool
prop_Int xs ys
  =  toAscList (intersection (fromList xs) (fromList ys))
    == List.sort (nub ((List.intersect) (xs)  (ys)))

{--------------------------------------------------------------------
  Lists
--------------------------------------------------------------------}
prop_Ordered
  = forAll (choose (5,100)) $ \n ->
    let xs = [0..n::Int]
    in fromAscList xs == fromList xs

prop_List :: [Int] -> Bool
prop_List xs
  = (sort (nub xs) == toList (fromList xs))
-}