--------------------------------------------------------------------------------

{-| 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

, disjoint

-- * Filter

, filter

, partition

, split

, splitMember

, all

, any

-- * Fold

, fold

, image

-- * Min\/Max

, findMin

, findMax

, deleteMin

, deleteMax

, deleteFindMin

, deleteFindMax

-- * Conversion

-- ** List

, elems

, toList

, fromList

-- ** Maybe

, maybeToSet

, maybeSetToSet

-- ** Ordered list

, toAscList

, fromAscList

, fromDistinctAscList

-- * Debugging

, showTree

, showTreeWith

, valid

) where

import Prelude hiding (filter, all, any)

{-

-- 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

{--------------------------------------------------------------------

Disjointness

--------------------------------------------------------------------}

-- | /O(n+m)/. Are two sets disjoint?

disjoint :: Ord a => Set a -> Set a -> Bool

s1 `disjoint` s2 = s1 `intersection` s2 == empty

{--------------------------------------------------------------------

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

-- | /O(n)/. Do all elements satisfy the predicate?

all :: Ord a => (a -> Bool) -> Set a -> Bool

all p Tip = True

all p (Bin _ x l r)

| p x = (all p l) && (all p r)

| otherwise = False

-- | /O(n)/. Does some element satisfy the predicate?

any :: Ord a => (a -> Bool) -> Set a -> Bool

any p Tip = False

any p (Bin _ x l r)

| p x = True

| otherwise = (any p l) || (any 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

{--------------------------------------------------------------------

Maybe (Added by T.M.)

--------------------------------------------------------------------}

-- | Convert Maybe a to Set a; Nothing becomes the empty set

maybeToSet :: Maybe a -> Set a

maybeToSet Nothing = empty

maybeToSet (Just x) = single x

-- | Convert Maybe (Set a) to Set a; Nothing becomes the empty set

maybeSetToSet :: Maybe (Set a) -> Set a

maybeSetToSet Nothing = empty

maybeSetToSet (Just x) = x

{--------------------------------------------------------------------

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))

-}