{-# OPTIONS -cpp #-}

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

-- |

-- Module : Common.Lib.Set

-- Copyright : (c) Daan Leijen 2002

-- License : BSD-style

-- Maintainer : libraries@haskell.org

-- Stability : provisional

-- Portability : portable

--

-- An efficient implementation of sets.

--

-- This module is intended to be imported @qualified@, to avoid name

-- clashes with "Prelude" functions. eg.

--

-- > import Common.Lib.Set as Set

--

-- 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.

--

-- Note that the implementation is /left-biased/ -- the elements of a

-- first argument are always perferred to the second, for example in

-- 'union' or 'insert'. Of course, left-biasing can only be observed

-- when equality is an equivalence relation instead of structural

-- equality.

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

module Common.Lib.Set (

-- * Set type

Set -- instance Eq,Show

-- * Operators

, (\\)

-- * Query

, null

, size

, compareSize

, member

, lookupSingleton

, isSingleton

, isSubsetOf

, isProperSubsetOf

-- * Construction

, empty

, singleton

, insert

, delete

-- * Combine

, union, unions

, difference

, intersection

-- * Filter

, filter

, partition

, split

, splitMember

-- * Map

, map

, mapMonotonic

-- * Fold

, fold

-- * 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,foldr,null,map)

#if __GLASGOW_HASKELL__<=602

import qualified Data.List as List

{-

-- 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 {-# UNPACK #-} !Size a !(Set a) !(Set a)

type Size = Int

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

Query

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

-- | /O(1)/. Is this the empty set?

null :: Set a -> Bool

null 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.

singleton :: a -> Set a

singleton x

= Bin 1 x Tip Tip

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

Insertion, Deletion

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

-- | /O(log n)/. Insert an element in a set.

-- If the set already contains an element equal to the given value,

-- it is replaced with the new value.

insert :: Ord a => a -> Set a -> Set a

insert x t

= case t of

Tip -> singleton 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).

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

isProperSubsetOf s1 s2

= (size s1 < size s2) && (isSubsetOf s1 s2)

-- | /O(n+m)/. Is this a subset?

-- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.

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

isSubsetOf t1 t2

= (size t1 <= size t2) && (isSubsetOfX t1 t2)

isSubsetOfX Tip t = True

isSubsetOfX t Tip = False

isSubsetOfX (Bin _ x l r) t

= found && isSubsetOfX l lt && isSubsetOfX r gt

where

(lt,found,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, preferring the first set when

-- equal elements are encountered.

-- The implementation uses the efficient /hedge-union/ algorithm.

-- Hedge-union is more efficient on (bigset `union` smallset).

union :: Ord a => Set a -> Set a -> Set a

union Tip t2 = t2

union t1 Tip = t1

union t1 t2

| 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 is more efficient on (bigset `intersection` smallset).

intersection :: Ord a => Set a -> Set a -> Set a

intersection Tip t = Tip

intersection t Tip = Tip

intersection t1 t2

| 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

(lt,found,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

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

Map

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

-- | /O(n*log n)/.

-- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.

--

-- It's worth noting that the size of the result may be smaller if,

-- for some @(x,y)@, @x \/= y && f x == f y@

map :: (Ord a, Ord b) => (a->b) -> Set a -> Set b

map f = fromList . List.map f . toList

-- | /O(n)/. The

--

-- @'mapMonotonic' f s == 'map' f s@, but works only when @f@ is monotonic.

-- /The precondition is not checked./

-- Semi-formally, we have:

--

-- > and [x < y ==> f x < f y | x <- ls, y <- ls]

-- > ==> mapMonotonic f s == map f s

-- > where ls = toList s

mapMonotonic :: (a->b) -> Set a -> Set b

mapMonotonic f Tip = Tip

mapMonotonic f (Bin sz x l r) =

Bin sz (f x) (mapMonotonic f l) (mapMonotonic f r)

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

Fold

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

-- | /O(n)/. Fold over the elements of a set in an unspecified order.

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

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

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 set from an ascending list in linear time.

-- /The precondition (input list is ascending) is not checked./

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.

-- /The precondition (input list is strictly ascending) is not checked./

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 (singleton x1) (singleton x3)) (singleton 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)

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

Ord

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

instance Ord a => Ord (Set a) where

compare s1 s2 = compare (toAscList s1) (toAscList s2)

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

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@. @x@ is not found in neither @set1@ nor @set2@.

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 -> (Set a,Bool,Set a)

splitMember x Tip = (Tip,False,Tip)

splitMember x (Bin sy y l r)

= case compare x y of

LT -> let (lt,found,gt) = splitMember x l in (lt,found,join y gt r)

GT -> let (lt,found,gt) = splitMember x r in (join y l lt,found,gt)

EQ -> (l,True,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 -> singleton x

Bin sz y l r

-> balance y l (insertMax x r)

insertMin x t

= case t of

Tip -> singleton 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 = (findMin set, deleteMin set)

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 = (findMax set, deleteMax set)

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 == singleton 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 (singleton 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))

-}

#else

import Data.Set

#endif

-- | /O(1)/ try to extract the element from a singleton set

lookupSingleton :: Set a -> Maybe a

lookupSingleton s = if size s == 1 then Just $ findMin s else Nothing

-- | /O(1)/ test if set is a singleton

isSingleton :: Set a -> Bool

isSingleton s = size s == 1

-- | /O(1)/ compare the number with the set's size

compareSize :: Int -> Set a -> Ordering

compareSize i = compare i . size