+++ /dev/null
------------------------------------------------------------------------------
--- |
--- Module : Data.Set
--- Copyright : (c) Daan Leijen 2002
--- License : BSD-style
--- Maintainer : libraries@haskell.org
--- Stability : provisional
--- Portability : portable
---
--- An efficient implementation of sets.
---
--- Since many function names (but not the type name) clash with
--- "Prelude" names, this module is usually imported @qualified@, e.g.
---
--- > import Data.Set (Set)
--- > import qualified Data.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 preferred 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 Data.Set (
- -- * Set type
- Set -- instance Eq,Ord,Show,Read,Data,Typeable
-
- -- * Operators
- , (\\)
-
- -- * Query
- , null
- , size
- , member
- , notMember
- , 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
- , maxView
- , minView
-
- -- * Conversion
-
- -- ** List
- , elems
- , toList
- , fromList
-
- -- ** Ordered list
- , toAscList
- , fromAscList
- , fromDistinctAscList
-
- -- * Debugging
- , showTree
- , showTreeWith
- , valid
- ) where
-
-import Prelude hiding (filter,foldr,null,map)
-import qualified Data.List as List
-import Data.Monoid (Monoid(..))
-import Data.Typeable
-import Data.Foldable (Foldable(foldMap))
-
-{-
--- just for testing
-import QuickCheck
-import List (nub,sort)
-import qualified List
--}
-
-#if __GLASGOW_HASKELL__
-import Text.Read
-import Data.Generics.Basics
-import Data.Generics.Instances
-#endif
-
-{--------------------------------------------------------------------
- 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
-
-instance Ord a => Monoid (Set a) where
- mempty = empty
- mappend = union
- mconcat = unions
-
-instance Foldable Set where
- foldMap f Tip = mempty
- foldMap f (Bin _s k l r) = foldMap f l `mappend` f k `mappend` foldMap f r
-
-#if __GLASGOW_HASKELL__
-
-{--------------------------------------------------------------------
- A Data instance
---------------------------------------------------------------------}
-
--- This instance preserves data abstraction at the cost of inefficiency.
--- We omit reflection services for the sake of data abstraction.
-
-instance (Data a, Ord a) => Data (Set a) where
- gfoldl f z set = z fromList `f` (toList set)
- toConstr _ = error "toConstr"
- gunfold _ _ = error "gunfold"
- dataTypeOf _ = mkNorepType "Data.Set.Set"
- dataCast1 f = gcast1 f
-
-#endif
-
-{--------------------------------------------------------------------
- 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
-
--- | /O(log n)/. Is the element not in the set?
-notMember :: Ord a => a -> Set a -> Bool
-notMember x t = not $ member x t
-
-{--------------------------------------------------------------------
- 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 = hedgeUnion (const LT) (const GT) t1 t2
-
-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.
--- Elements of the result come from the first set, so for example
---
--- > import qualified Data.Set as S
--- > data AB = A | B deriving Show
--- > instance Ord AB where compare _ _ = EQ
--- > instance Eq AB where _ == _ = True
--- > main = print (S.singleton A `S.intersection` S.singleton B,
--- > S.singleton B `S.intersection` S.singleton A)
---
--- prints @(fromList [A],fromList [B])@.
-intersection :: Ord a => Set a -> Set a -> Set a
-intersection Tip t = Tip
-intersection t Tip = Tip
-intersection t1@(Bin s1 x1 l1 r1) t2@(Bin s2 x2 l2 r2) =
- if s1 >= s2 then
- let (lt,found,gt) = splitLookup x2 t1
- tl = intersection lt l2
- tr = intersection gt r2
- in case found of
- Just x -> join x tl tr
- Nothing -> merge tl tr
- else let (lt,found,gt) = splitMember x1 t2
- tl = intersection l1 lt
- tr = intersection r1 gt
- in if found then join x1 tl tr
- else merge tl tr
-
-{--------------------------------------------------------------------
- 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 p xs = showParen (p > 10) $
- showString "fromList " . shows (toList xs)
-
-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
-
-{--------------------------------------------------------------------
- Read
---------------------------------------------------------------------}
-instance (Read a, Ord a) => Read (Set a) where
-#ifdef __GLASGOW_HASKELL__
- readPrec = parens $ prec 10 $ do
- Ident "fromList" <- lexP
- xs <- readPrec
- return (fromList xs)
-
- readListPrec = readListPrecDefault
-#else
- readsPrec p = readParen (p > 10) $ \ r -> do
- ("fromList",s) <- lex r
- (xs,t) <- reads s
- return (fromList xs,t)
-#endif
-
-{--------------------------------------------------------------------
- Typeable/Data
---------------------------------------------------------------------}
-
-#include "Typeable.h"
-INSTANCE_TYPEABLE1(Set,setTc,"Set")
-
-{--------------------------------------------------------------------
- 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 t = let (l,m,r) = splitLookup x t in
- (l,maybe False (const True) m,r)
-
--- | /O(log n)/. Performs a 'split' but also returns the pivot
--- element that was found in the original set.
-splitLookup :: Ord a => a -> Set a -> (Set a,Maybe a,Set a)
-splitLookup x Tip = (Tip,Nothing,Tip)
-splitLookup x (Bin sy y l r)
- = case compare x y of
- LT -> let (lt,found,gt) = splitLookup x l in (lt,found,join y gt r)
- GT -> let (lt,found,gt) = splitLookup x r in (join y l lt,found,gt)
- EQ -> (l,Just y,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)
-
--- | /O(log n)/. Retrieves the minimal key of the set, and the set stripped from that element
--- @fail@s (in the monad) when passed an empty set.
-minView :: Monad m => Set a -> m (a, Set a)
-minView Tip = fail "Set.minView: empty set"
-minView x = return (deleteFindMin x)
-
--- | /O(log n)/. Retrieves the maximal key of the set, and the set stripped from that element
--- @fail@s (in the monad) when passed an empty set.
-maxView :: Monad m => Set a -> m (a, Set a)
-maxView Tip = fail "Set.maxView: empty set"
-maxView x = return (deleteFindMax x)
-
-
-{--------------------------------------------------------------------
- [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))
--}
projects / ghc-base.git / blobdiff