-----------------------------------------------------------------------------
-- |
-- Module : Data.Set
-- Copyright : (c) Daan Leijen 2002
-- License : BSD-style
-- Maintainer : [email protected]
-- 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))
-}
|
