{-# OPTIONS -cpp -fglasgow-exts #-}
-----------------------------------------------------------------------------
-- |
-- Module : Data.IntSet
-- Copyright : (c) Daan Leijen 2002
-- License : BSD-style
-- Maintainer : [email protected]
-- Stability : provisional
-- Portability : portable
--
-- An efficient implementation of integer sets.
--
-- Since many function names (but not the type name) clash with
-- "Prelude" names, this module is usually imported @qualified@, e.g.
--
-- > import Data.IntSet (IntSet)
-- > import qualified Data.IntSet as IntSet
--
-- The implementation is based on /big-endian patricia trees/. This data
-- structure performs especially well on binary operations like 'union'
-- and 'intersection'. However, my benchmarks show that it is also
-- (much) faster on insertions and deletions when compared to a generic
-- size-balanced set implementation (see "Data.Set").
--
-- * Chris Okasaki and Andy Gill, \"/Fast Mergeable Integer Maps/\",
-- Workshop on ML, September 1998, pages 77-86,
-- <http://www.cse.ogi.edu/~andy/pub/finite.htm>
--
-- * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve
-- Information Coded In Alphanumeric/\", Journal of the ACM, 15(4),
-- October 1968, pages 514-534.
--
-- Many operations have a worst-case complexity of /O(min(n,W))/.
-- This means that the operation can become linear in the number of
-- elements with a maximum of /W/ -- the number of bits in an 'Int'
-- (32 or 64).
-----------------------------------------------------------------------------
module Data.IntSet (
-- * Set type
IntSet -- instance Eq,Show
-- * Operators
, (\\)
-- * Query
, null
, size
, member
, notMember
, isSubsetOf
, isProperSubsetOf
-- * Construction
, empty
, singleton
, insert
, delete
-- * Combine
, union, unions
, difference
, intersection
-- * Filter
, filter
, partition
, split
, splitMember
-- * Min\/Max
, findMin
, findMax
, deleteMin
, deleteMax
, deleteFindMin
, deleteFindMax
, maxView
, minView
-- * Map
, map
-- * Fold
, fold
-- * Conversion
-- ** List
, elems
, toList
, fromList
-- ** Ordered list
, toAscList
, fromAscList
, fromDistinctAscList
-- * Debugging
, showTree
, showTreeWith
) where
import Prelude hiding (lookup,filter,foldr,foldl,null,map)
import Data.Bits
import qualified Data.List as List
import Data.Monoid (Monoid(..))
import Data.Typeable
{-
-- just for testing
import QuickCheck
import List (nub,sort)
import qualified List
-}
#if __GLASGOW_HASKELL__
import Text.Read
import Data.Generics.Basics (Data(..), mkNorepType)
import Data.Generics.Instances ()
#endif
#if __GLASGOW_HASKELL__ >= 503
import GHC.Exts ( Word(..), Int(..), shiftRL# )
#elif __GLASGOW_HASKELL__
import Word
import GlaExts ( Word(..), Int(..), shiftRL# )
#else
import Data.Word
#endif
infixl 9 \\{-This comment teaches CPP correct behaviour -}
-- A "Nat" is a natural machine word (an unsigned Int)
type Nat = Word
natFromInt :: Int -> Nat
natFromInt i = fromIntegral i
intFromNat :: Nat -> Int
intFromNat w = fromIntegral w
shiftRL :: Nat -> Int -> Nat
#if __GLASGOW_HASKELL__
{--------------------------------------------------------------------
GHC: use unboxing to get @shiftRL@ inlined.
--------------------------------------------------------------------}
shiftRL (W# x) (I# i)
= W# (shiftRL# x i)
#else
shiftRL x i = shiftR x i
#endif
{--------------------------------------------------------------------
Operators
--------------------------------------------------------------------}
-- | /O(n+m)/. See 'difference'.
(\\) :: IntSet -> IntSet -> IntSet
m1 \\ m2 = difference m1 m2
{--------------------------------------------------------------------
Types
--------------------------------------------------------------------}
-- | A set of integers.
data IntSet = Nil
| Tip {-# UNPACK #-} !Int
| Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet
-- Invariant: Nil is never found as a child of Bin.
type Prefix = Int
type Mask = Int
instance Monoid IntSet where
mempty = empty
mappend = union
mconcat = unions
#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 IntSet where
gfoldl f z is = z fromList `f` (toList is)
toConstr _ = error "toConstr"
gunfold _ _ = error "gunfold"
dataTypeOf _ = mkNorepType "Data.IntSet.IntSet"
#endif
{--------------------------------------------------------------------
Query
--------------------------------------------------------------------}
-- | /O(1)/. Is the set empty?
null :: IntSet -> Bool
null Nil = True
null other = False
-- | /O(n)/. Cardinality of the set.
size :: IntSet -> Int
size t
= case t of
Bin p m l r -> size l + size r
Tip y -> 1
Nil -> 0
-- | /O(min(n,W))/. Is the value a member of the set?
member :: Int -> IntSet -> Bool
member x t
= case t of
Bin p m l r
| nomatch x p m -> False
| zero x m -> member x l
| otherwise -> member x r
Tip y -> (x==y)
Nil -> False
-- | /O(min(n,W))/. Is the element not in the set?
notMember :: Int -> IntSet -> Bool
notMember k = not . member k
-- 'lookup' is used by 'intersection' for left-biasing
lookup :: Int -> IntSet -> Maybe Int
lookup k t
= let nk = natFromInt k in seq nk (lookupN nk t)
lookupN :: Nat -> IntSet -> Maybe Int
lookupN k t
= case t of
Bin p m l r
| zeroN k (natFromInt m) -> lookupN k l
| otherwise -> lookupN k r
Tip kx
| (k == natFromInt kx) -> Just kx
| otherwise -> Nothing
Nil -> Nothing
{--------------------------------------------------------------------
Construction
--------------------------------------------------------------------}
-- | /O(1)/. The empty set.
empty :: IntSet
empty
= Nil
-- | /O(1)/. A set of one element.
singleton :: Int -> IntSet
singleton x
= Tip x
{--------------------------------------------------------------------
Insert
--------------------------------------------------------------------}
-- | /O(min(n,W))/. Add a value to the set. When the value is already
-- an element of the set, it is replaced by the new one, ie. 'insert'
-- is left-biased.
insert :: Int -> IntSet -> IntSet
insert x t
= case t of
Bin p m l r
| nomatch x p m -> join x (Tip x) p t
| zero x m -> Bin p m (insert x l) r
| otherwise -> Bin p m l (insert x r)
Tip y
| x==y -> Tip x
| otherwise -> join x (Tip x) y t
Nil -> Tip x
-- right-biased insertion, used by 'union'
insertR :: Int -> IntSet -> IntSet
insertR x t
= case t of
Bin p m l r
| nomatch x p m -> join x (Tip x) p t
| zero x m -> Bin p m (insert x l) r
| otherwise -> Bin p m l (insert x r)
Tip y
| x==y -> t
| otherwise -> join x (Tip x) y t
Nil -> Tip x
-- | /O(min(n,W))/. Delete a value in the set. Returns the
-- original set when the value was not present.
delete :: Int -> IntSet -> IntSet
delete x t
= case t of
Bin p m l r
| nomatch x p m -> t
| zero x m -> bin p m (delete x l) r
| otherwise -> bin p m l (delete x r)
Tip y
| x==y -> Nil
| otherwise -> t
Nil -> Nil
{--------------------------------------------------------------------
Union
--------------------------------------------------------------------}
-- | The union of a list of sets.
unions :: [IntSet] -> IntSet
unions xs
= foldlStrict union empty xs
-- | /O(n+m)/. The union of two sets.
union :: IntSet -> IntSet -> IntSet
union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
| shorter m1 m2 = union1
| shorter m2 m1 = union2
| p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2)
| otherwise = join p1 t1 p2 t2
where
union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2
| zero p2 m1 = Bin p1 m1 (union l1 t2) r1
| otherwise = Bin p1 m1 l1 (union r1 t2)
union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2
| zero p1 m2 = Bin p2 m2 (union t1 l2) r2
| otherwise = Bin p2 m2 l2 (union t1 r2)
union (Tip x) t = insert x t
union t (Tip x) = insertR x t -- right bias
union Nil t = t
union t Nil = t
{--------------------------------------------------------------------
Difference
--------------------------------------------------------------------}
-- | /O(n+m)/. Difference between two sets.
difference :: IntSet -> IntSet -> IntSet
difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
| shorter m1 m2 = difference1
| shorter m2 m1 = difference2
| p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2)
| otherwise = t1
where
difference1 | nomatch p2 p1 m1 = t1
| zero p2 m1 = bin p1 m1 (difference l1 t2) r1
| otherwise = bin p1 m1 l1 (difference r1 t2)
difference2 | nomatch p1 p2 m2 = t1
| zero p1 m2 = difference t1 l2
| otherwise = difference t1 r2
difference t1@(Tip x) t2
| member x t2 = Nil
| otherwise = t1
difference Nil t = Nil
difference t (Tip x) = delete x t
difference t Nil = t
{--------------------------------------------------------------------
Intersection
--------------------------------------------------------------------}
-- | /O(n+m)/. The intersection of two sets.
intersection :: IntSet -> IntSet -> IntSet
intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
| shorter m1 m2 = intersection1
| shorter m2 m1 = intersection2
| p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
| otherwise = Nil
where
intersection1 | nomatch p2 p1 m1 = Nil
| zero p2 m1 = intersection l1 t2
| otherwise = intersection r1 t2
intersection2 | nomatch p1 p2 m2 = Nil
| zero p1 m2 = intersection t1 l2
| otherwise = intersection t1 r2
intersection t1@(Tip x) t2
| member x t2 = t1
| otherwise = Nil
intersection t (Tip x)
= case lookup x t of
Just y -> Tip y
Nothing -> Nil
intersection Nil t = Nil
intersection t Nil = Nil
{--------------------------------------------------------------------
Subset
--------------------------------------------------------------------}
-- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
isProperSubsetOf :: IntSet -> IntSet -> Bool
isProperSubsetOf t1 t2
= case subsetCmp t1 t2 of
LT -> True
ge -> False
subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
| shorter m1 m2 = GT
| shorter m2 m1 = subsetCmpLt
| p1 == p2 = subsetCmpEq
| otherwise = GT -- disjoint
where
subsetCmpLt | nomatch p1 p2 m2 = GT
| zero p1 m2 = subsetCmp t1 l2
| otherwise = subsetCmp t1 r2
subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
(GT,_ ) -> GT
(_ ,GT) -> GT
(EQ,EQ) -> EQ
other -> LT
subsetCmp (Bin p m l r) t = GT
subsetCmp (Tip x) (Tip y)
| x==y = EQ
| otherwise = GT -- disjoint
subsetCmp (Tip x) t
| member x t = LT
| otherwise = GT -- disjoint
subsetCmp Nil Nil = EQ
subsetCmp Nil t = LT
-- | /O(n+m)/. Is this a subset?
-- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
isSubsetOf :: IntSet -> IntSet -> Bool
isSubsetOf t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
| shorter m1 m2 = False
| shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2
else isSubsetOf t1 r2)
| otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
isSubsetOf (Bin p m l r) t = False
isSubsetOf (Tip x) t = member x t
isSubsetOf Nil t = True
{--------------------------------------------------------------------
Filter
--------------------------------------------------------------------}
-- | /O(n)/. Filter all elements that satisfy some predicate.
filter :: (Int -> Bool) -> IntSet -> IntSet
filter pred t
= case t of
Bin p m l r
-> bin p m (filter pred l) (filter pred r)
Tip x
| pred x -> t
| otherwise -> Nil
Nil -> Nil
-- | /O(n)/. partition the set according to some predicate.
partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
partition pred t
= case t of
Bin p m l r
-> let (l1,l2) = partition pred l
(r1,r2) = partition pred r
in (bin p m l1 r1, bin p m l2 r2)
Tip x
| pred x -> (t,Nil)
| otherwise -> (Nil,t)
Nil -> (Nil,Nil)
-- | /O(min(n,W))/. The expression (@'split' x set@) is a pair @(set1,set2)@
-- where all elements in @set1@ are lower than @x@ and all elements in
-- @set2@ larger than @x@.
--
-- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
split :: Int -> IntSet -> (IntSet,IntSet)
split x t
= case t of
Bin p m l r
| m < 0 -> if x >= 0 then let (lt,gt) = split' x l in (union r lt, gt)
else let (lt,gt) = split' x r in (lt, union gt l)
-- handle negative numbers.
| otherwise -> split' x t
Tip y
| x>y -> (t,Nil)
| x<y -> (Nil,t)
| otherwise -> (Nil,Nil)
Nil -> (Nil, Nil)
split' :: Int -> IntSet -> (IntSet,IntSet)
split' x t
= case t of
Bin p m l r
| match x p m -> if zero x m then let (lt,gt) = split' x l in (lt,union gt r)
else let (lt,gt) = split' x r in (union l lt,gt)
| otherwise -> if x < p then (Nil, t)
else (t, Nil)
Tip y
| x>y -> (t,Nil)
| x<y -> (Nil,t)
| otherwise -> (Nil,Nil)
Nil -> (Nil,Nil)
-- | /O(min(n,W))/. Performs a 'split' but also returns whether the pivot
-- element was found in the original set.
splitMember :: Int -> IntSet -> (IntSet,Bool,IntSet)
splitMember x t
= case t of
Bin p m l r
| m < 0 -> if x >= 0 then let (lt,found,gt) = splitMember' x l in (union r lt, found, gt)
else let (lt,found,gt) = splitMember' x r in (lt, found, union gt l)
-- handle negative numbers.
| otherwise -> splitMember' x t
Tip y
| x>y -> (t,False,Nil)
| x<y -> (Nil,False,t)
| otherwise -> (Nil,True,Nil)
Nil -> (Nil,False,Nil)
splitMember' :: Int -> IntSet -> (IntSet,Bool,IntSet)
splitMember' x t
= case t of
Bin p m l r
| match x p m -> if zero x m then let (lt,found,gt) = splitMember x l in (lt,found,union gt r)
else let (lt,found,gt) = splitMember x r in (union l lt,found,gt)
| otherwise -> if x < p then (Nil, False, t)
else (t, False, Nil)
Tip y
| x>y -> (t,False,Nil)
| x<y -> (Nil,False,t)
| otherwise -> (Nil,True,Nil)
Nil -> (Nil,False,Nil)
{----------------------------------------------------------------------
Min/Max
----------------------------------------------------------------------}
-- | /O(min(n,W))/. 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) => IntSet -> m (Int, IntSet)
maxView t
= case t of
Bin p m l r | m < 0 -> let (result,t') = maxViewUnsigned l in return (result, bin p m t' r)
Bin p m l r -> let (result,t') = maxViewUnsigned r in return (result, bin p m l t')
Tip y -> return (y,Nil)
Nil -> fail "maxView: empty set has no maximal element"
maxViewUnsigned :: IntSet -> (Int, IntSet)
maxViewUnsigned t
= case t of
Bin p m l r -> let (result,t') = maxViewUnsigned r in (result, bin p m l t')
Tip y -> (y, Nil)
-- | /O(min(n,W))/. 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) => IntSet -> m (Int, IntSet)
minView t
= case t of
Bin p m l r | m < 0 -> let (result,t') = minViewUnsigned r in return (result, bin p m l t')
Bin p m l r -> let (result,t') = minViewUnsigned l in return (result, bin p m t' r)
Tip y -> return (y, Nil)
Nil -> fail "minView: empty set has no minimal element"
minViewUnsigned :: IntSet -> (Int, IntSet)
minViewUnsigned t
= case t of
Bin p m l r -> let (result,t') = minViewUnsigned l in (result, bin p m t' r)
Tip y -> (y, Nil)
-- Duplicate the Identity monad here because base < mtl.
newtype Identity a = Identity { runIdentity :: a }
instance Monad Identity where
return a = Identity a
m >>= k = k (runIdentity m)
-- | /O(min(n,W))/. Delete and find the minimal element.
--
-- > deleteFindMin set = (findMin set, deleteMin set)
deleteFindMin :: IntSet -> (Int, IntSet)
deleteFindMin = runIdentity . minView
-- | /O(min(n,W))/. Delete and find the maximal element.
--
-- > deleteFindMax set = (findMax set, deleteMax set)
deleteFindMax :: IntSet -> (Int, IntSet)
deleteFindMax = runIdentity . maxView
-- | /O(min(n,W))/. The minimal element of a set.
findMin :: IntSet -> Int
findMin = fst . runIdentity . minView
-- | /O(min(n,W))/. The maximal element of a set.
findMax :: IntSet -> Int
findMax = fst . runIdentity . maxView
-- | /O(min(n,W))/. Delete the minimal element.
deleteMin :: IntSet -> IntSet
deleteMin = snd . runIdentity . minView
-- | /O(min(n,W))/. Delete the maximal element.
deleteMax :: IntSet -> IntSet
deleteMax = snd . runIdentity . maxView
{----------------------------------------------------------------------
Map
----------------------------------------------------------------------}
-- | /O(n*min(n,W))/.
-- @'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 :: (Int->Int) -> IntSet -> IntSet
map f = fromList . List.map f . toList
{--------------------------------------------------------------------
Fold
--------------------------------------------------------------------}
-- | /O(n)/. Fold over the elements of a set in an unspecified order.
--
-- > sum set == fold (+) 0 set
-- > elems set == fold (:) [] set
fold :: (Int -> b -> b) -> b -> IntSet -> b
fold f z t
= case t of
Bin 0 m l r | m < 0 -> foldr f (foldr f z l) r
-- put negative numbers before.
Bin p m l r -> foldr f z t
Tip x -> f x z
Nil -> z
foldr :: (Int -> b -> b) -> b -> IntSet -> b
foldr f z t
= case t of
Bin p m l r -> foldr f (foldr f z r) l
Tip x -> f x z
Nil -> z
{--------------------------------------------------------------------
List variations
--------------------------------------------------------------------}
-- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
elems :: IntSet -> [Int]
elems s
= toList s
{--------------------------------------------------------------------
Lists
--------------------------------------------------------------------}
-- | /O(n)/. Convert the set to a list of elements.
toList :: IntSet -> [Int]
toList t
= fold (:) [] t
-- | /O(n)/. Convert the set to an ascending list of elements.
toAscList :: IntSet -> [Int]
toAscList t = toList t
-- | /O(n*min(n,W))/. Create a set from a list of integers.
fromList :: [Int] -> IntSet
fromList xs
= foldlStrict ins empty xs
where
ins t x = insert x t
-- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
fromAscList :: [Int] -> IntSet
fromAscList xs
= fromList xs
-- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
fromDistinctAscList :: [Int] -> IntSet
fromDistinctAscList xs
= fromList xs
{--------------------------------------------------------------------
Eq
--------------------------------------------------------------------}
instance Eq IntSet where
t1 == t2 = equal t1 t2
t1 /= t2 = nequal t1 t2
equal :: IntSet -> IntSet -> Bool
equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
= (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
equal (Tip x) (Tip y)
= (x==y)
equal Nil Nil = True
equal t1 t2 = False
nequal :: IntSet -> IntSet -> Bool
nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
= (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
nequal (Tip x) (Tip y)
= (x/=y)
nequal Nil Nil = False
nequal t1 t2 = True
{--------------------------------------------------------------------
Ord
--------------------------------------------------------------------}
instance Ord IntSet where
compare s1 s2 = compare (toAscList s1) (toAscList s2)
-- tentative implementation. See if more efficient exists.
{--------------------------------------------------------------------
Show
--------------------------------------------------------------------}
instance Show IntSet where
showsPrec p xs = showParen (p > 10) $
showString "fromList " . shows (toList xs)
showSet :: [Int] -> 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 IntSet 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
--------------------------------------------------------------------}
#include "Typeable.h"
INSTANCE_TYPEABLE0(IntSet,intSetTc,"IntSet")
{--------------------------------------------------------------------
Debugging
--------------------------------------------------------------------}
-- | /O(n)/. Show the tree that implements the set. The tree is shown
-- in a compressed, hanging format.
showTree :: IntSet -> 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.
-}
showTreeWith :: Bool -> Bool -> IntSet -> String
showTreeWith hang wide t
| hang = (showsTreeHang wide [] t) ""
| otherwise = (showsTree wide [] [] t) ""
showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS
showsTree wide lbars rbars t
= case t of
Bin p m l r
-> showsTree wide (withBar rbars) (withEmpty rbars) r .
showWide wide rbars .
showsBars lbars . showString (showBin p m) . showString "\n" .
showWide wide lbars .
showsTree wide (withEmpty lbars) (withBar lbars) l
Tip x
-> showsBars lbars . showString " " . shows x . showString "\n"
Nil -> showsBars lbars . showString "|\n"
showsTreeHang :: Bool -> [String] -> IntSet -> ShowS
showsTreeHang wide bars t
= case t of
Bin p m l r
-> showsBars bars . showString (showBin p m) . showString "\n" .
showWide wide bars .
showsTreeHang wide (withBar bars) l .
showWide wide bars .
showsTreeHang wide (withEmpty bars) r
Tip x
-> showsBars bars . showString " " . shows x . showString "\n"
Nil -> showsBars bars . showString "|\n"
showBin p m
= "*" -- ++ show (p,m)
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
{--------------------------------------------------------------------
Helpers
--------------------------------------------------------------------}
{--------------------------------------------------------------------
Join
--------------------------------------------------------------------}
join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
join p1 t1 p2 t2
| zero p1 m = Bin p m t1 t2
| otherwise = Bin p m t2 t1
where
m = branchMask p1 p2
p = mask p1 m
{--------------------------------------------------------------------
@bin@ assures that we never have empty trees within a tree.
--------------------------------------------------------------------}
bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet
bin p m l Nil = l
bin p m Nil r = r
bin p m l r = Bin p m l r
{--------------------------------------------------------------------
Endian independent bit twiddling
--------------------------------------------------------------------}
zero :: Int -> Mask -> Bool
zero i m
= (natFromInt i) .&. (natFromInt m) == 0
nomatch,match :: Int -> Prefix -> Mask -> Bool
nomatch i p m
= (mask i m) /= p
match i p m
= (mask i m) == p
mask :: Int -> Mask -> Prefix
mask i m
= maskW (natFromInt i) (natFromInt m)
zeroN :: Nat -> Nat -> Bool
zeroN i m = (i .&. m) == 0
{--------------------------------------------------------------------
Big endian operations
--------------------------------------------------------------------}
maskW :: Nat -> Nat -> Prefix
maskW i m
= intFromNat (i .&. (complement (m-1) `xor` m))
shorter :: Mask -> Mask -> Bool
shorter m1 m2
= (natFromInt m1) > (natFromInt m2)
branchMask :: Prefix -> Prefix -> Mask
branchMask p1 p2
= intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
{----------------------------------------------------------------------
Finding the highest bit (mask) in a word [x] can be done efficiently in
three ways:
* convert to a floating point value and the mantissa tells us the
[log2(x)] that corresponds with the highest bit position. The mantissa
is retrieved either via the standard C function [frexp] or by some bit
twiddling on IEEE compatible numbers (float). Note that one needs to
use at least [double] precision for an accurate mantissa of 32 bit
numbers.
* use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
* use processor specific assembler instruction (asm).
The most portable way would be [bit], but is it efficient enough?
I have measured the cycle counts of the different methods on an AMD
Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
highestBitMask: method cycles
--------------
frexp 200
float 33
bit 11
asm 12
highestBit: method cycles
--------------
frexp 195
float 33
bit 11
asm 11
Wow, the bit twiddling is on today's RISC like machines even faster
than a single CISC instruction (BSR)!
----------------------------------------------------------------------}
{----------------------------------------------------------------------
[highestBitMask] returns a word where only the highest bit is set.
It is found by first setting all bits in lower positions than the
highest bit and than taking an exclusive or with the original value.
Allthough the function may look expensive, GHC compiles this into
excellent C code that subsequently compiled into highly efficient
machine code. The algorithm is derived from Jorg Arndt's FXT library.
----------------------------------------------------------------------}
highestBitMask :: Nat -> Nat
highestBitMask x
= case (x .|. shiftRL x 1) of
x -> case (x .|. shiftRL x 2) of
x -> case (x .|. shiftRL x 4) of
x -> case (x .|. shiftRL x 8) of
x -> case (x .|. shiftRL x 16) of
x -> case (x .|. shiftRL x 32) of -- for 64 bit platforms
x -> (x `xor` (shiftRL x 1))
{--------------------------------------------------------------------
Utilities
--------------------------------------------------------------------}
foldlStrict f z xs
= case xs of
[] -> z
(x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
{-
{--------------------------------------------------------------------
Testing
--------------------------------------------------------------------}
testTree :: [Int] -> IntSet
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 Arbitrary IntSet where
arbitrary = do{ xs <- arbitrary
; return (fromList xs)
}
{--------------------------------------------------------------------
Single, Insert, Delete
--------------------------------------------------------------------}
prop_Single :: Int -> Bool
prop_Single x
= (insert x empty == singleton x)
prop_InsertDelete :: Int -> IntSet -> Property
prop_InsertDelete k t
= not (member k t) ==> delete k (insert k t) == t
{--------------------------------------------------------------------
Union
--------------------------------------------------------------------}
prop_UnionInsert :: Int -> IntSet -> Bool
prop_UnionInsert x t
= union t (singleton x) == insert x t
prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool
prop_UnionAssoc t1 t2 t3
= union t1 (union t2 t3) == union (union t1 t2) t3
prop_UnionComm :: IntSet -> IntSet -> Bool
prop_UnionComm t1 t2
= (union t1 t2 == union t2 t1)
prop_Diff :: [Int] -> [Int] -> Bool
prop_Diff xs ys
= toAscList (difference (fromList xs) (fromList ys))
== List.sort ((List.\\) (nub xs) (nub ys))
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) == toAscList (fromList xs))
-}
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