{-# LANGUAGE ScopedTypeVariables, TemplateHaskell, GADTs #-}
module Main where
--------------------------------------------------------------------------
-- imports
import Test.QuickCheck
import Test.QuickCheck.Poly
import Data.List
( sort
, nub
, (\\)
)
import Data.Maybe
( fromJust
)
import Control.Monad
( liftM
, liftM2
)
--------------------------------------------------------------------------
-- skew heaps
data Heap a
= Node a (Heap a) (Heap a)
| Nil
deriving ( Eq, Ord, Show )
empty :: Heap a
empty = Nil
isEmpty :: Heap a -> Bool
isEmpty Nil = True
isEmpty _ = False
unit :: a -> Heap a
unit x = Node x empty empty
size :: Heap a -> Int
size Nil = 0
size (Node _ h1 h2) = 1 + size h1 + size h2
insert :: Ord a => a -> Heap a -> Heap a
insert x h = unit x `merge` h
removeMin :: Ord a => Heap a -> Maybe (a, Heap a)
removeMin Nil = Nothing
removeMin (Node x h1 h2) = Just (x, h1 `merge` h2)
merge :: Ord a => Heap a -> Heap a -> Heap a
h1 `merge` Nil = h1
Nil `merge` h2 = h2
h1@(Node x h11 h12) `merge` h2@(Node y h21 h22)
| x <= y = Node x (h12 `merge` h2) h11
| otherwise = Node y (h22 `merge` h1) h21
fromList :: Ord a => [a] -> Heap a
fromList xs = merging [ unit x | x <- xs ] []
where
merging [] [] = empty
merging [p] [] = p
merging (p:q:ps) qs = merging ps ((p`merge`q):qs)
merging ps qs = merging (ps ++ reverse qs) []
toList :: Heap a -> [a]
toList h = toList' [h]
where
toList' [] = []
toList' (Nil : hs) = toList' hs
toList' (Node x h1 h2 : hs) = x : toList' (h1:h2:hs)
toSortedList :: Ord a => Heap a -> [a]
toSortedList Nil = []
toSortedList (Node x h1 h2) = x : toSortedList (h1 `merge` h2)
--------------------------------------------------------------------------
-- heap programs
data HeapP a
= Empty
| Unit a
| Insert a (HeapP a)
| SafeRemoveMin (HeapP a)
| Merge (HeapP a) (HeapP a)
| FromList [a]
deriving (Show)
safeRemoveMin :: Ord a => Heap a -> Heap a
safeRemoveMin h = case removeMin h of
Nothing -> empty -- arbitrary choice
Just (_,h) -> h
heap :: Ord a => HeapP a -> Heap a
heap Empty = empty
heap (Unit x) = unit x
heap (Insert x p) = insert x (heap p)
heap (SafeRemoveMin p) = safeRemoveMin (heap p)
heap (Merge p q) = heap p `merge` heap q
heap (FromList xs) = fromList xs
instance (Ord a, Arbitrary a) => Arbitrary (HeapP a) where
arbitrary = sized arbHeapP
where
arbHeapP s =
frequency
[ (1, do return Empty)
, (1, do x <- arbitrary
return (Unit x))
, (s, do x <- arbitrary
p <- arbHeapP s1
return (Insert x p))
, (s, do p <- arbHeapP s1
return (SafeRemoveMin p))
, (s, do p <- arbHeapP s2
q <- arbHeapP s2
return (Merge p q))
, (1, do xs <- arbitrary
return (FromList xs))
]
where
s1 = s-1
s2 = s`div`2
shrink Empty = []
shrink (Unit x) = [ Unit x' | x' <- shrink x ]
shrink (FromList xs) = [ Unit x | x <- xs ]
++ [ FromList xs' | xs' <- shrink xs ]
shrink p =
[ FromList (toList (heap p)) ] ++
case p of
Insert x p -> [ p ]
++ [ Insert x p' | p' <- shrink p ]
++ [ Insert x' p | x' <- shrink x ]
SafeRemoveMin p -> [ p ]
++ [ SafeRemoveMin p' | p' <- shrink p ]
Merge p q -> [ p, q ]
++ [ Merge p' q | p' <- shrink p ]
++ [ Merge p q' | q' <- shrink q ]
data HeapPP a = HeapPP (HeapP a) (Heap a)
deriving (Show)
instance (Ord a, Arbitrary a) => Arbitrary (HeapPP a) where
arbitrary =
do p <- arbitrary
return (HeapPP p (heap p))
shrink (HeapPP p _) =
[ HeapPP p' (heap p') | p' <- shrink p ]
--------------------------------------------------------------------------
-- properties
data Context a where
Context :: Eq b => (Heap a -> b) -> Context a
instance (Ord a, Arbitrary a) => Arbitrary (Context a) where
arbitrary =
do f <- sized arbContext
let vec h = (size h, toSortedList h, isEmpty h)
return (Context (vec . f))
where
arbContext s =
frequency
[ (1, do return id)
, (s, do x <- arbitrary
f <- arbContext (s-1)
return (insert x . f))
, (s, do f <- arbContext (s-1)
return (safeRemoveMin . f))
, (s, do HeapPP _ h <- arbitrary
f <- arbContext (s`div`2)
elements [ (h `merge`) . f, (`merge` h) . f ])
]
instance Show (Context a) where
show _ = "*"
(=~) :: Heap Char -> Heap Char -> Property
--h1 =~ h2 = sort (toList h1) == sort (toList h2)
--h1 =~ h2 = property (nub (sort (toList h1)) == nub (sort (toList h2))) -- bug!
h1 =~ h2 = property (\(Context c) -> c h1 == c h2)
{-
The normal form is:
insert x1 (insert x2 (... empty)...)
where x1 <= x2 <= ...
-}
-- heap creating operations
prop_Unit x =
unit x =~ insert x empty
prop_RemoveMin_Empty =
removeMin (empty :: Heap OrdA) == Nothing
prop_RemoveMin_Insert1 x =
removeMin (insert x empty :: Heap OrdA) == Just (x, empty)
prop_RemoveMin_Insert2 x y (HeapPP _ h) =
removeMin (insert x (insert y h)) ==~
(insert (max x y) `maph` removeMin (insert (min x y) h))
where
f `maph` Just (x,h) = Just (x, f h)
f `maph` Nothing = Nothing
Nothing ==~ Nothing = property True
Just (x,h1) ==~ Just (y,h2) = x==y .&&. h1 =~ h2
prop_InsertSwap x y (HeapPP _ h) =
insert x (insert y h) =~ insert y (insert x h)
prop_MergeInsertLeft x (HeapPP _ h1) (HeapPP _ h2) =
(insert x h1 `merge` h2) =~ insert x (h1 `merge` h2)
prop_MergeInsertRight x (HeapPP _ h1) (HeapPP _ h2) =
(h1 `merge` insert x h2) =~ insert x (h1 `merge` h2)
-- heap observing operations
prop_Size_Empty =
size empty == 0
prop_Size_Insert x (HeapPP _ (h :: Heap OrdA)) =
size (insert x h) == 1 + size h
prop_ToList_Empty =
toList empty == ([] :: [OrdA])
prop_ToList_Insert x (HeapPP _ (h :: Heap OrdA)) =
sort (toList (insert x h)) == sort (x : toList h)
prop_ToSortedList (HeapPP _ (h :: Heap OrdA)) =
toSortedList h == sort (toList h)
--------------------------------------------------------------------------
-- main
return []
main = $(quickCheckAll)
--------------------------------------------------------------------------
-- the end.