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|
{-# LANGUAGE ScopedTypeVariables, TemplateHaskell #-}
module Main where
--------------------------------------------------------------------------
-- imports
import Test.QuickCheck
import Test.QuickCheck.Poly
import Data.List
( sort
, (\\)
)
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 [h] = h
merging hs = merging (sweep hs)
sweep [] = []
sweep [h] = [h]
sweep (h1:h2:hs) = (h1 `merge` h2) : sweep hs
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 : toList (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)
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) = case removeMin (heap p) of
Nothing -> empty -- arbitrary choice
Just (_,h) -> h
heap (Merge p q) = heap p `merge` heap q
heap (FromList xs) = fromList xs
instance 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 (Unit x) = [ Unit x' | x' <- shrink x ]
shrink (FromList xs) = [ Unit x | x <- xs ]
++ [ FromList xs' | xs' <- shrink xs ]
shrink (Insert x p) = [ p ]
++ [ Insert x p' | p' <- shrink p ]
++ [ Insert x' p | x' <- shrink x ]
shrink (SafeRemoveMin p) = [ p ]
++ [ SafeRemoveMin p' | p' <- shrink p ]
shrink (Merge p q) = [ p, q ]
++ [ Merge p' q | p' <- shrink p ]
++ [ Merge p q' | q' <- shrink q ]
shrink _ = []
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
(==?) :: Heap OrdA -> [OrdA] -> Bool
h ==? xs = sort (toList h) == sort xs
prop_Empty =
empty ==? []
prop_IsEmpty (HeapPP _ h) =
isEmpty h == null (toList h)
prop_Unit x =
unit x ==? [x]
prop_Size (HeapPP _ h) =
size h == length (toList h)
prop_Insert x (HeapPP _ h) =
insert x h ==? (x : toList h)
prop_RemoveMin (HeapPP _ h) =
cover 80 (size h > 1) "non-trivial" $
case removeMin h of
Nothing -> h ==? []
Just (x,h') -> x == minimum (toList h) && h' ==? (toList h \\ [x])
prop_Merge (HeapPP _ h1) (HeapPP _ h2) =
(h1 `merge` h2) ==? (toList h1 ++ toList h2)
prop_FromList xs =
fromList xs ==? xs
prop_ToSortedList (HeapPP _ h) =
h ==? xs && xs == sort xs
where
xs = toSortedList h
--------------------------------------------------------------------------
-- main
return []
main = $(quickCheckAll)
--------------------------------------------------------------------------
-- the end.
-- toSortedList (Node x h1 h2) = x : toSortedList (h1 `merge` h2)
|
