{-# 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 (size h > 1) 80 "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)