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module Graph where
import qualified Data.Map as M
import Data.Map( Map, (!) )
import qualified Data.Set as S
import Data.Set( Set )
import Data.List( nub, sort, (\\) )
--import Test.QuickCheck hiding ( generate )
-- == almost everything in this module is inspired by King & Launchbury ==
--------------------------------------------------------------------------------
-- depth-first trees
data Tree a
= Node a [Tree a]
| Cut a
deriving ( Eq, Show )
type Forest a
= [Tree a]
top :: Tree a -> a
top (Node x _) = x
top (Cut x) = x
-- pruning a possibly infinite forest
prune :: Ord a => Forest a -> Forest a
prune ts = go S.empty ts
where
go seen [] = []
go seen (Cut x :ts) = Cut x : go seen ts
go seen (Node x vs:ts)
| x `S.member` seen = Cut x : go seen ts
| otherwise = Node x (take n ws) : drop n ws
where
n = length vs
ws = go (S.insert x seen) (vs ++ ts)
-- pre- and post-order traversals
preorder :: Tree a -> [a]
preorder t = preorderF [t]
preorderF :: Forest a -> [a]
preorderF ts = go ts []
where
go [] xs = xs
go (Cut x : ts) xs = go ts xs
go (Node x vs : ts) xs = x : go vs (go ts xs)
postorder :: Tree a -> [a]
postorder t = postorderF [t]
postorderF :: Forest a -> [a]
postorderF ts = go ts []
where
go [] xs = xs
go (Cut x : ts) xs = go ts xs
go (Node x vs : ts) xs = go vs (x : go ts xs)
-- computing back-arrows
backs :: Ord a => Tree a -> Set a
backs t = S.fromList (go S.empty t)
where
go ups (Node x ts) = concatMap (go (S.insert x ups)) ts
go ups (Cut x) = [x | x `S.member` ups ]
--------------------------------------------------------------------------------
-- graphs
type Graph a
= Map a [a]
vertices :: Graph a -> [a]
vertices g = [ x | (x,_) <- M.toList g ]
transposeG :: Ord a => Graph a -> Graph a
transposeG g =
M.fromListWith (++) $
[ (y,[x]) | (x,ys) <- M.toList g, y <- ys ] ++
[ (x,[]) | x <- vertices g ]
--------------------------------------------------------------------------------
-- graphs and trees
generate :: Ord a => Graph a -> a -> Tree a
generate g x = Node x (map (generate g) (g!x))
dfs :: Ord a => Graph a -> [a] -> Forest a
dfs g xs = prune (map (generate g) xs)
reach :: Ord a => Graph a -> [a] -> Graph a
reach g xs = M.fromList [ (x,g!x) | x <- preorderF (dfs g xs) ]
dff :: Ord a => Graph a -> Forest a
dff g = dfs g (vertices g)
preOrd :: Ord a => Graph a -> [a]
preOrd g = preorderF (dff g)
postOrd :: Ord a => Graph a -> [a]
postOrd g = postorderF (dff g)
scc1 :: Ord a => Graph a -> Forest a
scc1 g = reverse (dfs (transposeG g) (reverse (postOrd g)))
scc2 :: Ord a => Graph a -> Forest a
scc2 g = dfs g (reverse (postOrd (transposeG g)))
scc :: Ord a => Graph a -> Forest a
scc g = scc2 g
sccs :: Ord a => Graph a -> [[a]]
sccs = map preorder . scc
--------------------------------------------------------------------------------
-- testing correctness
{-
newtype G = G (Graph Int) deriving ( Show )
set :: (Ord a, Num a, Arbitrary a) => Gen [a]
set = (nub . sort . map abs) `fmap` arbitrary
instance Arbitrary G where
arbitrary =
do xs <- set `suchThat` (not . null)
yss <- sequence [ listOf (elements xs) | x <- xs ]
return (G (M.fromList (xs `zip` yss)))
shrink (G g) =
[ G (delNode x g)
| (x,_) <- M.toList g
] ++
[ G (delEdge x y g)
| (x,ys) <- M.toList g
, y <- ys
]
where
delNode v g =
M.fromList
[ (x,filter (v/=) ys)
| (x,ys) <- M.toList g
, x /= v
]
delEdge v w g =
M.insert v ((g!v) \\ [w]) g
-- all vertices in a component can reach each other
prop_Scc_StronglyConnected (G g) =
whenFail (print cs) $
and [ y `S.member` r | c <- cs, x <- c, let r = reach x, y <- c ]
where
cs = sccs g
reach x = go S.empty [x]
where
go seen [] = seen
go seen (x:xs)
| x `S.member` seen = go seen xs
| otherwise = go (S.insert x seen) ((g!x) ++ xs)
-- vertices cannot forward-reach to other components
prop_Scc_NotConnected (G g) =
whenFail (print cs) $
-- every vertex is somewhere
and [ or [ x `elem` c | c <- cs ]
| x <- vertices g
] &&
-- cannot foward-reach
and [ y `S.notMember` rx
| (c,d) <- pairs cs
, x <- c
, let rx = reach x
, y <- d
]
where
cs = sccs g
pairs (x:xs) = [ (x,y) | y <- xs ] ++ pairs xs
pairs [] = []
reach x = go S.empty [x]
where
go seen [] = seen
go seen (x:xs)
| x `S.member` seen = go seen xs
| otherwise = go (S.insert x seen) ((g!x) ++ xs)
-}
--------------------------------------------------------------------------------
|
