{- | Module : Util.Digraph Copyright : Maintainer : lib@galois.com Stability : Portability : Functional graph algorithms; code taken from King and Launchbury's POPL paper (via GHC sources.) -} module Util.Digraph( -- At present the only one with a "nice" external interface stronglyConnComp, stronglyConnCompR, SCC(..), Graph, Vertex, graphFromEdges, buildG, transposeG, reverseE, outdegree, indegree, Tree(..), Forest, showTree, showForest, dfs, dff, topSort, components, scc, back, cross, forward, reachable, path, bcc ) where ------------------------------------------------------------------------------ -- A version of the graph algorithms described in: -- -- ``Lazy Depth-First Search and Linear Graph Algorithms in Haskell'' -- by David King and John Launchbury -- -- Also included is some additional code for printing tree structures ... ------------------------------------------------------------------------------ import Util.Sort ( sortLe ) -- merge sosrt import Control.Monad.ST import Data.Array.ST ( STArray, newArray, writeArray, readArray ) -- std interfaces import Data.Maybe import Data.Array import Data.List ( (\\) ) {- %************************************************************************ %* * %* External interface %* * %************************************************************************ -} data SCC vertex = AcyclicSCC vertex | CyclicSCC [vertex] deriving Show stronglyConnComp :: Ord key => [(node, key, [key])] -- The graph; its ok for the -- out-list to contain keys which arent -- a vertex key, they are ignored -> [SCC node] stronglyConnComp edges1 = map get_node (stronglyConnCompR edges1) where get_node (AcyclicSCC (n, _, _)) = AcyclicSCC n get_node (CyclicSCC triples) = CyclicSCC [n | (n,_,_) <- triples] -- The "R" interface is used when you expect to apply SCC to -- the (some of) the result of SCC, so you dont want to lose the dependency info stronglyConnCompR :: Ord key => [(node, key, [key])] -- The graph; its ok for the -- out-list to contain keys which arent -- a vertex key, they are ignored -> [SCC (node, key, [key])] stronglyConnCompR [] = [] -- added to avoid creating empty array in graphFromEdges -- SOF stronglyConnCompR edges1 = map decode forest where (graph, vertex_fn) = graphFromEdges edges1 forest = scc graph decode (Node v []) | mentions_itself v = CyclicSCC [vertex_fn v] | otherwise = AcyclicSCC (vertex_fn v) decode other = CyclicSCC (dec other []) where dec (Node v ts) vs = vertex_fn v : foldr dec vs ts mentions_itself v = v `elem` (graph ! v) {- %************************************************************************ %* * %* Graphs %* * %************************************************************************ -} type Vertex = Int type Table a = Array Vertex a type Graph = Table [Vertex] type Bounds = (Vertex, Vertex) type Edge = (Vertex, Vertex) vertices :: Graph -> [Vertex] vertices = indices edges :: Graph -> [Edge] edges g = [ (v, w) | v <- vertices g, w <- g!v ] mapT :: (Vertex -> a -> b) -> Table a -> Table b mapT f t = array (bounds t) [ (,) v (f v (t!v)) | v <- indices t ] buildG :: Bounds -> [Edge] -> Graph buildG bounds1 edges1 = accumArray (flip (:)) [] bounds1 [(,) k v | (k,v) <- edges1] transposeG :: Graph -> Graph transposeG g = buildG (bounds g) (reverseE g) reverseE :: Graph -> [Edge] reverseE g = [ (w, v) | (v, w) <- edges g ] outdegree :: Graph -> Table Int outdegree = mapT numEdges where numEdges _ ws = length ws indegree :: Graph -> Table Int indegree = outdegree . transposeG graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key])) graphFromEdges edgs = (graph, \v -> vertex_map ! v) where max_v = length edgs - 1 bounds1 = (0,max_v) :: (Vertex, Vertex) sorted_edges = sortLe le edgs where (_,k1,_) `le` (_,k2,_) = case k1 `compare` k2 of { GT -> False; _other -> True } edges1 = zipWith (,) [0..] sorted_edges graph = array bounds1 [(,) v (mapMaybe key_vertex ks) | (,) v (_, _, ks) <- edges1] key_map = array bounds1 [(,) v k | (,) v (_, k, _ ) <- edges1] vertex_map = array bounds1 edges1 -- key_vertex :: key -> Maybe Vertex -- returns Nothing for non-interesting vertices key_vertex k = find 0 max_v where find a b | a > b = Nothing find a b = case compare k (key_map ! mid) of LT -> find a (mid-1) EQ -> Just mid GT -> find (mid+1) b where mid = (a + b) `div` 2 {- %************************************************************************ %* * %* Trees and forests %* * %************************************************************************ -} data Tree a = Node a (Forest a) type Forest a = [Tree a] mapTree :: (a -> b) -> (Tree a -> Tree b) mapTree f (Node x ts) = Node (f x) (map (mapTree f) ts) instance Show a => Show (Tree a) where show t = showTree t showTree :: Show a => Tree a -> String showTree = drawTree . mapTree show showForest :: Show a => Forest a -> String showForest = unlines . map showTree drawTree :: Tree String -> String drawTree = unlines . draw where draw (Node x ts) = grp this (space (length this)) (stLoop ts) where this = s1 ++ x ++ " " space n = take n (repeat ' ') stLoop [] = [""] stLoop [t] = grp s2 " " (draw t) stLoop (t:xs) = grp s3 s4 (draw t) ++ [s4] ++ rsLoop xs rsLoop [] = [] rsLoop [t] = grp s5 " " (draw t) rsLoop (t:xs) = grp s6 s4 (draw t) ++ [s4] ++ rsLoop xs grp a rst = zipWith (++) (a:repeat rst) [s1,s2,s3,s4,s5,s6] = ["- ", "--", "-+", " |", " `", " +"] {- %************************************************************************ %* * %* Depth first search %* * %************************************************************************ -} --type Set s = MutableArray s Vertex Bool type Set s = STArray s Vertex Bool mkEmpty :: Bounds -> ST s (Set s) mkEmpty bnds = newArray bnds False contains :: Set s -> Vertex -> ST s Bool contains m v = readArray m v include :: Set s -> Vertex -> ST s () include m v = writeArray m v True dff :: Graph -> Forest Vertex dff g = dfs g (vertices g) dfs :: Graph -> [Vertex] -> Forest Vertex dfs g vs = prune (bounds g) (map (generate g) vs) generate :: Graph -> Vertex -> Tree Vertex generate g v = Node v (map (generate g) (g!v)) prune :: Bounds -> Forest Vertex -> Forest Vertex prune bnds ts = runST (mkEmpty bnds >>= \m -> chop m ts) chop :: Set s -> Forest Vertex -> ST s (Forest Vertex) chop _ [] = return [] chop m (Node v ts : us) = contains m v >>= \visited -> if visited then chop m us else include m v >>= \_ -> chop m ts >>= \as -> chop m us >>= \bs -> return (Node v as : bs) {- %************************************************************************ %* * %* Algorithms %* * %************************************************************************ -} ------------------------------------------------------------ -- Algorithm 1: depth first search numbering ------------------------------------------------------------ preorder :: Tree a -> [a] preorder (Node a ts) = a : preorderF ts preorderF :: Forest a -> [a] preorderF ts = concat (map preorder ts) {- UNUSED: preOrd :: Graph -> [Vertex] preOrd = preorderF . dff -} tabulate :: Bounds -> [Vertex] -> Table Int tabulate bnds vs = array bnds (zipWith (,) vs [1..]) preArr :: Bounds -> Forest Vertex -> Table Int preArr bnds = tabulate bnds . preorderF ------------------------------------------------------------ -- Algorithm 2: topological sorting ------------------------------------------------------------ postorder :: Tree a -> [a] postorder (Node a ts) = postorderF ts ++ [a] postorderF :: Forest a -> [a] postorderF ts = concat (map postorder ts) postOrd :: Graph -> [Vertex] postOrd = postorderF . dff topSort :: Graph -> [Vertex] topSort = reverse . postOrd ------------------------------------------------------------ -- Algorithm 3: connected components ------------------------------------------------------------ components :: Graph -> Forest Vertex components = dff . undirected undirected :: Graph -> Graph undirected g = buildG (bounds g) (edges g ++ reverseE g) ------------------------------------------------------------ -- Algorithm 4: strongly connected components ------------------------------------------------------------ scc :: Graph -> Forest Vertex scc g = dfs g (reverse (postOrd (transposeG g))) ------------------------------------------------------------ -- Algorithm 5: Classifying edges ------------------------------------------------------------ {- UNUSED: tree :: Bounds -> Forest Vertex -> Graph tree bnds ts = buildG bnds (concat (map flat ts)) where flat (Node v rs) = [ (v, w) | Node w us <- ts ] ++ concat (map flat ts) -} back :: Graph -> Table Int -> Graph back g post = mapT select g where select v ws = [ w | w <- ws, post!v < post!w ] cross :: Graph -> Table Int -> Table Int -> Graph cross g pre post = mapT select g where select v ws = [ w | w <- ws, post!v > post!w, pre!v > pre!w ] forward :: Graph -> Graph -> Table Int -> Graph forward g tree pre = mapT select g where select v ws = [ w | w <- ws, pre!v < pre!w ] \\ tree!v ------------------------------------------------------------ -- Algorithm 6: Finding reachable vertices ------------------------------------------------------------ reachable :: Graph -> Vertex -> [Vertex] reachable g v = preorderF (dfs g [v]) path :: Graph -> Vertex -> Vertex -> Bool path g v w = w `elem` (reachable g v) ------------------------------------------------------------ -- Algorithm 7: Biconnected components ------------------------------------------------------------ bcc :: Graph -> Forest [Vertex] bcc g = (concat . map bicomps . map (label g dnum)) forest where forest = dff g dnum = preArr (bounds g) forest label :: Graph -> Table Int -> Tree Vertex -> Tree (Vertex,Int,Int) label g dnum (Node v ts) = Node (v,dnum!v,lv) us where us = map (label g dnum) ts lv = minimum ([dnum!v] ++ [dnum!w | w <- g!v] ++ [lu | Node (_, _, lu) _ <- us]) bicomps :: Tree (Vertex,Int,Int) -> Forest [Vertex] bicomps (Node (v,_,_) ts) = [ Node (v:vs) us | (_, Node vs us) <- map collect ts] collect :: Tree (Vertex,Int,Int) -> (Int, Tree [Vertex]) collect (Node (v,dv,lv) ts) = (lv, Node (v:vs) cs) where collected = map collect ts vs = concat [ ws | (lw, Node ws _) <- collected, lw