diff options
Diffstat (limited to 'compiler/utils/Digraph.lhs')
| -rw-r--r-- | compiler/utils/Digraph.lhs | 426 |
1 files changed, 426 insertions, 0 deletions
diff --git a/compiler/utils/Digraph.lhs b/compiler/utils/Digraph.lhs new file mode 100644 index 0000000000..c49087c8f3 --- /dev/null +++ b/compiler/utils/Digraph.lhs @@ -0,0 +1,426 @@ +\begin{code} +module Digraph( + + -- At present the only one with a "nice" external interface + stronglyConnComp, stronglyConnCompR, SCC(..), flattenSCC, flattenSCCs, + + Graph, Vertex, + graphFromEdges, graphFromEdges', + buildG, transposeG, reverseE, outdegree, indegree, + + Tree(..), Forest, + showTree, showForest, + + dfs, dff, + topSort, + components, + scc, + back, cross, forward, + reachable, path, + bcc + + ) where + +# include "HsVersions.h" + +------------------------------------------------------------------------------ +-- 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 ( sortLe ) + +-- Extensions +import MONAD_ST + +-- std interfaces +import Maybe +import Array +import List +import Outputable + +#if __GLASGOW_HASKELL__ >= 504 +import Data.Array.ST hiding ( indices, bounds ) +#else +import ST +#endif +\end{code} + + +%************************************************************************ +%* * +%* External interface +%* * +%************************************************************************ + +\begin{code} +data SCC vertex = AcyclicSCC vertex + | CyclicSCC [vertex] + +flattenSCCs :: [SCC a] -> [a] +flattenSCCs = concatMap flattenSCC + +flattenSCC (AcyclicSCC v) = [v] +flattenSCC (CyclicSCC vs) = vs + +instance Outputable a => Outputable (SCC a) where + ppr (AcyclicSCC v) = text "NONREC" $$ (nest 3 (ppr v)) + ppr (CyclicSCC vs) = text "REC" $$ (nest 3 (vcat (map ppr vs))) +\end{code} + +\begin{code} +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] -- Returned in topologically sorted order + -- Later components depend on earlier ones, but not vice versa + +stronglyConnComp edges + = map get_node (stronglyConnCompR edges) + 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])] -- Topologically sorted + +stronglyConnCompR [] = [] -- added to avoid creating empty array in graphFromEdges -- SOF +stronglyConnCompR edges + = map decode forest + where + (graph, vertex_fn) = _scc_ "graphFromEdges" graphFromEdges edges + forest = _scc_ "Digraph.scc" 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) +\end{code} + +%************************************************************************ +%* * +%* Graphs +%* * +%************************************************************************ + + +\begin{code} +type Vertex = Int +type Table a = Array Vertex a +type Graph = Table [Vertex] +type Bounds = (Vertex, Vertex) +type Edge = (Vertex, Vertex) +\end{code} + +\begin{code} +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 bounds edges = accumArray (flip (:)) [] bounds edges + +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 v ws = length ws + +indegree :: Graph -> Table Int +indegree = outdegree . transposeG +\end{code} + + +\begin{code} +graphFromEdges + :: Ord key + => [(node, key, [key])] + -> (Graph, Vertex -> (node, key, [key])) +graphFromEdges edges = + case graphFromEdges' edges of (graph, vertex_fn, _) -> (graph, vertex_fn) + +graphFromEdges' + :: Ord key + => [(node, key, [key])] + -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex) +graphFromEdges' edges + = (graph, \v -> vertex_map ! v, key_vertex) + where + max_v = length edges - 1 + bounds = (0,max_v) :: (Vertex, Vertex) + sorted_edges = let + (_,k1,_) `le` (_,k2,_) = case k1 `compare` k2 of { GT -> False; other -> True } + in + sortLe le edges + edges1 = zipWith (,) [0..] sorted_edges + + graph = array bounds [(,) v (mapMaybe key_vertex ks) | (,) v (_, _, ks) <- edges1] + key_map = array bounds [(,) v k | (,) v (_, k, _ ) <- edges1] + vertex_map = array bounds 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 +\end{code} + +%************************************************************************ +%* * +%* Trees and forests +%* * +%************************************************************************ + +\begin{code} +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) +\end{code} + +\begin{code} +instance Show a => Show (Tree a) where + showsPrec p t s = showTree t ++ s + +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 + +draw (Node x ts) = grp this (space (length this)) (stLoop ts) + where this = s1 ++ x ++ " " + + space n = replicate n ' ' + + stLoop [] = [""] + stLoop [t] = grp s2 " " (draw t) + stLoop (t:ts) = grp s3 s4 (draw t) ++ [s4] ++ rsLoop ts + + rsLoop [t] = grp s5 " " (draw t) + rsLoop (t:ts) = grp s6 s4 (draw t) ++ [s4] ++ rsLoop ts + + grp fst rst = zipWith (++) (fst:repeat rst) + + [s1,s2,s3,s4,s5,s6] = ["- ", "--", "-+", " |", " `", " +"] +\end{code} + + +%************************************************************************ +%* * +%* Depth first search +%* * +%************************************************************************ + +\begin{code} +#if __GLASGOW_HASKELL__ >= 504 +newSTArray :: Ix i => (i,i) -> e -> ST s (STArray s i e) +newSTArray = newArray + +readSTArray :: Ix i => STArray s i e -> i -> ST s e +readSTArray = readArray + +writeSTArray :: Ix i => STArray s i e -> i -> e -> ST s () +writeSTArray = writeArray +#endif + +type Set s = STArray s Vertex Bool + +mkEmpty :: Bounds -> ST s (Set s) +mkEmpty bnds = newSTArray bnds False + +contains :: Set s -> Vertex -> ST s Bool +contains m v = readSTArray m v + +include :: Set s -> Vertex -> ST s () +include m v = writeSTArray m v True +\end{code} + +\begin{code} +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 m [] = 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) +\end{code} + + +%************************************************************************ +%* * +%* Algorithms +%* * +%************************************************************************ + +------------------------------------------------------------ +-- Algorithm 1: depth first search numbering +------------------------------------------------------------ + +\begin{code} +--preorder :: Tree a -> [a] +preorder (Node a ts) = a : preorderF ts + +preorderF :: Forest a -> [a] +preorderF ts = concat (map preorder ts) + +tabulate :: Bounds -> [Vertex] -> Table Int +tabulate bnds vs = array bnds (zipWith (,) vs [1..]) + +preArr :: Bounds -> Forest Vertex -> Table Int +preArr bnds = tabulate bnds . preorderF +\end{code} + + +------------------------------------------------------------ +-- Algorithm 2: topological sorting +------------------------------------------------------------ + +\begin{code} +--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 +\end{code} + + +------------------------------------------------------------ +-- Algorithm 3: connected components +------------------------------------------------------------ + +\begin{code} +components :: Graph -> Forest Vertex +components = dff . undirected + +undirected :: Graph -> Graph +undirected g = buildG (bounds g) (edges g ++ reverseE g) +\end{code} + + +-- Algorithm 4: strongly connected components + +\begin{code} +scc :: Graph -> Forest Vertex +scc g = dfs g (reverse (postOrd (transposeG g))) +\end{code} + + +------------------------------------------------------------ +-- Algorithm 5: Classifying edges +------------------------------------------------------------ + +\begin{code} +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 +\end{code} + + +------------------------------------------------------------ +-- Algorithm 6: Finding reachable vertices +------------------------------------------------------------ + +\begin{code} +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) +\end{code} + + +------------------------------------------------------------ +-- Algorithm 7: Biconnected components +------------------------------------------------------------ + +\begin{code} +bcc :: Graph -> Forest [Vertex] +bcc g = (concat . map bicomps . map (do_label g dnum)) forest + where forest = dff g + dnum = preArr (bounds g) forest + +do_label :: Graph -> Table Int -> Tree Vertex -> Tree (Vertex,Int,Int) +do_label g dnum (Node v ts) = Node (v,dnum!v,lv) us + where us = map (do_label g dnum) ts + lv = minimum ([dnum!v] ++ [dnum!w | w <- g!v] + ++ [lu | Node (u,du,lu) xs <- us]) + +bicomps :: Tree (Vertex,Int,Int) -> Forest [Vertex] +bicomps (Node (v,dv,lv) ts) + = [ Node (v:vs) us | (l,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 us) <- collected, lw<dv] + cs = concat [ if lw<dv then us else [Node (v:ws) us] + | (lw, Node ws us) <- collected ] +\end{code} + |