X-Git-Url: http://git.megacz.com/?a=blobdiff_plain;f=ghc%2Fcompiler%2Futils%2FDigraph.lhs;h=0eff6da6980e6d87e24d3fb45c506325e61c1ed0;hb=59c796f8e77325d35f29ddd3e724bfa780466d40;hp=84cf220919988cf1d1adbcd999ba6fde061e00ef;hpb=e7d21ee4f8ac907665a7e170c71d59e13a01da09;p=ghc-hetmet.git diff --git a/ghc/compiler/utils/Digraph.lhs b/ghc/compiler/utils/Digraph.lhs index 84cf220..0eff6da 100644 --- a/ghc/compiler/utils/Digraph.lhs +++ b/ghc/compiler/utils/Digraph.lhs @@ -1,159 +1,418 @@ -% -% (c) The GRASP/AQUA Project, Glasgow University, 1992-1995 -% -\section[Digraph]{An implementation of directed graphs} - -\begin{code} -module Digraph ( - stronglyConnComp, ---OLD: whichCycle, -- MOVED: isCyclic, - topologicalSort, - dfs, -- deforester - MaybeErr +\begin{code} +module Digraph( + + -- At present the only one with a "nice" external interface + stronglyConnComp, stronglyConnCompR, SCC(..), flattenSCC, flattenSCCs, + + Graph, Vertex, + graphFromEdges, buildG, transposeG, reverseE, outdegree, indegree, + + Tree(..), Forest, + showTree, showForest, + + dfs, dff, + topSort, + components, + scc, + back, cross, forward, + reachable, path, + bcc + ) where -import Maybes ( MaybeErr(..) ) -import Util +# 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} -This module implements at least part of an abstract data type for -directed graphs. The part implemented is what we need for doing -dependency analyses. ->type Edge vertex = (vertex, vertex) ->type Cycle vertex = [vertex] +%************************************************************************ +%* * +%* 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} %************************************************************************ %* * -%* Strongly connected components * +%* Graphs %* * %************************************************************************ -John Launchbury provided the basic code for doing strongly-connected -components. -The result is a list of cycles (each of which is a list of vertices), -and these cycles are topologically sorted, so that if there is an edge from -cycle A to cycle B, then A occurs after B in the result list. +\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} -stronglyConnComp :: (vertex->vertex->Bool) -> [Edge vertex] -> [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 bounds edges = accumArray (flip (:)) [] bounds edges + +transposeG :: Graph -> Graph +transposeG g = buildG (bounds g) (reverseE g) -stronglyConnComp eq edges vertices - = snd (span_tree (new_range reversed_edges) - ([],[]) - ( snd (dfs (new_range edges) ([],[]) vertices) ) - ) +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 + = (graph, \v -> vertex_map ! v) where - reversed_edges = map swap edges + 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 - swap (x,y) = (y, x) - -- new_range :: Eq v => [Edge v] -> v -> [v] + -- 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} - new_range [] w = [] - new_range ((x,y):xys) w - = if x `eq` w - then (y : (new_range xys w)) - else (new_range xys w) +%************************************************************************ +%* * +%* Trees and forests +%* * +%************************************************************************ - elem x [] = False - elem x (y:ys) = x `eq` y || x `elem` ys +\begin{code} +data Tree a = Node a (Forest a) +type Forest a = [Tree a] -{- span_tree :: Eq v => (v -> [v]) - -> ([v], [[v]]) - -> [v] - -> ([v], [[v]]) --} - span_tree r (vs,ns) [] = (vs,ns) - span_tree r (vs,ns) (x:xs) - | x `elem` vs = span_tree r (vs,ns) xs - | True = case (dfs r (x:vs,[]) (r x)) of { (vs',ns') -> - span_tree r (vs',(x:ns'):ns) xs } - -{- dfs :: Eq v => (v -> [v]) - -> ([v], [v]) - -> [v] - -> ([v], [v]) --} - dfs r (vs,ns) [] = (vs,ns) - dfs r (vs,ns) (x:xs) | x `elem` vs = dfs r (vs,ns) xs - | True = case (dfs r (x:vs,[]) (r x)) of { (vs',ns') -> - dfs r (vs',(x:ns')++ns) xs } +mapTree :: (a -> b) -> (Tree a -> Tree b) +mapTree f (Node x ts) = Node (f x) (map (mapTree f) ts) \end{code} \begin{code} -dfs :: (v -> v -> Bool) - -> (v -> [v]) - -> ([v], [v]) - -> [v] - -> ([v], [v]) +instance Show a => Show (Tree a) where + showsPrec p t s = showTree t ++ s + +showTree :: Show a => Tree a -> String +showTree = drawTree . mapTree show -dfs eq r (vs,ns) [] = (vs,ns) -dfs eq r (vs,ns) (x:xs) - | any (eq x) vs = dfs eq r (vs,ns) xs - | True = case (dfs eq r (x:vs,[]) (r x)) of - (vs',ns') -> dfs eq r (vs',(x:ns')++ns) xs +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} - -@isCyclic@ expects to be applied to an element of the result of a -stronglyConnComp; it tells whether such an element is a cycle. The -answer is True if it is not a singleton, of course, but if it is a -singleton we have to look up in the edges to see if it refers to -itself. + +%************************************************************************ +%* * +%* Depth first search +%* * +%************************************************************************ \begin{code} -{- MOVED TO POINT OF SINGLE USE: RenameBinds4 (WDP 95/02) +#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 -isCyclic :: Eq vertex => [Edge vertex] -> [vertex] -> Bool +mkEmpty :: Bounds -> ST s (Set s) +mkEmpty bnds = newSTArray bnds False -isCyclic edges [] = panic "isCyclic: empty component" -isCyclic edges [v] = (v,v) `is_elem` edges where { is_elem = isIn "isCyclic" } -isCyclic edges vs = True --} +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} -OLD: The following @whichCycle@ should be called only when the given -@vertex@ is known to be in one of the cycles. This isn't difficult to -achieve if the call follows the creation of the list of components by -@cycles@ (NB: strictness analyser) with all vertices of interest in -them. +\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} ->{- UNUSED: ->whichCycle :: Eq vertex => [Cycle vertex] -> vertex -> (Cycle vertex) ->whichCycle vss v = head [vs | vs <-vss, v `is_elem` vs] where { is_elem = isIn "whichCycle" } ->-} %************************************************************************ %* * -%* Topological sort * +%* Algorithms %* * %************************************************************************ -Topological sort fails if it finds any cycles, returning the offending cycles. +------------------------------------------------------------ +-- Algorithm 1: depth first search numbering +------------------------------------------------------------ -If it succeeds, the result is a list of vertices, such that if there is -an edge from vertex A to vertex B then A occurs after B in the result list. +\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} -topologicalSort :: (vertex->vertex->Bool) -> [Edge vertex] -> [vertex] - -> MaybeErr [vertex] -- Success: the sorted list - [[vertex]] -- Failure: the cycles +--postorder :: Tree a -> [a] +postorder (Node a ts) = postorderF ts ++ [a] -topologicalSort eq edges vertices - = case (stronglyConnComp eq edges vertices) of { sccs -> - case (partition (is_cyclic edges) sccs) of { (cycles, singletons) -> - if null cycles - then Succeeded [ v | [v] <- singletons ] - else Failed cycles - }} - where - is_cyclic es [] = panic "is_cyclic: empty component" - is_cyclic es [v] = (v,v) `elem` es - is_cyclic es vs = True +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} - elem (x,y) [] = False - elem z@(x,y) ((a,b):cs) = (x `eq` a && y `eq` b) || z `elem` cs + +------------------------------------------------------------ +-- 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