% % (c) The University of Glasgow 2006 % \begin{code}
module Digraph(

SCC(..), flattenSCC, flattenSCCs,
stronglyConnCompG, topologicalSortG,
verticesG, edgesG, hasVertexG,
reachableG, transposeG,
outdegreeG, indegreeG,
vertexGroupsG, emptyG,
componentsG,

-- For backwards compatability with the simpler version of Digraph
stronglyConnCompFromEdgedVertices, stronglyConnCompFromEdgedVerticesR,

-- No friendly interface yet, not used but exported to avoid warnings
tabulate, preArr,
components, undirected,
back, cross, forward,
path,
bcc, do_label, bicomps, collect
) where

#include "HsVersions.h"

------------------------------------------------------------------------------
-- A version of the graph algorithms described in:
--
-- Lazy Depth-First Search and Linear IntGraph Algorithms in Haskell''
--   by David King and John Launchbury
--
-- Also included is some additional code for printing tree structures ...
------------------------------------------------------------------------------

import Util        ( sortLe )
import Outputable
import Maybes      ( expectJust )

-- Extensions
import Control.Monad    ( filterM, liftM, liftM2 )

-- std interfaces
import Data.Maybe
import Data.Array
import Data.List   ( (\\) )

import Data.Array.ST
#else
import Data.Array.ST  hiding ( indices, bounds )
#endif

\end{code} %************************************************************************ %* * %* Graphs and Graph Construction %* * %************************************************************************ Note [Nodes, keys, vertices] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ * A 'node' is a big blob of client-stuff * Each 'node' has a unique (client) 'key', but the latter is in Ord and has fast comparison * Digraph then maps each 'key' to a Vertex (Int) which is arranged densely in 0.n \begin{code}
data Graph node = Graph {
gr_int_graph      :: IntGraph,
gr_vertex_to_node :: Vertex -> node,
gr_node_to_vertex :: node -> Maybe Vertex
}

data Edge node = Edge node node

emptyGraph :: Graph a
emptyGraph = Graph (array (1, 0) []) (error "emptyGraph") (const Nothing)

:: Ord key
=> [(node, key)]
-> [(key, key)]  -- First component is source vertex key,
-- second is target vertex key (thing depended on)
-- Unlike the other interface I insist they correspond to
-- actual vertices because the alternative hides bugs. I can't
-- do the same thing for the other one for backcompat reasons.
-> Graph (node, key)
graphFromVerticesAndAdjacency vertices edges = Graph graph vertex_node (key_vertex . key_extractor)
where key_extractor = snd
(bounds, vertex_node, key_vertex, _) = reduceNodesIntoVertices vertices key_extractor
key_vertex_pair (a, b) = (expectJust "graphFromVerticesAndAdjacency" $key_vertex a, expectJust "graphFromVerticesAndAdjacency"$ key_vertex b)
reduced_edges = map key_vertex_pair edges
graph = buildG bounds reduced_edges

graphFromEdgedVertices
:: Ord key
=> [(node, key, [key])]         -- The graph; its ok for the
-- out-list to contain keys which arent
-- a vertex key, they are ignored
-> Graph (node, key, [key])
graphFromEdgedVertices []             = emptyGraph
graphFromEdgedVertices edged_vertices = Graph graph vertex_fn (key_vertex . key_extractor)
where key_extractor (_, k, _) = k
(bounds, vertex_fn, key_vertex, numbered_nodes) = reduceNodesIntoVertices edged_vertices key_extractor
graph = array bounds [(v, mapMaybe key_vertex ks) | (v, (_, _, ks)) <- numbered_nodes]

reduceNodesIntoVertices
:: Ord key
=> [node]
-> (node -> key)
-> (Bounds, Vertex -> node, key -> Maybe Vertex, [(Int, node)])
reduceNodesIntoVertices nodes key_extractor = (bounds, (!) vertex_map, key_vertex, numbered_nodes)
where
max_v           = length nodes - 1
bounds          = (0, max_v) :: (Vertex, Vertex)

sorted_nodes    = let n1 le n2 = (key_extractor n1 compare key_extractor n2) /= GT
in sortLe le nodes
numbered_nodes  = zipWith (,) [0..] sorted_nodes

key_map         = array bounds [(i, key_extractor node) | (i, node) <- numbered_nodes]
vertex_map      = array bounds numbered_nodes

--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
| otherwise = let mid = (a + b) div 2
in case compare k (key_map ! mid) of
LT -> find a (mid - 1)
EQ -> Just mid
GT -> find (mid + 1) b

\end{code} %************************************************************************ %* * %* SCC %* * %************************************************************************ \begin{code}
data SCC vertex = AcyclicSCC vertex
| CyclicSCC  [vertex]

instance Functor SCC where
fmap f (AcyclicSCC v) = AcyclicSCC (f v)
fmap f (CyclicSCC vs) = CyclicSCC (fmap f vs)

flattenSCCs :: [SCC a] -> [a]
flattenSCCs = concatMap flattenSCC

flattenSCC :: SCC a -> [a]
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} %************************************************************************ %* * %* Strongly Connected Component wrappers for Graph %* * %************************************************************************ Note: the components are returned topologically sorted: later components depend on earlier ones, but not vice versa i.e. later components only have edges going from them to earlier ones. \begin{code}
stronglyConnCompG :: Graph node -> [SCC node]
stronglyConnCompG (Graph { gr_int_graph = graph, gr_vertex_to_node = vertex_fn }) = map decode forest
where
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)

-- The following two versions are provided for backwards compatability:
stronglyConnCompFromEdgedVertices
:: Ord key
=> [(node, key, [key])]
-> [SCC node]
stronglyConnCompFromEdgedVertices = map (fmap get_node) . stronglyConnCompFromEdgedVerticesR
where get_node (n, _, _) = n

-- 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
stronglyConnCompFromEdgedVerticesR
:: Ord key
=> [(node, key, [key])]
-> [SCC (node, key, [key])]
stronglyConnCompFromEdgedVerticesR = stronglyConnCompG . graphFromEdgedVertices

\end{code} %************************************************************************ %* * %* Misc wrappers for Graph %* * %************************************************************************ \begin{code}
topologicalSortG :: Graph node -> [node]
topologicalSortG graph = map (gr_vertex_to_node graph) result
where result = {-# SCC "Digraph.topSort" #-} topSort (gr_int_graph graph)

reachableG :: Graph node -> node -> [node]
reachableG graph from = map (gr_vertex_to_node graph) result
where from_vertex = expectJust "reachableG" (gr_node_to_vertex graph from)
result = {-# SCC "Digraph.reachable" #-} reachable (gr_int_graph graph) from_vertex

hasVertexG :: Graph node -> node -> Bool
hasVertexG graph node = isJust $gr_node_to_vertex graph node verticesG :: Graph node -> [node] verticesG graph = map (gr_vertex_to_node graph)$ vertices (gr_int_graph graph)

edgesG :: Graph node -> [Edge node]
edgesG graph = map (\(v1, v2) -> Edge (v2n v1) (v2n v2)) $edges (gr_int_graph graph) where v2n = gr_vertex_to_node graph transposeG :: Graph node -> Graph node transposeG graph = Graph (transpose (gr_int_graph graph)) (gr_vertex_to_node graph) (gr_node_to_vertex graph) outdegreeG :: Graph node -> node -> Maybe Int outdegreeG = degreeG outdegree indegreeG :: Graph node -> node -> Maybe Int indegreeG = degreeG indegree degreeG :: (IntGraph -> Table Int) -> Graph node -> node -> Maybe Int degreeG degree graph node = let table = degree (gr_int_graph graph) in fmap ((!) table)$ gr_node_to_vertex graph node

vertexGroupsG :: Graph node -> [[node]]
vertexGroupsG graph = map (map (gr_vertex_to_node graph)) result
where result = vertexGroups (gr_int_graph graph)

emptyG :: Graph node -> Bool
emptyG g = graphEmpty (gr_int_graph g)

componentsG :: Graph node -> [[node]]
componentsG graph = map (map (gr_vertex_to_node graph) . flattenTree) $components (gr_int_graph graph)  \end{code} %************************************************************************ %* * %* Showing Graphs %* * %************************************************************************ \begin{code}  instance Outputable node => Outputable (Graph node) where ppr graph = vcat [ hang (text "Vertices:") 2 (vcat (map ppr$ verticesG graph)),
hang (text "Edges:") 2 (vcat (map ppr $edgesG graph)) ] instance Outputable node => Outputable (Edge node) where ppr (Edge from to) = ppr from <+> text "->" <+> ppr to  \end{code} %************************************************************************ %* * %* IntGraphs %* * %************************************************************************ \begin{code} type Vertex = Int type Table a = Array Vertex a type IntGraph = Table [Vertex] type Bounds = (Vertex, Vertex) type IntEdge = (Vertex, Vertex)  \end{code} \begin{code} vertices :: IntGraph -> [Vertex] vertices = indices edges :: IntGraph -> [IntEdge] 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 -> [IntEdge] -> IntGraph buildG bounds edges = accumArray (flip (:)) [] bounds edges transpose :: IntGraph -> IntGraph transpose g = buildG (bounds g) (reverseE g) reverseE :: IntGraph -> [IntEdge] reverseE g = [ (w, v) | (v, w) <- edges g ] outdegree :: IntGraph -> Table Int outdegree = mapT numEdges where numEdges _ ws = length ws indegree :: IntGraph -> Table Int indegree = outdegree . transpose graphEmpty :: IntGraph -> Bool graphEmpty g = lo > hi where (lo, hi) = bounds g  \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) flattenTree :: Tree a -> [a] flattenTree (Node x ts) = x : concatMap flattenTree ts  \end{code} \begin{code} instance Show a => Show (Tree a) where showsPrec _ t s = showTree t ++ s showTree :: Show a => Tree a -> String showTree = drawTree . mapTree show instance Show a => Show (Forest a) where showsPrec _ f s = showForest f ++ s showForest :: Show a => Forest a -> String showForest = unlines . map showTree drawTree :: Tree String -> String drawTree = unlines . draw draw :: Tree String -> [String] 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 [] = [] 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} 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  \end{code} \begin{code} dff :: IntGraph -> Forest Vertex dff g = dfs g (vertices g) dfs :: IntGraph -> [Vertex] -> Forest Vertex dfs g vs = prune (bounds g) (map (generate g) vs) generate :: IntGraph -> 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)  \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 (zip 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] -> [a] postorder (Node a ts) = postorderF ts . (a :) postorderF :: Forest a -> [a] -> [a] postorderF ts = foldr (.) id$ map postorder ts

postOrd :: IntGraph -> [Vertex]
postOrd g = postorderF (dff g) []

topSort :: IntGraph -> [Vertex]
topSort = reverse . postOrd

\end{code} ------------------------------------------------------------ -- Algorithm 3: connected components ------------------------------------------------------------ \begin{code}
components   :: IntGraph -> Forest Vertex
components    = dff . undirected

undirected   :: IntGraph -> IntGraph
undirected g  = buildG (bounds g) (edges g ++ reverseE g)

\end{code} ------------------------------------------------------------ -- Algorithm 4: strongly connected components ------------------------------------------------------------ \begin{code}
scc  :: IntGraph -> Forest Vertex
scc g = dfs g (reverse (postOrd (transpose g)))

\end{code} ------------------------------------------------------------ -- Algorithm 5: Classifying edges ------------------------------------------------------------ \begin{code}
back              :: IntGraph -> Table Int -> IntGraph
back g post        = mapT select g
where select v ws = [ w | w <- ws, post!v < post!w ]

cross             :: IntGraph -> Table Int -> Table Int -> IntGraph
cross g pre post   = mapT select g
where select v ws = [ w | w <- ws, post!v > post!w, pre!v > pre!w ]

forward           :: IntGraph -> IntGraph -> Table Int -> IntGraph
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    :: IntGraph -> Vertex -> [Vertex]
reachable g v = preorderF (dfs g [v])

path         :: IntGraph -> Vertex -> Vertex -> Bool
path g v w    = w elem (reachable g v)

\end{code} ------------------------------------------------------------ -- Algorithm 7: Biconnected components ------------------------------------------------------------ \begin{code}
bcc :: IntGraph -> Forest [Vertex]
bcc g = (concat . map bicomps . map (do_label g dnum)) forest
where forest = dff g
dnum   = preArr (bounds g) forest

do_label :: IntGraph -> 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 (_,_,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<dv]
cs = concat [ if lw<dv then us else [Node (v:ws) us]
| (lw, Node ws us) <- collected ]

\end{code} ------------------------------------------------------------ -- Algorithm 8: Total ordering on groups of vertices ------------------------------------------------------------ The plan here is to extract a list of groups of elements of the graph such that each group has no dependence except on nodes in previous groups (i.e. in particular they may not depend on nodes in their own group) and is maximal such group. Clearly we cannot provide a solution for cyclic graphs. We proceed by iteratively removing elements with no outgoing edges and their associated edges from the graph. This probably isn't very efficient and certainly isn't very clever. \begin{code}

vertexGroups :: IntGraph -> [[Vertex]]
vertexGroups g = runST (mkEmpty (bounds g) >>= \provided -> vertexGroupsS provided g next_vertices)
where next_vertices = noOutEdges g

noOutEdges :: IntGraph -> [Vertex]
noOutEdges g = [ v | v <- vertices g, null (g!v)]

vertexGroupsS :: Set s -> IntGraph -> [Vertex] -> ST s [[Vertex]]
vertexGroupsS provided g to_provide
= if null to_provide
then do {
all_provided <- allM (provided contains) (vertices g)
; if all_provided
then return []
else error "vertexGroup: cyclic graph"
}
else do {
mapM_ (include provided) to_provide
; to_provide' <- filterM (vertexReady provided g) (vertices g)
; rest <- vertexGroupsS provided g to_provide'
; return $to_provide : rest } vertexReady :: Set s -> IntGraph -> Vertex -> ST s Bool vertexReady provided g v = liftM2 (&&) (liftM not$ provided contains v) (allM (provided contains) (g!v))
`
\end{code}