\begin{code}
module 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 ...
------------------------------------------------------------------------------

-- GHC extensions
import Data.Array.ST
import GHC.Arr

-- std interfaces
import Maybe
import Array
import List ( sortBy, (\\) )

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

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 es
= map get_node (stronglyConnCompR es)
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 es
= map decode forest
where
(graph, vertex_fn) = graphFromEdges es
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)

\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 bnds es = accumArray (flip (:)) [] bnds [(,) k v | (k,v) <- es]

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

\end{code} \begin{code}
graphFromEdges
:: Ord key
=> [(node, key, [key])]
-> (Graph, Vertex -> (node, key, [key]))
graphFromEdges es
= (graph, \v -> vertex_map ! v)
where
max_v      	    = length es - 1
bnds            = (0,max_v) :: (Vertex, Vertex)
sorted_edges    = sortBy lt es
edges1	    = zipWith (,) [0..] sorted_edges

graph	    = array bnds [(,) v (mapMaybe key_vertex ks) | (,) v (_,    _, ks) <- edges1]
key_map	    = array bnds [(,) v k			       | (,) v (_,    k, _ ) <- edges1]
vertex_map	    = array bnds edges1

(_,k1,_) lt (_,k2,_) = k1 compare k2 --of { LT -> True; other -> False }

-- 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 _ 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 :: Tree String -> [String]
draw (Node x xs) = grp this (space (length this)) (stLoop xs)
where this          = s1 ++ x ++ " "

space n       = take n (repeat ' ')

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 first rst = zipWith (++) (first: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  = 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 _ []     = 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)

{- 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

\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}
{- 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

\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 (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<dv]
cs = concat [ if lw<dv then us else [Node (v:ws) us]
| (lw, Node ws us) <- collected ]
`
\end{code}