```#if __GLASGOW_HASKELL__ >= 703
{-# LANGUAGE Trustworthy #-}
#endif
-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Graph
-- Copyright   :  (c) The University of Glasgow 2002
--
-- Stability   :  experimental
-- Portability :  portable
--
-- A version of the graph algorithms described in:
--
--   /Lazy Depth-First Search and Linear Graph Algorithms in Haskell/,
--   by David King and John Launchbury.
--
-----------------------------------------------------------------------------

module Data.Graph(

-- * External interface

-- At present the only one with a "nice" external interface
stronglyConnComp, stronglyConnCompR, SCC(..), flattenSCC, flattenSCCs,

-- * Graphs

Graph, Table, Bounds, Edge, Vertex,

-- ** Building graphs

graphFromEdges, graphFromEdges', buildG, transposeG,
-- reverseE,

-- ** Graph properties

vertices, edges,
outdegree, indegree,

-- * Algorithms

dfs, dff,
topSort,
components,
scc,
bcc,
-- tree, back, cross, forward,
reachable, path,

module Data.Tree

) where

#endif

-- Extensions
import Data.Array.ST (STArray, newArray, readArray, writeArray)
#else
import Data.IntSet (IntSet)
import qualified Data.IntSet as Set
#endif
import Data.Tree (Tree(Node), Forest)

-- std interfaces
import Data.Maybe
import Data.Array
import Data.List

-------------------------------------------------------------------------
--									-
--	External interface
--									-
-------------------------------------------------------------------------

-- | Strongly connected component.
data SCC vertex = AcyclicSCC vertex	-- ^ A single vertex that is not
-- in any cycle.
| CyclicSCC  [vertex]	-- ^ A maximal set of mutually
-- reachable vertices.

-- | The vertices of a list of strongly connected components.
flattenSCCs :: [SCC a] -> [a]
flattenSCCs = concatMap flattenSCC

-- | The vertices of a strongly connected component.
flattenSCC :: SCC vertex -> [vertex]
flattenSCC (AcyclicSCC v) = [v]
flattenSCC (CyclicSCC vs) = vs

-- | The strongly connected components of a directed graph, topologically
-- sorted.
stronglyConnComp
:: Ord key
=> [(node, key, [key])]
-- ^ The graph: a list of nodes uniquely identified by keys,
-- with a list of keys of nodes this node has edges to.
-- The out-list may contain keys that don't correspond to
-- nodes of the graph; such edges are ignored.
-> [SCC node]

stronglyConnComp edges0
= map get_node (stronglyConnCompR edges0)
where
get_node (AcyclicSCC (n, _, _)) = AcyclicSCC n
get_node (CyclicSCC triples)     = CyclicSCC [n | (n,_,_) <- triples]

-- | The strongly connected components of a directed graph, topologically
-- sorted.  The function is the same as 'stronglyConnComp', except that
-- all the information about each node retained.
-- This interface is used when you expect to apply 'SCC' to
-- (some of) the result of 'SCC', so you don't want to lose the
-- dependency information.
stronglyConnCompR
:: Ord key
=> [(node, key, [key])]
-- ^ The graph: a list of nodes uniquely identified by keys,
-- with a list of keys of nodes this node has edges to.
-- The out-list may contain keys that don't correspond to
-- nodes of the graph; such edges are ignored.
-> [SCC (node, key, [key])]	-- ^ Topologically sorted

stronglyConnCompR [] = []  -- added to avoid creating empty array in graphFromEdges -- SOF
stronglyConnCompR edges0
= map decode forest
where
(graph, vertex_fn,_) = graphFromEdges edges0
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
--									-
-------------------------------------------------------------------------

-- | Abstract representation of vertices.
type Vertex  = Int
-- | Table indexed by a contiguous set of vertices.
type Table a = Array Vertex a
-- | Adjacency list representation of a graph, mapping each vertex to its
-- list of successors.
type Graph   = Table [Vertex]
-- | The bounds of a 'Table'.
type Bounds  = (Vertex, Vertex)
-- | An edge from the first vertex to the second.
type Edge    = (Vertex, Vertex)

-- | All vertices of a graph.
vertices :: Graph -> [Vertex]
vertices  = indices

-- | All edges of a graph.
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 ]

-- | Build a graph from a list of edges.
buildG :: Bounds -> [Edge] -> Graph
buildG bounds0 edges0 = accumArray (flip (:)) [] bounds0 edges0

-- | The graph obtained by reversing all edges.
transposeG  :: Graph -> Graph
transposeG g = buildG (bounds g) (reverseE g)

reverseE    :: Graph -> [Edge]
reverseE g   = [ (w, v) | (v, w) <- edges g ]

-- | A table of the count of edges from each node.
outdegree :: Graph -> Table Int
outdegree  = mapT numEdges
where numEdges _ ws = length ws

-- | A table of the count of edges into each node.
indegree :: Graph -> Table Int
indegree  = outdegree . transposeG

-- | Identical to 'graphFromEdges', except that the return value
-- does not include the function which maps keys to vertices.  This
-- version of 'graphFromEdges' is for backwards compatibility.
graphFromEdges'
:: Ord key
=> [(node, key, [key])]
-> (Graph, Vertex -> (node, key, [key]))
graphFromEdges' x = (a,b) where
(a,b,_) = graphFromEdges x

-- | Build a graph from a list of nodes uniquely identified by keys,
-- with a list of keys of nodes this node should have edges to.
-- The out-list may contain keys that don't correspond to
-- nodes of the graph; they are ignored.
graphFromEdges
:: Ord key
=> [(node, key, [key])]
-> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex)
graphFromEdges edges0
= (graph, \v -> vertex_map ! v, key_vertex)
where
max_v      	    = length edges0 - 1
bounds0         = (0,max_v) :: (Vertex, Vertex)
sorted_edges    = sortBy lt edges0
edges1	    = zipWith (,) [0..] sorted_edges

graph	    = array bounds0 [(,) v (mapMaybe key_vertex ks) | (,) v (_,    _, ks) <- edges1]
key_map	    = array bounds0 [(,) v k			   | (,) v (_,    k, _ ) <- edges1]
vertex_map	    = array bounds0 edges1

(_,k1,_) `lt` (_,k2,_) = k1 `compare` k2

-- key_vertex :: key -> Maybe Vertex
-- 	returns Nothing for non-interesting vertices
key_vertex k   = findVertex 0 max_v
where
findVertex a b | a > b
= Nothing
findVertex a b = case compare k (key_map ! mid) of
LT -> findVertex a (mid-1)
EQ -> Just mid
GT -> findVertex (mid+1) b
where
mid = (a + b) `div` 2

-------------------------------------------------------------------------
--									-
--	Depth first search
--									-
-------------------------------------------------------------------------

-- | A spanning forest of the graph, obtained from a depth-first search of
-- the graph starting from each vertex in an unspecified order.
dff          :: Graph -> Forest Vertex
dff g         = dfs g (vertices g)

-- | A spanning forest of the part of the graph reachable from the listed
-- vertices, obtained from a depth-first search of the graph starting at
-- each of the listed vertices in order.
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 = run bnds (chop ts)

chop         :: Forest Vertex -> SetM s (Forest Vertex)
chop []       = return []
chop (Node v ts : us)
= do
visited <- contains v
if visited then
chop us
else do
include v
as <- chop ts
bs <- chop us
return (Node v as : bs)

-- A monad holding a set of vertices visited so far.

-- Use the ST monad if available, for constant-time primitives.

newtype SetM s a = SetM { runSetM :: STArray s Vertex Bool -> ST s a }

return x     = SetM \$ const (return x)
SetM v >>= f = SetM \$ \ s -> do { x <- v s; runSetM (f x) s }

run          :: Bounds -> (forall s. SetM s a) -> a
run bnds act  = runST (newArray bnds False >>= runSetM act)

contains     :: Vertex -> SetM s Bool
contains v    = SetM \$ \ m -> readArray m v

include      :: Vertex -> SetM s ()
include v     = SetM \$ \ m -> writeArray m v True

-- Portable implementation using IntSet.

newtype SetM s a = SetM { runSetM :: IntSet -> (a, IntSet) }

return x     = SetM \$ \ s -> (x, s)
SetM v >>= f = SetM \$ \ s -> case v s of (x, s') -> runSetM (f x) s'

run          :: Bounds -> SetM s a -> a
run _ act     = fst (runSetM act Set.empty)

contains     :: Vertex -> SetM s Bool
contains v    = SetM \$ \ m -> (Set.member v m, m)

include      :: Vertex -> SetM s ()
include v     = SetM \$ \ m -> ((), Set.insert v m)

-------------------------------------------------------------------------
--									-
--	Algorithms
--									-
-------------------------------------------------------------------------

------------------------------------------------------------
-- Algorithm 1: depth first search numbering
------------------------------------------------------------

preorder' :: Tree a -> [a] -> [a]
preorder' (Node a ts) = (a :) . preorderF' ts

preorderF' :: Forest a -> [a] -> [a]
preorderF' ts = foldr (.) id \$ map preorder' ts

preorderF :: Forest a -> [a]
preorderF ts = preorderF' 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

------------------------------------------------------------
-- Algorithm 2: topological sorting
------------------------------------------------------------

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

-- | A topological sort of the graph.
-- The order is partially specified by the condition that a vertex /i/
-- precedes /j/ whenever /j/ is reachable from /i/ but not vice versa.
topSort      :: Graph -> [Vertex]
topSort       = reverse . postOrd

------------------------------------------------------------
-- Algorithm 3: connected components
------------------------------------------------------------

-- | The connected components of a graph.
-- Two vertices are connected if there is a path between them, traversing
-- edges in either direction.
components   :: Graph -> Forest Vertex
components    = dff . undirected

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

-- Algorithm 4: strongly connected components

-- | The strongly connected components of a graph.
scc  :: Graph -> Forest Vertex
scc g = dfs g (reverse (postOrd (transposeG g)))

------------------------------------------------------------
-- Algorithm 5: Classifying edges
------------------------------------------------------------

{-
XXX unused code

tree              :: Bounds -> Forest Vertex -> Graph
tree bnds ts       = buildG bnds (concat (map flat ts))
where flat (Node v ts') = [ (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
------------------------------------------------------------

-- | A list of vertices reachable from a given vertex.
reachable    :: Graph -> Vertex -> [Vertex]
reachable g v = preorderF (dfs g [v])

-- | Is the second vertex reachable from the first?
path         :: Graph -> Vertex -> Vertex -> Bool
path g v w    = w `elem` (reachable g v)

------------------------------------------------------------
-- Algorithm 7: Biconnected components
------------------------------------------------------------

-- | The biconnected components of a graph.
-- An undirected graph is biconnected if the deletion of any vertex
-- leaves it connected.
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 (_,_,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 ]
```