-----------------------------------------------------------------------------
-- |
-- Module : Data.Graph
-- Copyright : (c) The University of Glasgow 2002
-- License : BSD-style (see the file libraries/base/LICENSE)
--
-- Maintainer : libraries@haskell.org
-- 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
#if __GLASGOW_HASKELL__
# define USE_ST_MONAD 1
#endif
-- Extensions
#if USE_ST_MONAD
import Control.Monad.ST
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
#ifdef __HADDOCK__
import Prelude
#endif
-------------------------------------------------------------------------
-- -
-- 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.
#if USE_ST_MONAD
-- Use the ST monad if available, for constant-time primitives.
newtype SetM s a = SetM { runSetM :: STArray s Vertex Bool -> ST s a }
instance Monad (SetM s) where
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
#else /* !USE_ST_MONAD */
-- Portable implementation using IntSet.
newtype SetM s a = SetM { runSetM :: IntSet -> (a, IntSet) }
instance Monad (SetM s) where
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)
#endif /* !USE_ST_MONAD */
-------------------------------------------------------------------------
-- -
-- Algorithms
-- -
-------------------------------------------------------------------------
------------------------------------------------------------
-- Algorithm 1: depth first search numbering
------------------------------------------------------------
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
------------------------------------------------------------
-- Algorithm 2: topological sorting
------------------------------------------------------------
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
-- | 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
------------------------------------------------------------
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 (u,du,lu) xs <- us])
bicomps :: Tree (Vertex,Int,Int) -> Forest [Vertex]
bicomps (Node (v,_,_) 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 ]
|
