Copyright	(c) The University of Glasgow 2002
License	BSD-style (see the file libraries/base/LICENSE)
Maintainer	libraries@haskell.org
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell98

Data.Graph

Contents

External interface
Graphs
- Building graphs
- Graph properties
Algorithms

Description

A version of the graph algorithms described in:

Structuring Depth-First Search Algorithms in Haskell, by David King and John Launchbury.

Synopsis

stronglyConnComp :: Ord key => [(node, key, [key])] -> [SCC node]
stronglyConnCompR :: Ord key => [(node, key, [key])] -> [SCC (node, key, [key])]
data SCC vertex
- = AcyclicSCC vertex
- | CyclicSCC [vertex]
flattenSCC :: SCC vertex -> [vertex]
flattenSCCs :: [SCC a] -> [a]
type Graph = Table [Vertex]
type Table a = Array Vertex a
type Bounds = (Vertex, Vertex)
type Edge = (Vertex, Vertex)
type Vertex = Int
graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex)
graphFromEdges' :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]))
buildG :: Bounds -> [Edge] -> Graph
transposeG :: Graph -> Graph
vertices :: Graph -> [Vertex]
edges :: Graph -> [Edge]
outdegree :: Graph -> Table Int
indegree :: Graph -> Table Int
dfs :: Graph -> [Vertex] -> Forest Vertex
dff :: Graph -> Forest Vertex
topSort :: Graph -> [Vertex]
components :: Graph -> Forest Vertex
scc :: Graph -> Forest Vertex
bcc :: Graph -> Forest [Vertex]
reachable :: Graph -> Vertex -> [Vertex]
path :: Graph -> Vertex -> Vertex -> Bool
type Forest a = [Tree a]
data Tree a = Node a (Forest a)

External interface

stronglyConnComp Source #

Arguments

:: 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]

The strongly connected components of a directed graph, topologically sorted.

stronglyConnCompR Source #

Arguments

:: 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

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.

data SCC vertex Source #

Strongly connected component.

Constructors

AcyclicSCC vertex	A single vertex that is not in any cycle.
CyclicSCC [vertex]	A maximal set of mutually reachable vertices.

Instances

Functor SCC Source #	Since: 0.5.4
Instance details Methods fmap :: (a -> b) -> SCC a -> SCC b Source # (<$) :: a -> SCC b -> SCC a Source #
Foldable SCC Source #	Since: 0.5.9
Instance details Methods fold :: Monoid m => SCC m -> m Source # foldMap :: Monoid m => (a -> m) -> SCC a -> m Source # foldr :: (a -> b -> b) -> b -> SCC a -> b Source # foldr' :: (a -> b -> b) -> b -> SCC a -> b Source # foldl :: (b -> a -> b) -> b -> SCC a -> b Source # foldl' :: (b -> a -> b) -> b -> SCC a -> b Source # foldr1 :: (a -> a -> a) -> SCC a -> a Source # foldl1 :: (a -> a -> a) -> SCC a -> a Source # toList :: SCC a -> [a] Source # null :: SCC a -> Bool Source # length :: SCC a -> Int Source # elem :: Eq a => a -> SCC a -> Bool Source # maximum :: Ord a => SCC a -> a Source # minimum :: Ord a => SCC a -> a Source # sum :: Num a => SCC a -> a Source # product :: Num a => SCC a -> a Source #
Traversable SCC Source #	Since: 0.5.9
Instance details Methods traverse :: Applicative f => (a -> f b) -> SCC a -> f (SCC b) Source # sequenceA :: Applicative f => SCC (f a) -> f (SCC a) Source # mapM :: Monad m => (a -> m b) -> SCC a -> m (SCC b) Source # sequence :: Monad m => SCC (m a) -> m (SCC a) Source #
Eq1 SCC Source #	Since: 0.5.9
Instance details Methods liftEq :: (a -> b -> Bool) -> SCC a -> SCC b -> Bool Source #
Read1 SCC Source #	Since: 0.5.9
Instance details Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (SCC a) Source # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [SCC a] Source # liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (SCC a) Source # liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [SCC a] Source #
Show1 SCC Source #	Since: 0.5.9
Instance details Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> SCC a -> ShowS Source # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [SCC a] -> ShowS Source #
Eq vertex => Eq (SCC vertex) Source #	Since: 0.5.9
Instance details Methods (==) :: SCC vertex -> SCC vertex -> Bool Source # (/=) :: SCC vertex -> SCC vertex -> Bool Source #
Data vertex => Data (SCC vertex) Source #	Since: 0.5.9
Instance details Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> SCC vertex -> c (SCC vertex) Source # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (SCC vertex) Source # toConstr :: SCC vertex -> Constr Source # dataTypeOf :: SCC vertex -> DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (SCC vertex)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (SCC vertex)) Source # gmapT :: (forall b. Data b => b -> b) -> SCC vertex -> SCC vertex Source # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> SCC vertex -> r Source # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> SCC vertex -> r Source # gmapQ :: (forall d. Data d => d -> u) -> SCC vertex -> [u] Source # gmapQi :: Int -> (forall d. Data d => d -> u) -> SCC vertex -> u Source # gmapM :: Monad m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex) Source # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex) Source # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex) Source #
Read vertex => Read (SCC vertex) Source #	Since: 0.5.9
Instance details Methods readsPrec :: Int -> ReadS (SCC vertex) Source # readList :: ReadS [SCC vertex] Source # readPrec :: ReadPrec (SCC vertex) Source # readListPrec :: ReadPrec [SCC vertex] Source #
Show vertex => Show (SCC vertex) Source #	Since: 0.5.9
Instance details Methods showsPrec :: Int -> SCC vertex -> ShowS Source # show :: SCC vertex -> String Source # showList :: [SCC vertex] -> ShowS Source #
Generic (SCC vertex) Source #
Instance details Associated Types type Rep (SCC vertex) :: * -> * Source # Methods from :: SCC vertex -> Rep (SCC vertex) x Source # to :: Rep (SCC vertex) x -> SCC vertex Source #
NFData a => NFData (SCC a) Source #
Instance details Methods rnf :: SCC a -> () Source #
Generic1 SCC Source #
Instance details Associated Types type Rep1 SCC :: k -> * Source # Methods from1 :: SCC a -> Rep1 SCC a Source # to1 :: Rep1 SCC a -> SCC a Source #
type Rep (SCC vertex) Source #	Since: 0.5.9
Instance details type Rep (SCC vertex) = D1 (MetaData "SCC" "Data.Graph" "containers-0.5.11.0" False) (C1 (MetaCons "AcyclicSCC" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 vertex)) :+: C1 (MetaCons "CyclicSCC" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 [vertex])))
type Rep1 SCC Source #	Since: 0.5.9
Instance details type Rep1 SCC = D1 (MetaData "SCC" "Data.Graph" "containers-0.5.11.0" False) (C1 (MetaCons "AcyclicSCC" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1) :+: C1 (MetaCons "CyclicSCC" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 [])))

flattenSCC :: SCC vertex -> [vertex] Source #

The vertices of a strongly connected component.

flattenSCCs :: [SCC a] -> [a] Source #

The vertices of a list of strongly connected components.

Graphs

type Graph = Table [Vertex] Source #

Adjacency list representation of a graph, mapping each vertex to its list of successors.

type Table a = Array Vertex a Source #

Table indexed by a contiguous set of vertices.

type Bounds = (Vertex, Vertex) Source #

The bounds of a Table.

type Edge = (Vertex, Vertex) Source #

An edge from the first vertex to the second.

type Vertex = Int Source #

Abstract representation of vertices.

Building graphs

graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex) Source #

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])) Source #

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.

buildG :: Bounds -> [Edge] -> Graph Source #

Build a graph from a list of edges.

transposeG :: Graph -> Graph Source #

The graph obtained by reversing all edges.

Graph properties

vertices :: Graph -> [Vertex] Source #

All vertices of a graph.

edges :: Graph -> [Edge] Source #

All edges of a graph.

outdegree :: Graph -> Table Int Source #

A table of the count of edges from each node.

indegree :: Graph -> Table Int Source #

A table of the count of edges into each node.

Algorithms

dfs :: Graph -> [Vertex] -> Forest Vertex Source #

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.

dff :: Graph -> Forest Vertex Source #

A spanning forest of the graph, obtained from a depth-first search of the graph starting from each vertex in an unspecified order.

topSort :: Graph -> [Vertex] Source #

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.

components :: Graph -> Forest Vertex Source #

The connected components of a graph. Two vertices are connected if there is a path between them, traversing edges in either direction.

scc :: Graph -> Forest Vertex Source #

The strongly connected components of a graph.

bcc :: Graph -> Forest [Vertex] Source #

The biconnected components of a graph. An undirected graph is biconnected if the deletion of any vertex leaves it connected.

reachable :: Graph -> Vertex -> [Vertex] Source #

A list of vertices reachable from a given vertex.

path :: Graph -> Vertex -> Vertex -> Bool Source #

Is the second vertex reachable from the first?

type Forest a = [Tree a] Source #

data Tree a Source #

Multi-way trees, also known as rose trees.

Constructors

Node a (Forest a)

Instances

Monad Tree Source #
Instance details Methods (>>=) :: Tree a -> (a -> Tree b) -> Tree b Source # (>>) :: Tree a -> Tree b -> Tree b Source # return :: a -> Tree a Source # fail :: String -> Tree a Source #
Functor Tree Source #
Instance details Methods fmap :: (a -> b) -> Tree a -> Tree b Source # (<$) :: a -> Tree b -> Tree a Source #
MonadFix Tree Source #	Since: 0.5.11
Instance details Methods mfix :: (a -> Tree a) -> Tree a Source #
Applicative Tree Source #
Instance details Methods pure :: a -> Tree a Source # (<>) :: Tree (a -> b) -> Tree a -> Tree b Source # liftA2 :: (a -> b -> c) -> Tree a -> Tree b -> Tree c Source # (>) :: Tree a -> Tree b -> Tree b Source # (<*) :: Tree a -> Tree b -> Tree a Source #
Foldable Tree Source #
Instance details Methods fold :: Monoid m => Tree m -> m Source # foldMap :: Monoid m => (a -> m) -> Tree a -> m Source # foldr :: (a -> b -> b) -> b -> Tree a -> b Source # foldr' :: (a -> b -> b) -> b -> Tree a -> b Source # foldl :: (b -> a -> b) -> b -> Tree a -> b Source # foldl' :: (b -> a -> b) -> b -> Tree a -> b Source # foldr1 :: (a -> a -> a) -> Tree a -> a Source # foldl1 :: (a -> a -> a) -> Tree a -> a Source # toList :: Tree a -> [a] Source # null :: Tree a -> Bool Source # length :: Tree a -> Int Source # elem :: Eq a => a -> Tree a -> Bool Source # maximum :: Ord a => Tree a -> a Source # minimum :: Ord a => Tree a -> a Source # sum :: Num a => Tree a -> a Source # product :: Num a => Tree a -> a Source #
Traversable Tree Source #
Instance details Methods traverse :: Applicative f => (a -> f b) -> Tree a -> f (Tree b) Source # sequenceA :: Applicative f => Tree (f a) -> f (Tree a) Source # mapM :: Monad m => (a -> m b) -> Tree a -> m (Tree b) Source # sequence :: Monad m => Tree (m a) -> m (Tree a) Source #
Eq1 Tree Source #	Since: 0.5.9
Instance details Methods liftEq :: (a -> b -> Bool) -> Tree a -> Tree b -> Bool Source #
Ord1 Tree Source #	Since: 0.5.9
Instance details Methods liftCompare :: (a -> b -> Ordering) -> Tree a -> Tree b -> Ordering Source #
Read1 Tree Source #	Since: 0.5.9
Instance details Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Tree a) Source # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Tree a] Source # liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Tree a) Source # liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Tree a] Source #
Show1 Tree Source #	Since: 0.5.9
Instance details Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Tree a -> ShowS Source # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Tree a] -> ShowS Source #
MonadZip Tree Source #
Instance details Methods mzip :: Tree a -> Tree b -> Tree (a, b) Source # mzipWith :: (a -> b -> c) -> Tree a -> Tree b -> Tree c Source # munzip :: Tree (a, b) -> (Tree a, Tree b) Source #
Eq a => Eq (Tree a) Source #
Instance details Methods (==) :: Tree a -> Tree a -> Bool Source # (/=) :: Tree a -> Tree a -> Bool Source #
Data a => Data (Tree a) Source #
Instance details Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Tree a -> c (Tree a) Source # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Tree a) Source # toConstr :: Tree a -> Constr Source # dataTypeOf :: Tree a -> DataType Source # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Tree a)) Source # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Tree a)) Source # gmapT :: (forall b. Data b => b -> b) -> Tree a -> Tree a Source # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Tree a -> r Source # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Tree a -> r Source # gmapQ :: (forall d. Data d => d -> u) -> Tree a -> [u] Source # gmapQi :: Int -> (forall d. Data d => d -> u) -> Tree a -> u Source # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a) Source # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a) Source # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a) Source #
Read a => Read (Tree a) Source #
Instance details Methods readsPrec :: Int -> ReadS (Tree a) Source # readList :: ReadS [Tree a] Source # readPrec :: ReadPrec (Tree a) Source # readListPrec :: ReadPrec [Tree a] Source #
Show a => Show (Tree a) Source #
Instance details Methods showsPrec :: Int -> Tree a -> ShowS Source # show :: Tree a -> String Source # showList :: [Tree a] -> ShowS Source #
Generic (Tree a) Source #
Instance details Associated Types type Rep (Tree a) :: * -> * Source # Methods from :: Tree a -> Rep (Tree a) x Source # to :: Rep (Tree a) x -> Tree a Source #
NFData a => NFData (Tree a) Source #
Instance details Methods rnf :: Tree a -> () Source #
Generic1 Tree Source #
Instance details Associated Types type Rep1 Tree :: k -> * Source # Methods from1 :: Tree a -> Rep1 Tree a Source # to1 :: Rep1 Tree a -> Tree a Source #
type Rep (Tree a) Source #	Since: 0.5.8
Instance details type Rep (Tree a) = D1 (MetaData "Tree" "Data.Tree" "containers-0.5.11.0" False) (C1 (MetaCons "Node" PrefixI True) (S1 (MetaSel (Just "rootLabel") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a) :*: S1 (MetaSel (Just "subForest") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Forest a))))
type Rep1 Tree Source #	Since: 0.5.8
Instance details type Rep1 Tree = D1 (MetaData "Tree" "Data.Tree" "containers-0.5.11.0" False) (C1 (MetaCons "Node" PrefixI True) (S1 (MetaSel (Just "rootLabel") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1 :*: S1 (MetaSel (Just "subForest") NoSourceUnpackedness NoSourceStrictness DecidedLazy) ([] :.: Rec1 Tree)))