Data.Graph

Finite Graphs

The Graph type is an adjacency list representation of a finite, directed graph with vertices of type Int.

The SCC type represents a strongly-connected component of a graph.

Implementation

The implementation is based on

• Structuring Depth-First Search Algorithms in Haskell, by David King and John Launchbury, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.6526

Graphs

type Graph = Array Vertex [Vertex]

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

type Bounds = (Vertex, Vertex)

The bounds of an Array.

type Edge = (Vertex, Vertex)

An edge from the first vertex to the second.

type Vertex = Int

Abstract representation of vertices.

type Table a = Array Vertex a

Table indexed by a contiguous set of vertices.

Note: This is included for backwards compatibility.

Graph Construction

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

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.

This function takes an adjacency list representing a graph with vertices of type key labeled by values of type node and produces a Graph-based representation of that list. The Graph result represents the shape of the graph, and the functions describe a) how to retrieve the label and adjacent vertices of a given vertex, and b) how to retrieve a vertex given a key.

(graph, nodeFromVertex, vertexFromKey) = graphFromEdges edgeList

• graph :: Graph is the raw, array based adjacency list for the graph.
• nodeFromVertex :: Vertex -> (node, key, [key]) returns the node associated with the given 0-based Int vertex; see warning below.
• vertexFromKey :: key -> Maybe Vertex returns the Int vertex for the key if it exists in the graph, Nothing otherwise.

To safely use this API you must either extract the list of vertices directly from the graph or first call vertexFromKey k to check if a vertex corresponds to the key k. Once it is known that a vertex exists you can use nodeFromVertex to access the labelled node and adjacent vertices. See below for examples.

Note: The out-list may contain keys that don't correspond to nodes of the graph; they are ignored.

Warning: The nodeFromVertex function will cause a runtime exception if the given Vertex does not exist.

Examples

An empty graph.

(graph, nodeFromVertex, vertexFromKey) = graphFromEdges []
graph = array (0,-1) []

A graph where the out-list references unspecified nodes ('c'), these are ignored.

(graph, _, _) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c'])]
array (0,1) [(0,[1]),(1,[])]

A graph with 3 vertices: ("a") -> ("b") -> ("c")

(graph, nodeFromVertex, vertexFromKey) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c']), ("c", 'c', [])]
graph == array (0,2) [(0,[1]),(1,[2]),(2,[])]
nodeFromVertex 0 == ("a",'a',"b")
vertexFromKey 'a' == Just 0

Get the label for a given key.

let getNodePart (n, _, _) = n
(graph, nodeFromVertex, vertexFromKey) = graphFromEdges [("a", 'a', ['b']), ("b", 'b', ['c']), ("c", 'c', [])]
getNodePart . nodeFromVertex <$> vertexFromKey 'a' == Just "A"

graphFromEdges' :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]))

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

Build a graph from a list of edges.

Warning: This function will cause a runtime exception if a vertex in the edge list is not within the given Bounds.

Examples

buildG (0,-1) [] == array (0,-1) []
buildG (0,2) [(0,1), (1,2)] == array (0,1) [(0,[1]),(1,[2])]
buildG (0,2) [(0,1), (0,2), (1,2)] == array (0,2) [(0,[2,1]),(1,[2]),(2,[])]

Graph Properties

vertices :: Graph -> [Vertex]

Returns the list of vertices in the graph.

Examples

vertices (buildG (0,-1) []) == []

vertices (buildG (0,2) [(0,1),(1,2)]) == [0,1,2]

edges :: Graph -> [Edge]

Returns the list of edges in the graph.

Examples

edges (buildG (0,-1) []) == []

edges (buildG (0,2) [(0,1),(1,2)]) == [(0,1),(1,2)]

outdegree :: Graph -> Array Vertex Int

A table of the count of edges from each node.

Examples

outdegree (buildG (0,-1) []) == array (0,-1) []

outdegree (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,1),(1,1),(2,0)]

indegree :: Graph -> Array Vertex Int

A table of the count of edges into each node.

Examples

indegree (buildG (0,-1) []) == array (0,-1) []

indegree (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,0),(1,1),(2,1)]

Graph Transformations

transposeG :: Graph -> Graph

The graph obtained by reversing all edges.

Examples

transposeG (buildG (0,2) [(0,1), (1,2)]) == array (0,2) [(0,[]),(1,[0]),(2,[1])]

Graph Algorithms

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

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

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]

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

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

The strongly connected components of a graph, in reverse topological order.

Examples

scc (buildG (0,3) [(3,1),(1,2),(2,0),(0,1)])
  == [Node {rootLabel = 0, subForest = [Node {rootLabel = 1, subForest = [Node {rootLabel = 2, subForest = []}]}]}
     ,Node {rootLabel = 3, subForest = []}]

bcc :: Graph -> Forest [Vertex]

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

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

Returns the list of vertices reachable from a given vertex.

Examples

reachable (buildG (0,0) []) 0 == [0]

reachable (buildG (0,2) [(0,1), (1,2)]) 0 == [0,1,2]

path :: Graph -> Vertex -> Vertex -> Bool

Returns True if the second vertex reachable from the first.

Examples

path (buildG (0,0) []) 0 0 == True

path (buildG (0,2) [(0,1), (1,2)]) 0 2 == True

path (buildG (0,2) [(0,1), (1,2)]) 2 0 == False

Strongly Connected Components

data SCC vertex

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

Since: 0.5.4

Defined in Data.Graph

Methods

fmap :: (a -> b) -> SCC a -> SCC b

(<$) :: a -> SCC b -> SCC a

Foldable SCC

Since: 0.5.9

Defined in Data.Graph

Methods

fold :: Monoid m => SCC m -> m

foldMap :: Monoid m => (a -> m) -> SCC a -> m

foldr :: (a -> b -> b) -> b -> SCC a -> b

foldr' :: (a -> b -> b) -> b -> SCC a -> b

foldl :: (b -> a -> b) -> b -> SCC a -> b

foldl' :: (b -> a -> b) -> b -> SCC a -> b

foldr1 :: (a -> a -> a) -> SCC a -> a

foldl1 :: (a -> a -> a) -> SCC a -> a

toList :: SCC a -> [a]

null :: SCC a -> Bool

length :: SCC a -> Int

elem :: Eq a => a -> SCC a -> Bool

maximum :: Ord a => SCC a -> a

minimum :: Ord a => SCC a -> a

sum :: Num a => SCC a -> a

product :: Num a => SCC a -> a

Traversable SCC

Since: 0.5.9

Defined in Data.Graph

Methods

traverse :: Applicative f => (a -> f b) -> SCC a -> f (SCC b)

sequenceA :: Applicative f => SCC (f a) -> f (SCC a)

mapM :: Monad m => (a -> m b) -> SCC a -> m (SCC b)

sequence :: Monad m => SCC (m a) -> m (SCC a)

Eq1 SCC

Since: 0.5.9

Defined in Data.Graph

Methods

liftEq :: (a -> b -> Bool) -> SCC a -> SCC b -> Bool

Read1 SCC

Since: 0.5.9

Defined in Data.Graph

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (SCC a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [SCC a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (SCC a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [SCC a]

Show1 SCC

Since: 0.5.9

Defined in Data.Graph

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> SCC a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [SCC a] -> ShowS

Eq vertex => Eq (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

Methods

(==) :: SCC vertex -> SCC vertex -> Bool

(/=) :: SCC vertex -> SCC vertex -> Bool

Data vertex => Data (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> SCC vertex -> c (SCC vertex)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (SCC vertex)

toConstr :: SCC vertex -> Constr

dataTypeOf :: SCC vertex -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (SCC vertex))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (SCC vertex))

gmapT :: (forall b. Data b => b -> b) -> SCC vertex -> SCC vertex

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> SCC vertex -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> SCC vertex -> r

gmapQ :: (forall d. Data d => d -> u) -> SCC vertex -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> SCC vertex -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> SCC vertex -> m (SCC vertex)

Read vertex => Read (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

Methods

readsPrec :: Int -> ReadS (SCC vertex)

readList :: ReadS [SCC vertex]

readPrec :: ReadPrec (SCC vertex)

readListPrec :: ReadPrec [SCC vertex]

Show vertex => Show (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

Methods

showsPrec :: Int -> SCC vertex -> ShowS

show :: SCC vertex -> String

showList :: [SCC vertex] -> ShowS

Generic (SCC vertex)

Defined in Data.Graph

Associated Types

type Rep (SCC vertex) :: Type -> Type

Methods

from :: SCC vertex -> Rep (SCC vertex) x

to :: Rep (SCC vertex) x -> SCC vertex

NFData a => NFData (SCC a)

Defined in Data.Graph

Methods

rnf :: SCC a -> ()

Generic1 SCC

Defined in Data.Graph

Associated Types

type Rep1 SCC :: k -> Type

Methods

from1 :: SCC a -> Rep1 SCC a

to1 :: Rep1 SCC a -> SCC a

type Rep (SCC vertex)

Since: 0.5.9

Defined in Data.Graph

type Rep (SCC vertex) = D1 (MetaData "SCC" "Data.Graph" "containers-0.6.3.1-Ep5ysRbYnpLLpgWNCSLrS7" 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

Since: 0.5.9

Defined in Data.Graph

type Rep1 SCC = D1 (MetaData "SCC" "Data.Graph" "containers-0.6.3.1-Ep5ysRbYnpLLpgWNCSLrS7" 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 [])))

Construction

stronglyConnComp

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, reverse topologically sorted.

Examples

stronglyConnComp [("a",0,[1]),("b",1,[2,3]),("c",2,[1]),("d",3,[3])]
  == [CyclicSCC ["d"],CyclicSCC ["b","c"],AcyclicSCC "a"]

stronglyConnCompR

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])]

Reverse topologically sorted

The strongly connected components of a directed graph, reverse 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.

Examples

stronglyConnCompR [("a",0,[1]),("b",1,[2,3]),("c",2,[1]),("d",3,[3])]
 == [CyclicSCC [("d",3,[3])],CyclicSCC [("b",1,[2,3]),("c",2,[1])],AcyclicSCC ("a",0,[1])]

Conversion

flattenSCC :: SCC vertex -> [vertex]

The vertices of a strongly connected component.

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

The vertices of a list of strongly connected components.

Trees

type Forest a = [Tree a]

This type synonym exists primarily for historical reasons.

data Tree a

Non-empty, possibly infinite, multi-way trees; also known as rose trees.

Constructors

Node a [Tree a]

Instances

Monad Tree

Defined in Data.Tree

Methods

(>>=) :: Tree a -> (a -> Tree b) -> Tree b

(>>) :: Tree a -> Tree b -> Tree b

return :: a -> Tree a

fail :: String -> Tree a

Functor Tree

Defined in Data.Tree

Methods

fmap :: (a -> b) -> Tree a -> Tree b

(<$) :: a -> Tree b -> Tree a

MonadFix Tree

Since: 0.5.11

Defined in Data.Tree

Methods

mfix :: (a -> Tree a) -> Tree a

Applicative Tree

Defined in Data.Tree

Methods

pure :: a -> Tree a

(<*>) :: Tree (a -> b) -> Tree a -> Tree b

liftA2 :: (a -> b -> c) -> Tree a -> Tree b -> Tree c

(*>) :: Tree a -> Tree b -> Tree b

(<*) :: Tree a -> Tree b -> Tree a

Foldable Tree

Defined in Data.Tree

Methods

fold :: Monoid m => Tree m -> m

foldMap :: Monoid m => (a -> m) -> Tree a -> m

foldr :: (a -> b -> b) -> b -> Tree a -> b

foldr' :: (a -> b -> b) -> b -> Tree a -> b

foldl :: (b -> a -> b) -> b -> Tree a -> b

foldl' :: (b -> a -> b) -> b -> Tree a -> b

foldr1 :: (a -> a -> a) -> Tree a -> a

foldl1 :: (a -> a -> a) -> Tree a -> a

toList :: Tree a -> [a]

null :: Tree a -> Bool

length :: Tree a -> Int

elem :: Eq a => a -> Tree a -> Bool

maximum :: Ord a => Tree a -> a

minimum :: Ord a => Tree a -> a

sum :: Num a => Tree a -> a

product :: Num a => Tree a -> a

Traversable Tree

Defined in Data.Tree

Methods

traverse :: Applicative f => (a -> f b) -> Tree a -> f (Tree b)

sequenceA :: Applicative f => Tree (f a) -> f (Tree a)

mapM :: Monad m => (a -> m b) -> Tree a -> m (Tree b)

sequence :: Monad m => Tree (m a) -> m (Tree a)

Eq1 Tree

Since: 0.5.9

Defined in Data.Tree

Methods

liftEq :: (a -> b -> Bool) -> Tree a -> Tree b -> Bool

Ord1 Tree

Since: 0.5.9

Defined in Data.Tree

Methods

liftCompare :: (a -> b -> Ordering) -> Tree a -> Tree b -> Ordering

Read1 Tree

Since: 0.5.9

Defined in Data.Tree

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Tree a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Tree a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Tree a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Tree a]

Show1 Tree

Since: 0.5.9

Defined in Data.Tree

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Tree a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Tree a] -> ShowS

MonadZip Tree

Defined in Data.Tree

Methods

mzip :: Tree a -> Tree b -> Tree (a, b)

mzipWith :: (a -> b -> c) -> Tree a -> Tree b -> Tree c

munzip :: Tree (a, b) -> (Tree a, Tree b)

Eq a => Eq (Tree a)

Defined in Data.Tree

Methods

(==) :: Tree a -> Tree a -> Bool

(/=) :: Tree a -> Tree a -> Bool

Data a => Data (Tree a)

Defined in Data.Tree

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Tree a -> c (Tree a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Tree a)

toConstr :: Tree a -> Constr

dataTypeOf :: Tree a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Tree a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Tree a))

gmapT :: (forall b. Data b => b -> b) -> Tree a -> Tree a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Tree a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Tree a -> r

gmapQ :: (forall d. Data d => d -> u) -> Tree a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Tree a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Tree a -> m (Tree a)

Read a => Read (Tree a)

Defined in Data.Tree

Methods

readsPrec :: Int -> ReadS (Tree a)

readList :: ReadS [Tree a]

readPrec :: ReadPrec (Tree a)

readListPrec :: ReadPrec [Tree a]

Show a => Show (Tree a)

Defined in Data.Tree

Methods

showsPrec :: Int -> Tree a -> ShowS

show :: Tree a -> String

showList :: [Tree a] -> ShowS

Generic (Tree a)

Defined in Data.Tree

Associated Types

type Rep (Tree a) :: Type -> Type

Methods

from :: Tree a -> Rep (Tree a) x

to :: Rep (Tree a) x -> Tree a

NFData a => NFData (Tree a)

Defined in Data.Tree

Methods

rnf :: Tree a -> ()

Generic1 Tree

Defined in Data.Tree

Associated Types

type Rep1 Tree :: k -> Type

Methods

from1 :: Tree a -> Rep1 Tree a

to1 :: Rep1 Tree a -> Tree a

type Rep (Tree a)

Since: 0.5.8

Defined in Data.Tree

type Rep (Tree a) = D1 (MetaData "Tree" "Data.Tree" "containers-0.6.3.1-Ep5ysRbYnpLLpgWNCSLrS7" False) (C1 (MetaCons "Node" PrefixI True) (S1 (MetaSel (Just "rootLabel") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a) :*: S1 (MetaSel (Just "subForest") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 [Tree a])))

type Rep1 Tree

Since: 0.5.8

Defined in Data.Tree

type Rep1 Tree = D1 (MetaData "Tree" "Data.Tree" "containers-0.6.3.1-Ep5ysRbYnpLLpgWNCSLrS7" False) (C1 (MetaCons "Node" PrefixI True) (S1 (MetaSel (Just "rootLabel") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1 :*: S1 (MetaSel (Just "subForest") NoSourceUnpackedness NoSourceStrictness DecidedLazy) ([] :.: Rec1 Tree)))