containers-0.6.4.1: Assorted concrete container types
Copyright(c) The University of Glasgow 2002
LicenseBSD-style (see the file libraries/base/LICENSE)
Maintainerlibraries@haskell.org
Portabilityportable
Safe HaskellSafe
LanguageHaskell2010

Data.Graph

Description

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

Synopsis

Graphs

type Graph = Array Vertex [Vertex] Source #

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

type Bounds = (Vertex, Vertex) Source #

The bounds of an Array.

type Edge = (Vertex, Vertex) Source #

An edge from the first vertex to the second.

type Vertex = Int Source #

Abstract representation of vertices.

type Table a = Array Vertex a Source #

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) 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.

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

Expand

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])) 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.

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

Examples

Expand
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] Source #

Returns the list of vertices in the graph.

Examples

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

edges :: Graph -> [Edge] Source #

Returns the list of edges in the graph.

Examples

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

outdegree :: Graph -> Array Vertex Int Source #

A table of the count of edges from each node.

Examples

Expand
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 Source #

A table of the count of edges into each node.

Examples

Expand
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 Source #

The graph obtained by reversing all edges.

Examples

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

Graph 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.

reverseTopSort :: Graph -> [Vertex] Source #

Reverse ordering of topSort.

Since: containers-0.6.4

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, in reverse topological order.

Examples

Expand
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] 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 #

Returns the list of vertices reachable from a given vertex.

Examples

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

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

Returns True if the second vertex reachable from the first.

Examples

Expand
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 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

Instances details
Functor SCC #

Since: containers-0.5.4

Instance details

Defined in Data.Graph

Methods

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

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

Foldable SCC #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

Methods

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

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

Since: containers-0.5.9

Instance details

Defined in Data.Graph

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 #

Show1 SCC #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

Methods

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

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

Read1 SCC #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

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 #

Eq1 SCC #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

Methods

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

Generic1 SCC # 
Instance details

Defined in Data.Graph

Associated Types

type Rep1 SCC :: k -> Type Source #

Methods

from1 :: forall (a :: k). SCC a -> Rep1 SCC a Source #

to1 :: forall (a :: k). Rep1 SCC a -> SCC a Source #

Eq vertex => Eq (SCC vertex) #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

Methods

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

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

Data vertex => Data (SCC vertex) #

Since: containers-0.5.9

Instance details

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) 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 :: forall r r'. (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) #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

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) #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

Methods

showsPrec :: Int -> SCC vertex -> ShowS Source #

show :: SCC vertex -> String Source #

showList :: [SCC vertex] -> ShowS Source #

Generic (SCC vertex) # 
Instance details

Defined in Data.Graph

Associated Types

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

Methods

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

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

NFData a => NFData (SCC a) # 
Instance details

Defined in Data.Graph

Methods

rnf :: SCC a -> () Source #

type Rep1 SCC #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

type Rep (SCC vertex) #

Since: containers-0.5.9

Instance details

Defined in Data.Graph

type Rep (SCC vertex) = D1 ('MetaData "SCC" "Data.Graph" "containers-0.6.4.1" '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])))

Construction

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, reverse topologically sorted.

Examples

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

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])]

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

Expand
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] Source #

The vertices of a strongly connected component.

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

The vertices of a list of strongly connected components.

Trees

type Forest a = [Tree a] Source #

This type synonym exists primarily for historical reasons.

data Tree a Source #

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

Constructors

Node a [Tree a] 

Instances

Instances details
Monad Tree # 
Instance details

Defined in Data.Tree

Methods

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

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

return :: a -> Tree a Source #

Functor Tree # 
Instance details

Defined in Data.Tree

Methods

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

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

MonadFix Tree #

Since: containers-0.5.11

Instance details

Defined in Data.Tree

Methods

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

Applicative Tree # 
Instance details

Defined in Data.Tree

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 # 
Instance details

Defined in Data.Tree

Methods

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

foldMap :: Monoid m => (a -> m) -> Tree a -> 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 # 
Instance details

Defined in Data.Tree

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 #

Show1 Tree #

Since: containers-0.5.9

Instance details

Defined in Data.Tree

Methods

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

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

Read1 Tree #

Since: containers-0.5.9

Instance details

Defined in Data.Tree

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 #

Ord1 Tree #

Since: containers-0.5.9

Instance details

Defined in Data.Tree

Methods

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

Eq1 Tree #

Since: containers-0.5.9

Instance details

Defined in Data.Tree

Methods

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

MonadZip Tree # 
Instance details

Defined in Data.Tree

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 #

Generic1 Tree # 
Instance details

Defined in Data.Tree

Associated Types

type Rep1 Tree :: k -> Type Source #

Methods

from1 :: forall (a :: k). Tree a -> Rep1 Tree a Source #

to1 :: forall (a :: k). Rep1 Tree a -> Tree a Source #

Eq a => Eq (Tree a) # 
Instance details

Defined in Data.Tree

Methods

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

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

Data a => Data (Tree a) # 
Instance details

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) 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 :: forall r r'. (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) # 
Instance details

Defined in Data.Tree

Show a => Show (Tree a) # 
Instance details

Defined in Data.Tree

Methods

showsPrec :: Int -> Tree a -> ShowS Source #

show :: Tree a -> String Source #

showList :: [Tree a] -> ShowS Source #

Generic (Tree a) # 
Instance details

Defined in Data.Tree

Associated Types

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

Methods

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

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

NFData a => NFData (Tree a) # 
Instance details

Defined in Data.Tree

Methods

rnf :: Tree a -> () Source #

type Rep1 Tree #

Since: containers-0.5.8

Instance details

Defined in Data.Tree

type Rep1 Tree = D1 ('MetaData "Tree" "Data.Tree" "containers-0.6.4.1" 'False) (C1 ('MetaCons "Node" 'PrefixI 'True) (S1 ('MetaSel ('Just "rootLabel") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1 :*: S1 ('MetaSel ('Just "subForest") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) ([] :.: Rec1 Tree)))
type Rep (Tree a) #

Since: containers-0.5.8

Instance details

Defined in Data.Tree

type Rep (Tree a) = D1 ('MetaData "Tree" "Data.Tree" "containers-0.6.4.1" '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])))