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Data.Graph

Copyright (c) The University of Glasgow 2002
License BSD-style (see the file libraries/base/LICENSE)
Maintainer [email protected]
Portability portable
Safe Haskell Safe
Language Haskell98

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

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

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 retrive 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.

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

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
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

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

Eq1 SCC

Since: containers-0.5.9

Instance details

Defined in Data.Graph

Methods

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

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

Eq vertex => Eq (SCC vertex)

Since: containers-0.5.9

Instance details

Defined in Data.Graph

Methods

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

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

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 :: (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

Generic1 SCC
Instance details

Defined in Data.Graph

Associated Types

type Rep1 SCC :: k -> Type Source

Methods

from1 :: SCC a -> Rep1 SCC a Source

to1 :: Rep1 SCC a -> SCC a Source

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-0.6.0.1" 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])))
type Rep1 SCC

Since: containers-0.5.9

Instance details

Defined in Data.Graph

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, 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.

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

data Tree a Source

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

Constructors

Node a (Forest a)
Instances
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

fail :: String -> 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

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

Eq1 Tree

Since: containers-0.5.9

Instance details

Defined in Data.Tree

Methods

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

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

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

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

Eq a => Eq (Tree a)
Instance details

Defined in Data.Tree

Methods

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

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

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 :: (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

Generic1 Tree
Instance details

Defined in Data.Tree

Associated Types

type Rep1 Tree :: k -> Type Source

Methods

from1 :: Tree a -> Rep1 Tree a Source

to1 :: Rep1 Tree a -> Tree a Source

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-0.6.0.1" 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

Since: containers-0.5.8

Instance details

Defined in Data.Tree

type Rep1 Tree = D1 (MetaData "Tree" "Data.Tree" "containers-0.6.0.1" False) (C1 (MetaCons "Node" PrefixI True) (S1 (MetaSel (Just "rootLabel") NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1 :*: S1 (MetaSel (Just "subForest") NoSourceUnpackedness NoSourceStrictness DecidedLazy) ([] :.: Rec1 Tree)))

© The University of Glasgow and others
Licensed under a BSD-style license (see top of the page).
https://downloads.haskell.org/~ghc/8.6.1/docs/html/libraries/containers-0.6.0.1/Data-Graph.html