Copyright | (c) The University of Glasgow 2002 |
---|---|
License | BSD-style (see the file libraries/base/LICENSE) |
Maintainer | libraries@haskell.org |
Portability | portable |
Safe Haskell | Safe |
Language | Haskell2010 |
Finite Graphs
The
type is an adjacency list representation of a finite, directed
graph with vertices of type Graph
Int
.
The
type represents a
strongly-connected component
of a graph.SCC
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
Synopsis
- type Graph = Array Vertex [Vertex]
- type Bounds = (Vertex, Vertex)
- type Edge = (Vertex, Vertex)
- type Vertex = Int
- type Table a = Array Vertex a
- 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
- vertices :: Graph -> [Vertex]
- edges :: Graph -> [Edge]
- outdegree :: Graph -> Array Vertex Int
- indegree :: Graph -> Array Vertex Int
- transposeG :: Graph -> Graph
- dfs :: Graph -> [Vertex] -> Forest Vertex
- dff :: Graph -> [Tree Vertex]
- topSort :: Graph -> [Vertex]
- reverseTopSort :: Graph -> [Vertex]
- components :: Graph -> [Tree Vertex]
- scc :: Graph -> [Tree Vertex]
- bcc :: Graph -> [Tree [Vertex]]
- reachable :: Graph -> Vertex -> [Vertex]
- path :: Graph -> Vertex -> Vertex -> Bool
- data SCC vertex
- = AcyclicSCC vertex
- | CyclicSCC [vertex]
- stronglyConnComp :: Ord key => [(node, key, [key])] -> [SCC node]
- stronglyConnCompR :: Ord key => [(node, key, [key])] -> [SCC (node, key, [key])]
- flattenSCC :: SCC vertex -> [vertex]
- flattenSCCs :: [SCC a] -> [a]
- data Tree a = Node a [Tree a]
- type Forest a = [Tree a]
Graphs
type Graph = Array Vertex [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.
Note: This is included for backwards compatibility.
Graph Construction
graphFromEdges :: Ord key => [(node, key, [key])] -> (Graph, Vertex -> (node, key, [key]), key -> Maybe Vertex) Source #
\(O((V+E) \log V)\). 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-basedInt
vertex; see warning below. This runs in \(O(1)\) time.vertexFromKey :: key -> Maybe Vertex
returns theInt
vertex for the key if it exists in the graph,Nothing
otherwise. This runs in \(O(\log V)\) time.
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])) Source #
\(O((V+E) \log V)\). 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 #
\(O(V+E)\). 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] Source #
\(O(V)\). 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] Source #
\(O(V+E)\). 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 Source #
\(O(V+E)\). 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 Source #
\(O(V+E)\). 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 Source #
\(O(V+E)\). 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 Source #
\(O(V+E)\). 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 -> [Tree Vertex] Source #
\(O(V+E)\). 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 #
\(O(V+E)\). 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.
Note: A topological sort exists only when there are no cycles in the graph.
If the graph has cycles, the output of this function will not be a
topological sort. In such a case consider using scc
.
reverseTopSort :: Graph -> [Vertex] Source #
components :: Graph -> [Tree Vertex] Source #
\(O(V+E)\). The connected components of a graph. Two vertices are connected if there is a path between them, traversing edges in either direction.
scc :: Graph -> [Tree Vertex] Source #
\(O(V+E)\). 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 -> [Tree [Vertex]] Source #
\(O(V+E)\). The biconnected components of a graph. An undirected graph is biconnected if the deletion of any vertex leaves it connected.
The input graph is expected to be undirected, i.e. for every edge in the graph the reverse edge is also in the graph. If the graph is not undirected the output is arbitrary.
reachable :: Graph -> Vertex -> [Vertex] Source #
\(O(V+E)\). 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 Source #
\(O(V+E)\). 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
Strongly connected component.
AcyclicSCC vertex | A single vertex that is not in any cycle. |
CyclicSCC [vertex] | A maximal set of mutually reachable vertices. |
Instances
Foldable SCC Source # | Since: containers-0.5.9 |
Defined in Data.Graph 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 # | |
Eq1 SCC Source # | Since: containers-0.5.9 |
Read1 SCC Source # | Since: containers-0.5.9 |
Defined in Data.Graph | |
Show1 SCC Source # | Since: containers-0.5.9 |
Traversable SCC Source # | Since: containers-0.5.9 |
Functor SCC Source # | Since: containers-0.5.4 |
Generic1 SCC Source # | |
Lift vertex => Lift (SCC vertex :: Type) Source # | Since: containers-0.6.6 |
Data vertex => Data (SCC vertex) Source # | Since: containers-0.5.9 |
Defined in Data.Graph 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 # | |
Generic (SCC vertex) Source # | |
Read vertex => Read (SCC vertex) Source # | Since: containers-0.5.9 |
Show vertex => Show (SCC vertex) Source # | Since: containers-0.5.9 |
NFData a => NFData (SCC a) Source # | |
Defined in Data.Graph | |
Eq vertex => Eq (SCC vertex) Source # | Since: containers-0.5.9 |
type Rep1 SCC Source # | Since: containers-0.5.9 |
Defined in Data.Graph type Rep1 SCC = D1 ('MetaData "SCC" "Data.Graph" "containers-0.6.7" '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 List))) | |
type Rep (SCC vertex) Source # | Since: containers-0.5.9 |
Defined in Data.Graph type Rep (SCC vertex) = D1 ('MetaData "SCC" "Data.Graph" "containers-0.6.7" '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
:: 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] |
\(O((V+E) \log V)\). 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"]
:: 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 |
\(O((V+E) \log V)\). 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] Source #
The vertices of a strongly connected component.
flattenSCCs :: [SCC a] -> [a] Source #
The vertices of a list of strongly connected components.
Trees
Non-empty, possibly infinite, multi-way trees; also known as rose trees.
Instances
MonadFix Tree Source # | Since: containers-0.5.11 |
MonadZip Tree Source # | |
Foldable Tree Source # | Folds in preorder |
Defined in Data.Tree 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 # | |
Foldable1 Tree Source # | Folds in preorder Since: containers-0.6.7 |
Defined in Data.Tree fold1 :: Semigroup m => Tree m -> m Source # foldMap1 :: Semigroup m => (a -> m) -> Tree a -> m Source # foldMap1' :: Semigroup m => (a -> m) -> Tree a -> m Source # toNonEmpty :: Tree a -> NonEmpty a Source # maximum :: Ord a => Tree a -> a Source # minimum :: Ord a => Tree a -> a Source # foldrMap1 :: (a -> b) -> (a -> b -> b) -> Tree a -> b Source # foldlMap1' :: (a -> b) -> (b -> a -> b) -> Tree a -> b Source # foldlMap1 :: (a -> b) -> (b -> a -> b) -> Tree a -> b Source # foldrMap1' :: (a -> b) -> (a -> b -> b) -> Tree a -> b Source # | |
Eq1 Tree Source # | Since: containers-0.5.9 |
Ord1 Tree Source # | Since: containers-0.5.9 |
Read1 Tree Source # | Since: containers-0.5.9 |
Defined in Data.Tree 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: containers-0.5.9 |
Traversable Tree Source # | |
Applicative Tree Source # | |
Functor Tree Source # | |
Monad Tree Source # | |
Generic1 Tree Source # | |
Lift a => Lift (Tree a :: Type) Source # | Since: containers-0.6.6 |
Data a => Data (Tree a) Source # | |
Defined in Data.Tree 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 # | |
Generic (Tree a) Source # | |
Read a => Read (Tree a) Source # | |
Show a => Show (Tree a) Source # | |
NFData a => NFData (Tree a) Source # | |
Eq a => Eq (Tree a) Source # | |
Ord a => Ord (Tree a) Source # | Since: containers-0.6.5 |
type Rep1 Tree Source # | Since: containers-0.5.8 |
Defined in Data.Tree type Rep1 Tree = D1 ('MetaData "Tree" "Data.Tree" "containers-0.6.7" 'False) (C1 ('MetaCons "Node" 'PrefixI 'True) (S1 ('MetaSel ('Just "rootLabel") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1 :*: S1 ('MetaSel ('Just "subForest") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (List :.: Rec1 Tree))) | |
type Rep (Tree a) Source # | Since: containers-0.5.8 |
Defined in Data.Tree type Rep (Tree a) = D1 ('MetaData "Tree" "Data.Tree" "containers-0.6.7" '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]))) |