Algebra.Graph.AdjacencyMap

Alga is a library for algebraic construction and manipulation of graphs in Haskell. See this paper for the motivation behind the library, the underlying theory, and implementation details.

This module defines the AdjacencyMap data type and associated functions. See Algebra.Graph.AdjacencyMap.Algorithm for implementations of basic graph algorithms. AdjacencyMap is an instance of the Graph type class, which can be used for polymorphic graph construction and manipulation. Algebra.Graph.AdjacencyIntMap defines adjacency maps specialised to graphs with Int vertices.

Data structure

data AdjacencyMap a

The AdjacencyMap data type represents a graph by a map of vertices to their adjacency sets. We define a Num instance as a convenient notation for working with graphs:

0           == vertex 0
1 + 2       == overlay (vertex 1) (vertex 2)
1 * 2       == connect (vertex 1) (vertex 2)
1 + 2 * 3   == overlay (vertex 1) (connect (vertex 2) (vertex 3))
1 * (2 + 3) == connect (vertex 1) (overlay (vertex 2) (vertex 3))

Note: the Num instance does not satisfy several "customary laws" of Num, which dictate that fromInteger 0 and fromInteger 1 should act as additive and multiplicative identities, and negate as additive inverse. Nevertheless, overloading fromInteger, + and * is very convenient when working with algebraic graphs; we hope that in future Haskell's Prelude will provide a more fine-grained class hierarchy for algebraic structures, which we would be able to utilise without violating any laws.

The Show instance is defined using basic graph construction primitives:

show (empty     :: AdjacencyMap Int) == "empty"
show (1         :: AdjacencyMap Int) == "vertex 1"
show (1 + 2     :: AdjacencyMap Int) == "vertices [1,2]"
show (1 * 2     :: AdjacencyMap Int) == "edge 1 2"
show (1 * 2 * 3 :: AdjacencyMap Int) == "edges [(1,2),(1,3),(2,3)]"
show (1 * 2 + 3 :: AdjacencyMap Int) == "overlay (vertex 3) (edge 1 2)"

The Eq instance satisfies all axioms of algebraic graphs:

• overlay is commutative and associative:

        x + y == y + x
  x + (y + z) == (x + y) + z

• connect is associative and has empty as the identity:

    x * empty == x
    empty * x == x
  x * (y * z) == (x * y) * z

• connect distributes over overlay:

  x * (y + z) == x * y + x * z
  (x + y) * z == x * z + y * z

• connect can be decomposed:

  x * y * z == x * y + x * z + y * z

The following useful theorems can be proved from the above set of axioms.

• overlay has empty as the identity and is idempotent:

    x + empty == x
    empty + x == x
        x + x == x

• Absorption and saturation of connect:

  x * y + x + y == x * y
      x * x * x == x * x

When specifying the time and memory complexity of graph algorithms, n and m will denote the number of vertices and edges in the graph, respectively.

The total order on graphs is defined using size-lexicographic comparison:

• Compare the number of vertices. In case of a tie, continue.
• Compare the sets of vertices. In case of a tie, continue.
• Compare the number of edges. In case of a tie, continue.
• Compare the sets of edges.

Here are a few examples:

vertex 1 < vertex 2
vertex 3 < edge 1 2
vertex 1 < edge 1 1
edge 1 1 < edge 1 2
edge 1 2 < edge 1 1 + edge 2 2
edge 1 2 < edge 1 3

Note that the resulting order refines the isSubgraphOf relation and is compatible with overlay and connect operations:

isSubgraphOf x y ==> x <= y

empty <= x
x     <= x + y
x + y <= x * y

Instances

Eq a => Eq (AdjacencyMap a)

Defined in Algebra.Graph.AdjacencyMap

Methods

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

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

(Ord a, Num a) => Num (AdjacencyMap a)

Note: this does not satisfy the usual ring laws; see AdjacencyMap for more details.

Defined in Algebra.Graph.AdjacencyMap

Methods

(+) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

(-) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

(*) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

negate :: AdjacencyMap a -> AdjacencyMap a

abs :: AdjacencyMap a -> AdjacencyMap a

signum :: AdjacencyMap a -> AdjacencyMap a

fromInteger :: Integer -> AdjacencyMap a

Ord a => Ord (AdjacencyMap a)

Defined in Algebra.Graph.AdjacencyMap

Methods

compare :: AdjacencyMap a -> AdjacencyMap a -> Ordering

(<) :: AdjacencyMap a -> AdjacencyMap a -> Bool

(<=) :: AdjacencyMap a -> AdjacencyMap a -> Bool

(>) :: AdjacencyMap a -> AdjacencyMap a -> Bool

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

max :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

min :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

(Ord a, Show a) => Show (AdjacencyMap a)

Defined in Algebra.Graph.AdjacencyMap

Methods

showsPrec :: Int -> AdjacencyMap a -> ShowS

show :: AdjacencyMap a -> String

showList :: [AdjacencyMap a] -> ShowS

Generic (AdjacencyMap a)

Defined in Algebra.Graph.AdjacencyMap

Associated Types

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

Methods

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

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

NFData a => NFData (AdjacencyMap a)

Defined in Algebra.Graph.AdjacencyMap

Methods

rnf :: AdjacencyMap a -> ()

Ord a => ToGraph (AdjacencyMap a)

See Algebra.Graph.AdjacencyMap.

Defined in Algebra.Graph.ToGraph

Associated Types

type ToVertex (AdjacencyMap a) :: Type

Methods

toGraph :: AdjacencyMap a -> Graph (ToVertex (AdjacencyMap a))

foldg :: r -> (ToVertex (AdjacencyMap a) -> r) -> (r -> r -> r) -> (r -> r -> r) -> AdjacencyMap a -> r

isEmpty :: AdjacencyMap a -> Bool

hasVertex :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Bool

hasEdge :: ToVertex (AdjacencyMap a) -> ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Bool

vertexCount :: AdjacencyMap a -> Int

edgeCount :: AdjacencyMap a -> Int

vertexList :: AdjacencyMap a -> [ToVertex (AdjacencyMap a)]

edgeList :: AdjacencyMap a -> [(ToVertex (AdjacencyMap a), ToVertex (AdjacencyMap a))]

vertexSet :: AdjacencyMap a -> Set (ToVertex (AdjacencyMap a))

vertexIntSet :: AdjacencyMap a -> IntSet

edgeSet :: AdjacencyMap a -> Set (ToVertex (AdjacencyMap a), ToVertex (AdjacencyMap a))

preSet :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Set (ToVertex (AdjacencyMap a))

preIntSet :: Int -> AdjacencyMap a -> IntSet

postSet :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Set (ToVertex (AdjacencyMap a))

postIntSet :: Int -> AdjacencyMap a -> IntSet

adjacencyList :: AdjacencyMap a -> [(ToVertex (AdjacencyMap a), [ToVertex (AdjacencyMap a)])]

dfsForest :: AdjacencyMap a -> Forest (ToVertex (AdjacencyMap a))

dfsForestFrom :: [ToVertex (AdjacencyMap a)] -> AdjacencyMap a -> Forest (ToVertex (AdjacencyMap a))

dfs :: [ToVertex (AdjacencyMap a)] -> AdjacencyMap a -> [ToVertex (AdjacencyMap a)]

reachable :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> [ToVertex (AdjacencyMap a)]

topSort :: AdjacencyMap a -> Either (Cycle (ToVertex (AdjacencyMap a))) [ToVertex (AdjacencyMap a)]

isAcyclic :: AdjacencyMap a -> Bool

toAdjacencyMap :: AdjacencyMap a -> AdjacencyMap (ToVertex (AdjacencyMap a))

toAdjacencyMapTranspose :: AdjacencyMap a -> AdjacencyMap (ToVertex (AdjacencyMap a))

toAdjacencyIntMap :: AdjacencyMap a -> AdjacencyIntMap

toAdjacencyIntMapTranspose :: AdjacencyMap a -> AdjacencyIntMap

isDfsForestOf :: Forest (ToVertex (AdjacencyMap a)) -> AdjacencyMap a -> Bool

isTopSortOf :: [ToVertex (AdjacencyMap a)] -> AdjacencyMap a -> Bool

Ord a => Graph (AdjacencyMap a)

Defined in Algebra.Graph.Class

Associated Types

type Vertex (AdjacencyMap a) :: Type

Methods

empty :: AdjacencyMap a

vertex :: Vertex (AdjacencyMap a) -> AdjacencyMap a

overlay :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

connect :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

type Rep (AdjacencyMap a)

Defined in Algebra.Graph.AdjacencyMap

type Rep (AdjacencyMap a) = D1 (MetaData "AdjacencyMap" "Algebra.Graph.AdjacencyMap" "algebraic-graphs-0.5-4dnrALfehjHELqhQlGFoFS" True) (C1 (MetaCons "AM" PrefixI True) (S1 (MetaSel (Just "adjacencyMap") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Map a (Set a)))))

type ToVertex (AdjacencyMap a)

Defined in Algebra.Graph.ToGraph

type ToVertex (AdjacencyMap a) = a

type Vertex (AdjacencyMap a)

Defined in Algebra.Graph.Class

type Vertex (AdjacencyMap a) = a

adjacencyMap :: AdjacencyMap a -> Map a (Set a)

The adjacency map of a graph: each vertex is associated with a set of its direct successors. Complexity: O(1) time and memory.

adjacencyMap empty      == Map.empty
adjacencyMap (vertex x) == Map.singleton x Set.empty
adjacencyMap (edge 1 1) == Map.singleton 1 (Set.singleton 1)
adjacencyMap (edge 1 2) == Map.fromList [(1,Set.singleton 2), (2,Set.empty)]

Basic graph construction primitives

empty :: AdjacencyMap a

Construct the empty graph. Complexity: O(1) time and memory.

isEmpty     empty == True
hasVertex x empty == False
vertexCount empty == 0
edgeCount   empty == 0

vertex :: a -> AdjacencyMap a

Construct the graph comprising a single isolated vertex. Complexity: O(1) time and memory.

isEmpty     (vertex x) == False
hasVertex x (vertex y) == (x == y)
vertexCount (vertex x) == 1
edgeCount   (vertex x) == 0

edge :: Ord a => a -> a -> AdjacencyMap a

Construct the graph comprising a single edge. Complexity: O(1) time, memory.

edge x y               == connect (vertex x) (vertex y)
hasEdge x y (edge x y) == True
edgeCount   (edge x y) == 1
vertexCount (edge 1 1) == 1
vertexCount (edge 1 2) == 2

overlay :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

Overlay two graphs. This is a commutative, associative and idempotent operation with the identity empty. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

isEmpty     (overlay x y) == isEmpty   x   && isEmpty   y
hasVertex z (overlay x y) == hasVertex z x || hasVertex z y
vertexCount (overlay x y) >= vertexCount x
vertexCount (overlay x y) <= vertexCount x + vertexCount y
edgeCount   (overlay x y) >= edgeCount x
edgeCount   (overlay x y) <= edgeCount x   + edgeCount y
vertexCount (overlay 1 2) == 2
edgeCount   (overlay 1 2) == 0

connect :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

Connect two graphs. This is an associative operation with the identity empty, which distributes over overlay and obeys the decomposition axiom. Complexity: O((n + m) * log(n)) time and O(n + m) memory. Note that the number of edges in the resulting graph is quadratic with respect to the number of vertices of the arguments: m = O(m1 + m2 + n1 * n2).

isEmpty     (connect x y) == isEmpty   x   && isEmpty   y
hasVertex z (connect x y) == hasVertex z x || hasVertex z y
vertexCount (connect x y) >= vertexCount x
vertexCount (connect x y) <= vertexCount x + vertexCount y
edgeCount   (connect x y) >= edgeCount x
edgeCount   (connect x y) >= edgeCount y
edgeCount   (connect x y) >= vertexCount x * vertexCount y
edgeCount   (connect x y) <= vertexCount x * vertexCount y + edgeCount x + edgeCount y
vertexCount (connect 1 2) == 2
edgeCount   (connect 1 2) == 1

vertices :: Ord a => [a] -> AdjacencyMap a

Construct the graph comprising a given list of isolated vertices. Complexity: O(L * log(L)) time and O(L) memory, where L is the length of the given list.

vertices []            == empty
vertices [x]           == vertex x
hasVertex x . vertices == elem x
vertexCount . vertices == length . nub
vertexSet   . vertices == Set.fromList

edges :: Ord a => [(a, a)] -> AdjacencyMap a

Construct the graph from a list of edges. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

edges []          == empty
edges [(x,y)]     == edge x y
edges             == overlays . map (uncurry edge)
edgeCount . edges == length . nub
edgeList . edges  == nub . sort

overlays :: Ord a => [AdjacencyMap a] -> AdjacencyMap a

Overlay a given list of graphs. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

overlays []        == empty
overlays [x]       == x
overlays [x,y]     == overlay x y
overlays           == foldr overlay empty
isEmpty . overlays == all isEmpty

connects :: Ord a => [AdjacencyMap a] -> AdjacencyMap a

Connect a given list of graphs. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

connects []        == empty
connects [x]       == x
connects [x,y]     == connect x y
connects           == foldr connect empty
isEmpty . connects == all isEmpty

Relations on graphs

isSubgraphOf :: Ord a => AdjacencyMap a -> AdjacencyMap a -> Bool

The isSubgraphOf function takes two graphs and returns True if the first graph is a subgraph of the second. Complexity: O((n + m) * log(n)) time.

isSubgraphOf empty         x             ==  True
isSubgraphOf (vertex x)    empty         ==  False
isSubgraphOf x             (overlay x y) ==  True
isSubgraphOf (overlay x y) (connect x y) ==  True
isSubgraphOf (path xs)     (circuit xs)  ==  True
isSubgraphOf x y                         ==> x <= y

Graph properties

isEmpty :: AdjacencyMap a -> Bool

Check if a graph is empty. Complexity: O(1) time.

isEmpty empty                       == True
isEmpty (overlay empty empty)       == True
isEmpty (vertex x)                  == False
isEmpty (removeVertex x $ vertex x) == True
isEmpty (removeEdge x y $ edge x y) == False

hasVertex :: Ord a => a -> AdjacencyMap a -> Bool

Check if a graph contains a given vertex. Complexity: O(log(n)) time.

hasVertex x empty            == False
hasVertex x (vertex y)       == (x == y)
hasVertex x . removeVertex x == const False

hasEdge :: Ord a => a -> a -> AdjacencyMap a -> Bool

Check if a graph contains a given edge. Complexity: O(log(n)) time.

hasEdge x y empty            == False
hasEdge x y (vertex z)       == False
hasEdge x y (edge x y)       == True
hasEdge x y . removeEdge x y == const False
hasEdge x y                  == elem (x,y) . edgeList

vertexCount :: AdjacencyMap a -> Int

The number of vertices in a graph. Complexity: O(1) time.

vertexCount empty             ==  0
vertexCount (vertex x)        ==  1
vertexCount                   ==  length . vertexList
vertexCount x < vertexCount y ==> x < y

edgeCount :: AdjacencyMap a -> Int

The number of edges in a graph. Complexity: O(n) time.

edgeCount empty      == 0
edgeCount (vertex x) == 0
edgeCount (edge x y) == 1
edgeCount            == length . edgeList

vertexList :: AdjacencyMap a -> [a]

The sorted list of vertices of a given graph. Complexity: O(n) time and memory.

vertexList empty      == []
vertexList (vertex x) == [x]
vertexList . vertices == nub . sort

edgeList :: AdjacencyMap a -> [(a, a)]

The sorted list of edges of a graph. Complexity: O(n + m) time and O(m) memory.

edgeList empty          == []
edgeList (vertex x)     == []
edgeList (edge x y)     == [(x,y)]
edgeList (star 2 [3,1]) == [(2,1), (2,3)]
edgeList . edges        == nub . sort
edgeList . transpose    == sort . map swap . edgeList

adjacencyList :: AdjacencyMap a -> [(a, [a])]

The sorted adjacency list of a graph. Complexity: O(n + m) time and O(m) memory.

adjacencyList empty          == []
adjacencyList (vertex x)     == [(x, [])]
adjacencyList (edge 1 2)     == [(1, [2]), (2, [])]
adjacencyList (star 2 [3,1]) == [(1, []), (2, [1,3]), (3, [])]
stars . adjacencyList        == id

vertexSet :: AdjacencyMap a -> Set a

The set of vertices of a given graph. Complexity: O(n) time and memory.

vertexSet empty      == Set.empty
vertexSet . vertex   == Set.singleton
vertexSet . vertices == Set.fromList

edgeSet :: Eq a => AdjacencyMap a -> Set (a, a)

The set of edges of a given graph. Complexity: O((n + m) * log(m)) time and O(m) memory.

edgeSet empty      == Set.empty
edgeSet (vertex x) == Set.empty
edgeSet (edge x y) == Set.singleton (x,y)
edgeSet . edges    == Set.fromList

preSet :: Ord a => a -> AdjacencyMap a -> Set a

The preset of an element x is the set of its direct predecessors. Complexity: O(n * log(n)) time and O(n) memory.

preSet x empty      == Set.empty
preSet x (vertex x) == Set.empty
preSet 1 (edge 1 2) == Set.empty
preSet y (edge x y) == Set.fromList [x]

postSet :: Ord a => a -> AdjacencyMap a -> Set a

The postset of a vertex is the set of its direct successors. Complexity: O(log(n)) time and O(1) memory.

postSet x empty      == Set.empty
postSet x (vertex x) == Set.empty
postSet x (edge x y) == Set.fromList [y]
postSet 2 (edge 1 2) == Set.empty

Standard families of graphs

path :: Ord a => [a] -> AdjacencyMap a

The path on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

path []        == empty
path [x]       == vertex x
path [x,y]     == edge x y
path . reverse == transpose . path

circuit :: Ord a => [a] -> AdjacencyMap a

The circuit on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

circuit []        == empty
circuit [x]       == edge x x
circuit [x,y]     == edges [(x,y), (y,x)]
circuit . reverse == transpose . circuit

clique :: Ord a => [a] -> AdjacencyMap a

The clique on a list of vertices. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

clique []         == empty
clique [x]        == vertex x
clique [x,y]      == edge x y
clique [x,y,z]    == edges [(x,y), (x,z), (y,z)]
clique (xs ++ ys) == connect (clique xs) (clique ys)
clique . reverse  == transpose . clique

biclique :: Ord a => [a] -> [a] -> AdjacencyMap a

The biclique on two lists of vertices. Complexity: O(n * log(n) + m) time and O(n + m) memory.

biclique []      []      == empty
biclique [x]     []      == vertex x
biclique []      [y]     == vertex y
biclique [x1,x2] [y1,y2] == edges [(x1,y1), (x1,y2), (x2,y1), (x2,y2)]
biclique xs      ys      == connect (vertices xs) (vertices ys)

star :: Ord a => a -> [a] -> AdjacencyMap a

The star formed by a centre vertex connected to a list of leaves. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

star x []    == vertex x
star x [y]   == edge x y
star x [y,z] == edges [(x,y), (x,z)]
star x ys    == connect (vertex x) (vertices ys)

stars :: Ord a => [(a, [a])] -> AdjacencyMap a

The stars formed by overlaying a list of stars. An inverse of adjacencyList. Complexity: O(L * log(n)) time, memory and size, where L is the total size of the input.

stars []                      == empty
stars [(x, [])]               == vertex x
stars [(x, [y])]              == edge x y
stars [(x, ys)]               == star x ys
stars                         == overlays . map (uncurry star)
stars . adjacencyList         == id
overlay (stars xs) (stars ys) == stars (xs ++ ys)

fromAdjacencySets :: Ord a => [(a, Set a)] -> AdjacencyMap a

Construct a graph from a list of adjacency sets; a variation of stars. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

fromAdjacencySets []                                  == empty
fromAdjacencySets [(x, Set.empty)]                    == vertex x
fromAdjacencySets [(x, Set.singleton y)]              == edge x y
fromAdjacencySets . map (fmap Set.fromList)           == stars
overlay (fromAdjacencySets xs) (fromAdjacencySets ys) == fromAdjacencySets (xs ++ ys)

tree :: Ord a => Tree a -> AdjacencyMap a

The tree graph constructed from a given Tree data structure. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

tree (Node x [])                                         == vertex x
tree (Node x [Node y [Node z []]])                       == path [x,y,z]
tree (Node x [Node y [], Node z []])                     == star x [y,z]
tree (Node 1 [Node 2 [], Node 3 [Node 4 [], Node 5 []]]) == edges [(1,2), (1,3), (3,4), (3,5)]

forest :: Ord a => Forest a -> AdjacencyMap a

The forest graph constructed from a given Forest data structure. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

forest []                                                  == empty
forest [x]                                                 == tree x
forest [Node 1 [Node 2 [], Node 3 []], Node 4 [Node 5 []]] == edges [(1,2), (1,3), (4,5)]
forest                                                     == overlays . map tree

Graph transformation

removeVertex :: Ord a => a -> AdjacencyMap a -> AdjacencyMap a

Remove a vertex from a given graph. Complexity: O(n*log(n)) time.

removeVertex x (vertex x)       == empty
removeVertex 1 (vertex 2)       == vertex 2
removeVertex x (edge x x)       == empty
removeVertex 1 (edge 1 2)       == vertex 2
removeVertex x . removeVertex x == removeVertex x

removeEdge :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a

Remove an edge from a given graph. Complexity: O(log(n)) time.

removeEdge x y (edge x y)       == vertices [x,y]
removeEdge x y . removeEdge x y == removeEdge x y
removeEdge x y . removeVertex x == removeVertex x
removeEdge 1 1 (1 * 1 * 2 * 2)  == 1 * 2 * 2
removeEdge 1 2 (1 * 1 * 2 * 2)  == 1 * 1 + 2 * 2

replaceVertex :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a

The function replaceVertex x y replaces vertex x with vertex y in a given AdjacencyMap. If y already exists, x and y will be merged. Complexity: O((n + m) * log(n)) time.

replaceVertex x x            == id
replaceVertex x y (vertex x) == vertex y
replaceVertex x y            == mergeVertices (== x) y

mergeVertices :: Ord a => (a -> Bool) -> a -> AdjacencyMap a -> AdjacencyMap a

Merge vertices satisfying a given predicate into a given vertex. Complexity: O((n + m) * log(n)) time, assuming that the predicate takes O(1) to be evaluated.

mergeVertices (const False) x    == id
mergeVertices (== x) y           == replaceVertex x y
mergeVertices even 1 (0 * 2)     == 1 * 1
mergeVertices odd  1 (3 + 4 * 5) == 4 * 1

transpose :: Ord a => AdjacencyMap a -> AdjacencyMap a

Transpose a given graph. Complexity: O(m * log(n)) time, O(n + m) memory.

transpose empty       == empty
transpose (vertex x)  == vertex x
transpose (edge x y)  == edge y x
transpose . transpose == id
edgeList . transpose  == sort . map swap . edgeList

gmap :: (Ord a, Ord b) => (a -> b) -> AdjacencyMap a -> AdjacencyMap b

Transform a graph by applying a function to each of its vertices. This is similar to Functor's fmap but can be used with non-fully-parametric AdjacencyMap. Complexity: O((n + m) * log(n)) time.

gmap f empty      == empty
gmap f (vertex x) == vertex (f x)
gmap f (edge x y) == edge (f x) (f y)
gmap id           == id
gmap f . gmap g   == gmap (f . g)

induce :: (a -> Bool) -> AdjacencyMap a -> AdjacencyMap a

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(n + m) time, assuming that the predicate takes O(1) to be evaluated.

induce (const True ) x      == x
induce (const False) x      == empty
induce (/= x)               == removeVertex x
induce p . induce q         == induce (\x -> p x && q x)
isSubgraphOf (induce p x) x == True

induceJust :: Ord a => AdjacencyMap (Maybe a) -> AdjacencyMap a

Construct the induced subgraph of a given graph by removing the vertices that are Nothing. Complexity: O(n + m) time.

induceJust (vertex Nothing)                               == empty
induceJust (edge (Just x) Nothing)                        == vertex x
induceJust . gmap Just                                    == id
induceJust . gmap (\x -> if p x then Just x else Nothing) == induce p

Graph composition

compose :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a

Left-to-right relational composition of graphs: vertices x and z are connected in the resulting graph if there is a vertex y, such that x is connected to y in the first graph, and y is connected to z in the second graph. There are no isolated vertices in the result. This operation is associative, has empty and single-vertex graphs as annihilating zeroes, and distributes over overlay. Complexity: O(n * m * log(n)) time and O(n + m) memory.

compose empty            x                == empty
compose x                empty            == empty
compose (vertex x)       y                == empty
compose x                (vertex y)       == empty
compose x                (compose y z)    == compose (compose x y) z
compose x                (overlay y z)    == overlay (compose x y) (compose x z)
compose (overlay x y)    z                == overlay (compose x z) (compose y z)
compose (edge x y)       (edge y z)       == edge x z
compose (path    [1..5]) (path    [1..5]) == edges [(1,3), (2,4), (3,5)]
compose (circuit [1..5]) (circuit [1..5]) == circuit [1,3,5,2,4]

box :: (Ord a, Ord b) => AdjacencyMap a -> AdjacencyMap b -> AdjacencyMap (a, b)

Compute the Cartesian product of graphs. Complexity: O(n * m * log(n)^2) time.

box (path [0,1]) (path "ab") == edges [ ((0,'a'), (0,'b'))
                                      , ((0,'a'), (1,'a'))
                                      , ((0,'b'), (1,'b'))
                                      , ((1,'a'), (1,'b')) ]

Up to an isomorphism between the resulting vertex types, this operation is commutative, associative, distributes over overlay, has singleton graphs as identities and empty as the annihilating zero. Below ~~ stands for the equality up to an isomorphism, e.g. (x, ()) ~~ x.

box x y               ~~ box y x
box x (box y z)       ~~ box (box x y) z
box x (overlay y z)   == overlay (box x y) (box x z)
box x (vertex ())     ~~ x
box x empty           ~~ empty
transpose   (box x y) == box (transpose x) (transpose y)
vertexCount (box x y) == vertexCount x * vertexCount y
edgeCount   (box x y) <= vertexCount x * edgeCount y + edgeCount x * vertexCount y

Relational operations

closure :: Ord a => AdjacencyMap a -> AdjacencyMap a

Compute the reflexive and transitive closure of a graph. Complexity: O(n * m * log(n)^2) time.

closure empty           == empty
closure (vertex x)      == edge x x
closure (edge x x)      == edge x x
closure (edge x y)      == edges [(x,x), (x,y), (y,y)]
closure (path $ nub xs) == reflexiveClosure (clique $ nub xs)
closure                 == reflexiveClosure . transitiveClosure
closure                 == transitiveClosure . reflexiveClosure
closure . closure       == closure
postSet x (closure y)   == Set.fromList (reachable x y)

reflexiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a

Compute the reflexive closure of a graph by adding a self-loop to every vertex. Complexity: O(n * log(n)) time.

reflexiveClosure empty              == empty
reflexiveClosure (vertex x)         == edge x x
reflexiveClosure (edge x x)         == edge x x
reflexiveClosure (edge x y)         == edges [(x,x), (x,y), (y,y)]
reflexiveClosure . reflexiveClosure == reflexiveClosure

symmetricClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a

Compute the symmetric closure of a graph by overlaying it with its own transpose. Complexity: O((n + m) * log(n)) time.

symmetricClosure empty              == empty
symmetricClosure (vertex x)         == vertex x
symmetricClosure (edge x y)         == edges [(x,y), (y,x)]
symmetricClosure x                  == overlay x (transpose x)
symmetricClosure . symmetricClosure == symmetricClosure

transitiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a

Compute the transitive closure of a graph. Complexity: O(n * m * log(n)^2) time.

transitiveClosure empty               == empty
transitiveClosure (vertex x)          == vertex x
transitiveClosure (edge x y)          == edge x y
transitiveClosure (path $ nub xs)     == clique (nub xs)
transitiveClosure . transitiveClosure == transitiveClosure

Miscellaneous

consistent :: Ord a => AdjacencyMap a -> Bool

Check that the internal graph representation is consistent, i.e. that all edges refer to existing vertices. It should be impossible to create an inconsistent adjacency map, and we use this function in testing.

consistent empty         == True
consistent (vertex x)    == True
consistent (overlay x y) == True
consistent (connect x y) == True
consistent (edge x y)    == True
consistent (edges xs)    == True
consistent (stars xs)    == True