Algebra.Graph.Relation

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 Relation data type, as well as associated operations and algorithms. Relation is an instance of the Graph type class, which can be used for polymorphic graph construction and manipulation.

Data structure

data Relation a

The Relation data type represents a graph as a binary relation. 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))

The Show instance is defined using basic graph construction primitives:

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

Instances

Eq a => Eq (Relation a)

Defined in Algebra.Graph.Relation.Internal

Methods

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

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

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

Defined in Algebra.Graph.Relation.Internal

Methods

(+) :: Relation a -> Relation a -> Relation a

(-) :: Relation a -> Relation a -> Relation a

(*) :: Relation a -> Relation a -> Relation a

negate :: Relation a -> Relation a

abs :: Relation a -> Relation a

signum :: Relation a -> Relation a

fromInteger :: Integer -> Relation a

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

Defined in Algebra.Graph.Relation.Internal

Methods

showsPrec :: Int -> Relation a -> ShowS

show :: Relation a -> String

showList :: [Relation a] -> ShowS

NFData a => NFData (Relation a)

Defined in Algebra.Graph.Relation.Internal

Methods

rnf :: Relation a -> ()

Ord a => ToGraph (Relation a)

Defined in Algebra.Graph.ToGraph

Associated Types

type ToVertex (Relation a) :: *

Methods

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

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

isEmpty :: Relation a -> Bool

size :: Relation a -> Int

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

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

vertexCount :: Relation a -> Int

edgeCount :: Relation a -> Int

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

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

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

vertexIntSet :: Relation a -> IntSet

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

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

preIntSet :: Int -> Relation a -> IntSet

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

postIntSet :: Int -> Relation a -> IntSet

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

adjacencyMap :: Relation a -> Map (ToVertex (Relation a)) (Set (ToVertex (Relation a)))

adjacencyIntMap :: Relation a -> IntMap IntSet

adjacencyMapTranspose :: Relation a -> Map (ToVertex (Relation a)) (Set (ToVertex (Relation a)))

adjacencyIntMapTranspose :: Relation a -> IntMap IntSet

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

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

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

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

topSort :: Relation a -> Maybe [ToVertex (Relation a)]

isAcyclic :: Relation a -> Bool

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

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

toAdjacencyIntMap :: Relation a -> AdjacencyIntMap

toAdjacencyIntMapTranspose :: Relation a -> AdjacencyIntMap

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

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

Ord a => Graph (Relation a)

Defined in Algebra.Graph.Class

Associated Types

type Vertex (Relation a) :: *

Methods

empty :: Relation a

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

overlay :: Relation a -> Relation a -> Relation a

connect :: Relation a -> Relation a -> Relation a

type ToVertex (Relation a)

Defined in Algebra.Graph.ToGraph

type ToVertex (Relation a) = a

type Vertex (Relation a)

Defined in Algebra.Graph.Class

type Vertex (Relation a) = a

domain :: Relation a -> Set a

The domain of the relation.

relation :: Relation a -> Set (a, a)

The set of pairs of elements that are related. It is guaranteed that each element belongs to the domain.

Basic graph construction primitives

empty :: Relation 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 -> Relation a

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

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

edge :: Ord a => a -> a -> Relation a

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

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 => Relation a -> Relation a -> Relation 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   && 'iAlgebra.Graph.Relation.sEmpty'   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 => Relation a -> Relation a -> Relation 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] -> Relation 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)] -> Relation 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
edgeCount . edges == length . nub

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

Graph properties

isEmpty :: Relation a -> Bool

Check if a relation 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 -> Relation a -> Bool

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

hasVertex x empty            == False
hasVertex x (vertex x)       == True
hasVertex 1 (vertex 2)       == False
hasVertex x . removeVertex x == const False

hasEdge :: Ord a => a -> a -> Relation 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 :: Relation a -> Int

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

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

edgeCount :: Relation a -> Int

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

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

vertexList :: Relation 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 :: Relation 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 :: Eq a => Relation 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 :: Relation a -> Set a

The set of vertices of a given graph. Complexity: O(1) time.

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

vertexIntSet :: Relation Int -> IntSet

The set of vertices of a given graph. Like vertexSet but specialised for graphs with vertices of type Int. Complexity: O(n) time.

vertexIntSet empty      == IntSet.empty
vertexIntSet . vertex   == IntSet.singleton
vertexIntSet . vertices == IntSet.fromList
vertexIntSet . clique   == IntSet.fromList

edgeSet :: Relation a -> Set (a, a)

The set of edges of a given graph. Complexity: O(1) time.

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 -> Relation a -> Set a

The preset of an element x is the set of elements that are related to it on the left, i.e. preSet x == { a | aRx }. In the context of directed graphs, this corresponds to the set of direct predecessors of vertex x. Complexity: O(n + m) 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 -> Relation a -> Set a

The postset of an element x is the set of elements that are related to it on the right, i.e. postSet x == { a | xRa }. In the context of directed graphs, this corresponds to the set of direct successors of vertex x. Complexity: O(n + m) time and O(n) 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] -> Relation 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] -> Relation 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] -> Relation 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] -> Relation 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] -> Relation 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])] -> Relation 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)

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

Remove a vertex from a given graph. Complexity: O(n + m) 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 -> Relation a -> Relation a

Remove an edge from a given graph. Complexity: O(log(m)) 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 -> Relation a -> Relation 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 -> Relation a -> Relation 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 => Relation a -> Relation a

Transpose a given graph. Complexity: O(m * log(m)) time.

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 b => (a -> b) -> Relation a -> Relation 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 Relation. 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) -> Relation a -> Relation a

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(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

Operations on binary relations

compose :: Ord a => Relation a -> Relation a -> Relation a

Compose two relations: R = compose Q P. Two elements x and y are related in the resulting relation, i.e. xRy, if there exists an element z, such that xPz and zQy. This is an associative operation which has empty as the annihilating zero. Complexity: O(n * m * log(m)) time and O(n + m) memory.

compose empty            x                == empty
compose x                empty            == empty
compose x                (compose y z)    == compose (compose x y) z
compose (edge y z)       (edge x y)       == 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]

reflexiveClosure :: Ord a => Relation a -> Relation a

Compute the reflexive closure of a Relation. Complexity: O(n * log(m)) time.

reflexiveClosure empty      == empty
reflexiveClosure (vertex x) == edge x x

symmetricClosure :: Ord a => Relation a -> Relation a

Compute the symmetric closure of a Relation. Complexity: O(m * log(m)) time.

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

transitiveClosure :: Ord a => Relation a -> Relation a

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

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

preorderClosure :: Ord a => Relation a -> Relation a

Compute the preorder closure of a Relation. Complexity: O(n * m * log(m)) time.

preorderClosure empty           == empty
preorderClosure (vertex x)      == edge x x
preorderClosure (path $ nub xs) == reflexiveClosure (clique $ nub xs)