Algebra.Graph.Undirected

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 an undirected version of algebraic graphs. Undirected graphs satisfy all laws of the Undirected type class, including the commutativity of connect.

To avoid name clashes with Algebra.Graph, this module can be imported qualified:

import qualified Algebra.Graph.Undirected as Undirected

Algebraic data type for graphs

data Graph a

The Graph data type provides the four algebraic graph construction primitives empty, vertex, overlay and connect, as well as various derived functions. The only difference compared to the Graph data type defined in Algebra.Graph is that the connect operation is commutative. We define a Num instance as a convenient notation for working with undirected 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 Eq instance is currently implemented using the Relation as the canonical graph representation and satisfies all axioms of algebraic graphs:

• overlay is commutative and associative:

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

• connect is associative, commutative and has empty as the identity:

    x * empty == x
    empty * x == x
        x * y == y * 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 will denote the number of vertices in the graph, m will denote the number of edges in the graph, and s will denote the size of the corresponding Graph expression. For example, if g is a Graph then n, m and s can be computed as follows:

n == vertexCount g
m == edgeCount g
s == size g

Note that size counts all leaves of the expression:

vertexCount empty           == 0
size        empty           == 1
vertexCount (vertex x)      == 1
size        (vertex x)      == 1
vertexCount (empty + empty) == 0
size        (empty + empty) == 2

Converting an undirected Graph to the corresponding Relation takes O(s + m * log(m)) time and O(s + m) memory. This is also the complexity of the graph equality test, because it is currently implemented by converting graph expressions to canonical representations based on adjacency maps.

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
edge 1 2 == edge 2 1

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

Monad Graph

Defined in Algebra.Graph.Undirected

Methods

(>>=) :: Graph a -> (a -> Graph b) -> Graph b

(>>) :: Graph a -> Graph b -> Graph b

return :: a -> Graph a

Functor Graph

Defined in Algebra.Graph.Undirected

Methods

fmap :: (a -> b) -> Graph a -> Graph b

(<$) :: a -> Graph b -> Graph a

Applicative Graph

Defined in Algebra.Graph.Undirected

Methods

pure :: a -> Graph a

(<*>) :: Graph (a -> b) -> Graph a -> Graph b

liftA2 :: (a -> b -> c) -> Graph a -> Graph b -> Graph c

(*>) :: Graph a -> Graph b -> Graph b

(<*) :: Graph a -> Graph b -> Graph a

Alternative Graph

Defined in Algebra.Graph.Undirected

Methods

empty :: Graph a

(<|>) :: Graph a -> Graph a -> Graph a

some :: Graph a -> Graph [a]

many :: Graph a -> Graph [a]

MonadPlus Graph

Defined in Algebra.Graph.Undirected

Methods

mzero :: Graph a

mplus :: Graph a -> Graph a -> Graph a

Ord a => Eq (Graph a)

Defined in Algebra.Graph.Undirected

Methods

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

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

Num a => Num (Graph a)

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

Defined in Algebra.Graph.Undirected

Methods

(+) :: Graph a -> Graph a -> Graph a

(-) :: Graph a -> Graph a -> Graph a

(*) :: Graph a -> Graph a -> Graph a

negate :: Graph a -> Graph a

abs :: Graph a -> Graph a

signum :: Graph a -> Graph a

fromInteger :: Integer -> Graph a

Ord a => Ord (Graph a)

Defined in Algebra.Graph.Undirected

Methods

compare :: Graph a -> Graph a -> Ordering

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

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

(>) :: Graph a -> Graph a -> Bool

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

max :: Graph a -> Graph a -> Graph a

min :: Graph a -> Graph a -> Graph a

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

Defined in Algebra.Graph.Undirected

Methods

showsPrec :: Int -> Graph a -> ShowS

show :: Graph a -> String

showList :: [Graph a] -> ShowS

IsString a => IsString (Graph a)

Defined in Algebra.Graph.Undirected

Methods

fromString :: String -> Graph a

Generic (Graph a)

Defined in Algebra.Graph.Undirected

Associated Types

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

Methods

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

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

Semigroup (Graph a)

Defined via overlay.

Defined in Algebra.Graph.Undirected

Methods

(<>) :: Graph a -> Graph a -> Graph a

sconcat :: NonEmpty (Graph a) -> Graph a

stimes :: Integral b => b -> Graph a -> Graph a

Monoid (Graph a)

Defined via overlay and empty.

Defined in Algebra.Graph.Undirected

Methods

mempty :: Graph a

mappend :: Graph a -> Graph a -> Graph a

mconcat :: [Graph a] -> Graph a

NFData a => NFData (Graph a)

Defined in Algebra.Graph.Undirected

Methods

rnf :: Graph a -> ()

Undirected (Graph a)

Defined in Algebra.Graph.Class

Graph (Graph a)

Defined in Algebra.Graph.Class

Associated Types

type Vertex (Graph a)

Methods

empty :: Graph a

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

overlay :: Graph a -> Graph a -> Graph a

connect :: Graph a -> Graph a -> Graph a

type Rep (Graph a)

Defined in Algebra.Graph.Undirected

type Rep (Graph a) = D1 ('MetaData "Graph" "Algebra.Graph.Undirected" "algebraic-graphs-0.6-31pq0b3ITnUItSsb9kEuq8" 'True) (C1 ('MetaCons "UG" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a))))

type Vertex (Graph a)

Defined in Algebra.Graph.Class

type Vertex (Graph a) = a

fromUndirected :: Ord a => Graph a -> Graph a

Extract the underlying Algebra.Graph. Complexity: O(n + m) time.

fromUndirected (edge 1 2)     == edges [(1,2),(2,1)]
toUndirected . fromUndirected == id
vertexCount . fromUndirected  == vertexCount
edgeCount . fromUndirected    <= (*2) . edgeCount

toUndirected :: Graph a -> Graph a

Construct an undirected graph from a given Algebra.Graph. Complexity: O(1) time.

toUndirected (edge 1 2)         == edge 1 2
toUndirected . fromUndirected   == id
vertexCount . toUndirected      == vertexCount
(*2) . edgeCount . toUndirected >= edgeCount

Basic graph construction primitives

empty :: Graph a

Construct the empty graph.

isEmpty     empty == True
hasVertex x empty == False
vertexCount empty == 0
edgeCount   empty == 0
size        empty == 1

vertex :: a -> Graph a

Construct the graph comprising a single isolated vertex.

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

edge :: a -> a -> Graph a

Construct the graph comprising a single edge.

edge x y               == connect (vertex x) (vertex y)
edge x y               == edge y x
edge x y               == edges [(x,y), (y,x)]
hasEdge x y (edge x y) == True
edgeCount   (edge x y) == 1
vertexCount (edge 1 1) == 1
vertexCount (edge 1 2) == 2

overlay :: Graph a -> Graph a -> Graph a

Overlay two graphs. This is a commutative, associative and idempotent operation with the identity empty. Complexity: O(1) time and memory, O(s1 + s2) size.

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
size        (overlay x y) == size x        + size y
vertexCount (overlay 1 2) == 2
edgeCount   (overlay 1 2) == 0

connect :: Graph a -> Graph a -> Graph a

Connect two graphs. This is a commutative and associative operation with the identity empty, which distributes over overlay and obeys the decomposition axiom. Complexity: O(1) time and memory, O(s1 + s2) size. 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).

connect x y               == connect y x
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 div 2
size        (connect x y) == size x        + size y
vertexCount (connect 1 2) == 2
edgeCount   (connect 1 2) == 1

vertices :: [a] -> Graph a

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

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

edges :: [(a, a)] -> Graph a

Construct the graph from a list of edges. Complexity: O(L) time, memory and size, where L is the length of the given list.

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

overlays :: [Graph a] -> Graph a

Overlay a given list of graphs. Complexity: O(L) time and memory, and O(S) size, where L is the length of the given list, and S is the sum of sizes of the graphs in the list.

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

connects :: [Graph a] -> Graph a

Connect a given list of graphs. Complexity: O(L) time and memory, and O(S) size, where L is the length of the given list, and S is the sum of sizes of the graphs in the list.

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

Graph folding

foldg :: b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b

Generalised Graph folding: recursively collapse a Graph by applying the provided functions to the leaves and internal nodes of the expression. The order of arguments is: empty, vertex, overlay and connect. Complexity: O(s) applications of the given functions. As an example, the complexity of size is O(s), since const and + have constant costs.

foldg empty vertex        overlay connect        == id
foldg empty vertex        overlay (flip connect) == id
foldg 1     (const 1)     (+)     (+)            == size
foldg True  (const False) (&&)    (&&)           == isEmpty
foldg False (== x)        (||)    (||)           == hasVertex x

Relations on graphs

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

The isSubgraphOf function takes two graphs and returns True if the first graph is a subgraph of the second. Complexity: O(s + m * log(m)) time. Note that the number of edges m of a graph can be quadratic with respect to the expression size s.

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 (edge x y)    (edge y x)    ==  True
isSubgraphOf x y                         ==> x <= y

toRelation :: Ord a => Graph a -> Relation a

Convert an undirected graph to a symmetric Relation.

Graph properties

isEmpty :: Graph a -> Bool

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

size :: Graph a -> Int

The size of a graph, i.e. the number of leaves of the expression including empty leaves. Complexity: O(s) time.

size empty         == 1
size (vertex x)    == 1
size (overlay x y) == size x + size y
size (connect x y) == size x + size y
size x             >= 1
size x             >= vertexCount x

hasVertex :: Eq a => a -> Graph a -> Bool

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

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

hasEdge :: Eq a => a -> a -> Graph a -> Bool

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

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

vertexCount :: Ord a => Graph a -> Int

The number of vertices in a graph. Complexity: O(s * log(n)) time.

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

edgeCount :: Ord a => Graph a -> Int

The number of edges in a graph. Complexity: O(s + m * log(m)) time. Note that the number of edges m of a graph can be quadratic with respect to the expression size s.

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

vertexList :: Ord a => Graph a -> [a]

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

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

edgeList :: Ord a => Graph a -> [(a, a)]

The sorted list of edges of a graph. Complexity: O(s + m * log(m)) time and O(m) memory. Note that the number of edges m of a graph can be quadratic with respect to the expression size s.

edgeList empty          == []
edgeList (vertex x)     == []
edgeList (edge x y)     == [(min x y, max y x)]
edgeList (star 2 [3,1]) == [(1,2), (2,3)]

vertexSet :: Ord a => Graph a -> Set a

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

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

edgeSet :: Ord a => Graph a -> Set (a, a)

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

edgeSet empty      == Set.empty
edgeSet (vertex x) == Set.empty
edgeSet (edge x y) == Set.singleton (min x y, max x y)

adjacencyList :: Ord a => Graph a -> [(a, [a])]

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

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

neighbours :: Ord a => a -> Graph a -> Set a

The set of vertices adjacent to a given vertex.

neighbours x empty      == Set.empty
neighbours x (vertex x) == Set.empty
neighbours x (edge x y) == Set.fromList [y]
neighbours y (edge x y) == Set.fromList [x]

Standard families of graphs

path :: [a] -> Graph a

The path on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

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

circuit :: [a] -> Graph a

The circuit on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

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

clique :: [a] -> Graph a

The clique on a list of vertices. Complexity: O(L) time, memory and size, where L is the length of the given list.

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  == clique

biclique :: [a] -> [a] -> Graph a

The biclique on two lists of vertices. Complexity: O(L1 + L2) time, memory and size, where L1 and L2 are the lengths of the given lists.

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

star :: a -> [a] -> Graph a

The star formed by a centre vertex connected to a list of leaves. Complexity: O(L) time, memory and size, where L is the length of the given list.

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 :: [(a, [a])] -> Graph a

The stars formed by overlaying a list of stars. An inverse of adjacencyList. Complexity: O(L) 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 :: Tree a -> Graph a

The tree graph constructed from a given Tree data structure. Complexity: O(T) time, memory and size, where T is the size of the given tree (i.e. the number of vertices in the tree).

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 :: Forest a -> Graph a

The forest graph constructed from a given Forest data structure. Complexity: O(F) time, memory and size, where F is the size of the given forest (i.e. the number of vertices in the forest).

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 :: Eq a => a -> Graph a -> Graph a

Remove a vertex from a given graph. Complexity: O(s) time, memory and size.

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 :: Eq a => a -> a -> Graph a -> Graph a

Remove an edge from a given graph. Complexity: O(s) time, memory and size.

removeEdge x y (edge x y)       == vertices [x,y]
removeEdge x y . removeEdge x y == removeEdge x y
removeEdge x y                  == removeEdge y x
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 :: Eq a => a -> a -> Graph a -> Graph a

The function replaceVertex x y replaces vertex x with vertex y in a given Graph. If y already exists, x and y will be merged. Complexity: O(s) time, memory and size.

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

mergeVertices :: (a -> Bool) -> a -> Graph a -> Graph a

Merge vertices satisfying a given predicate into a given vertex. Complexity: O(s) time, memory and size, assuming that the predicate takes constant time.

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

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

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(s) time, memory and size, assuming that the predicate takes constant time.

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 :: Graph (Maybe a) -> Graph a

Construct the induced subgraph of a given graph by removing the vertices that are Nothing. Complexity: O(s) time, memory and size.

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

complement :: Ord a => Graph a -> Graph a

The edge complement of a graph. Note that, as can be seen from the examples below, this operation ignores self-loops. Complexity: O(n^2 * log n) time, O(n^2) memory.

complement empty           == empty
complement (vertex x)      == (vertex x)
complement (edge 1 2)      == (vertices [1, 2])
complement (edge 0 0)      == (edge 0 0)
complement (star 1 [2, 3]) == (overlay (vertex 1) (edge 2 3))
complement . complement    == id