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

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.

Instances

Eq a => Eq (AdjacencyMap a)

Methods

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

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

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

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, Show a) => Show (AdjacencyMap a)

Methods

showsPrec :: Int -> AdjacencyMap a -> ShowS

show :: AdjacencyMap a -> String

showList :: [AdjacencyMap a] -> ShowS

ToGraph (AdjacencyMap a)

Associated Types

type ToVertex (AdjacencyMap a) :: *

Methods

toGraph :: (Graph g, (* ~ Vertex g) (ToVertex (AdjacencyMap a))) => AdjacencyMap a -> g

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

Ord a => Graph (AdjacencyMap a)

Associated Types

type Vertex (AdjacencyMap a) :: *

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 ToVertex (AdjacencyMap a)

type ToVertex (AdjacencyMap a) = a

type Vertex (AdjacencyMap a)

type Vertex (AdjacencyMap a) = a

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

The adjacency map of the graph: each vertex is associated with a set of its direct successors.

Basic graph construction primitives

empty :: Ord a => 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 :: Ord a => a -> AdjacencyMap 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 -> 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
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

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

Construct a graph from an adjacency list. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

fromAdjacencyList []                                  == empty
fromAdjacencyList [(x, [])]                           == vertex x
fromAdjacencyList [(x, [y])]                          == edge x y
fromAdjacencyList . adjacencyList                     == id
overlay (fromAdjacencyList xs) (fromAdjacencyList ys) == fromAdjacencyList (xs ++ ys)

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

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 x)       == True
hasVertex 1 (vertex 2)       == False
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

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, [])]
fromAdjacencyList . 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
vertexSet . clique   == Set.fromList

edgeSet :: Ord 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

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

The postset (here postSet) of a vertex is the set of its direct successors.

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)

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

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

starTranspose x []    == vertex x
starTranspose x [y]   == edge y x
starTranspose x [y,z] == edges [(y,x), (z,x)]
starTranspose x ys    == connect (vertices ys) (vertex x)
starTranspose x ys    == transpose (star x 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 :: Ord a => (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(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

Algorithms

dfsForest :: AdjacencyMap a -> Forest a

Compute the depth-first search forest of a graph.

forest (dfsForest $ edge 1 1)         == vertex 1
forest (dfsForest $ edge 1 2)         == edge 1 2
forest (dfsForest $ edge 2 1)         == vertices [1, 2]
isSubgraphOf (forest $ dfsForest x) x == True
dfsForest . forest . dfsForest        == dfsForest
dfsForest (vertices vs)               == map (\v -> Node v []) (nub $ sort vs)
dfsForestFrom (vertexList x) x        == dfsForest x
dfsForest $ 3 * (1 + 4) * (1 + 5)     == [ Node { rootLabel = 1
                                                , subForest = [ Node { rootLabel = 5
                                                                     , subForest = [] }]}
                                         , Node { rootLabel = 3
                                                , subForest = [ Node { rootLabel = 4
                                                                     , subForest = [] }]}]

dfsForestFrom :: [a] -> AdjacencyMap a -> Forest a

Compute the depth-first search forest of a graph, searching from each of the given vertices in order. Note that the resulting forest does not necessarily span the whole graph, as some vertices may be unreachable.

forest (dfsForestFrom [1]    $ edge 1 1)     == vertex 1
forest (dfsForestFrom [1]    $ edge 1 2)     == edge 1 2
forest (dfsForestFrom [2]    $ edge 1 2)     == vertex 2
forest (dfsForestFrom [3]    $ edge 1 2)     == empty
forest (dfsForestFrom [2, 1] $ edge 1 2)     == vertices [1, 2]
isSubgraphOf (forest $ dfsForestFrom vs x) x == True
dfsForestFrom (vertexList x) x               == dfsForest x
dfsForestFrom vs             (vertices vs)   == map (\v -> Node v []) (nub vs)
dfsForestFrom []             x               == []
dfsForestFrom [1, 4] $ 3 * (1 + 4) * (1 + 5) == [ Node { rootLabel = 1
                                                       , subForest = [ Node { rootLabel = 5
                                                                            , subForest = [] }
                                                , Node { rootLabel = 4
                                                       , subForest = [] }]

dfs :: [a] -> AdjacencyMap a -> [a]

Compute the list of vertices visited by the depth-first search in a graph, when searching from each of the given vertices in order.

dfs [1]    $ edge 1 1                == [1]
dfs [1]    $ edge 1 2                == [1, 2]
dfs [2]    $ edge 1 2                == [2]
dfs [3]    $ edge 1 2                == []
dfs [1, 2] $ edge 1 2                == [1, 2]
dfs [2, 1] $ edge 1 2                == [2, 1]
dfs []     $ x                       == []
dfs [1, 4] $ 3 * (1 + 4) * (1 + 5)   == [1, 5, 4]
isSubgraphOf (vertices $ dfs vs x) x == True

topSort :: Ord a => AdjacencyMap a -> Maybe [a]

Compute the topological sort of a graph or return Nothing if the graph is cyclic.

topSort (1 * 2 + 3 * 1)             == Just [3,1,2]
topSort (1 * 2 + 2 * 1)             == Nothing
fmap (flip isTopSort x) (topSort x) /= Just False

isTopSort :: Ord a => [a] -> AdjacencyMap a -> Bool

Check if a given list of vertices is a valid topological sort of a graph.

isTopSort [3, 1, 2] (1 * 2 + 3 * 1) == True
isTopSort [1, 2, 3] (1 * 2 + 3 * 1) == False
isTopSort []        (1 * 2 + 3 * 1) == False
isTopSort []        empty           == True
isTopSort [x]       (vertex x)      == True
isTopSort [x]       (edge x x)      == False

scc :: Ord a => AdjacencyMap a -> AdjacencyMap (Set a)

Compute the condensation of a graph, where each vertex corresponds to a strongly-connected component of the original graph.

scc empty               == empty
scc (vertex x)          == vertex (Set.singleton x)
scc (edge x y)          == edge (Set.singleton x) (Set.singleton y)
scc (circuit (1:xs))    == edge (Set.fromList (1:xs)) (Set.fromList (1:xs))
scc (3 * 1 * 4 * 1 * 5) == edges [ (Set.fromList [1,4], Set.fromList [1,4])
                                 , (Set.fromList [1,4], Set.fromList [5]  )
                                 , (Set.fromList [3]  , Set.fromList [1,4])
                                 , (Set.fromList [3]  , Set.fromList [5]  )]