Algebra.Graph.AdjacencyIntMap

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 AdjacencyIntMap data type and associated functions. See Algebra.Graph.AdjacencyIntMap.Algorithm for implementations of basic graph algorithms. AdjacencyIntMap is an instance of the Graph type class, which can be used for polymorphic graph construction and manipulation. See Algebra.Graph.AdjacencyMap for graphs with non-Int vertices.

Data structure

data AdjacencyIntMap

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

Defined in Algebra.Graph.AdjacencyIntMap

Methods

(==) :: AdjacencyIntMap -> AdjacencyIntMap -> Bool

(/=) :: AdjacencyIntMap -> AdjacencyIntMap -> Bool

Num AdjacencyIntMap

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

Defined in Algebra.Graph.AdjacencyIntMap

Methods

(+) :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

(-) :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

(*) :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

negate :: AdjacencyIntMap -> AdjacencyIntMap

abs :: AdjacencyIntMap -> AdjacencyIntMap

signum :: AdjacencyIntMap -> AdjacencyIntMap

fromInteger :: Integer -> AdjacencyIntMap

Ord AdjacencyIntMap

Defined in Algebra.Graph.AdjacencyIntMap

Methods

compare :: AdjacencyIntMap -> AdjacencyIntMap -> Ordering

(<) :: AdjacencyIntMap -> AdjacencyIntMap -> Bool

(<=) :: AdjacencyIntMap -> AdjacencyIntMap -> Bool

(>) :: AdjacencyIntMap -> AdjacencyIntMap -> Bool

(>=) :: AdjacencyIntMap -> AdjacencyIntMap -> Bool

max :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

min :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

Show AdjacencyIntMap

Defined in Algebra.Graph.AdjacencyIntMap

Methods

showsPrec :: Int -> AdjacencyIntMap -> ShowS

show :: AdjacencyIntMap -> String

showList :: [AdjacencyIntMap] -> ShowS

Generic AdjacencyIntMap

Defined in Algebra.Graph.AdjacencyIntMap

Associated Types

type Rep AdjacencyIntMap :: Type -> Type

Methods

from :: AdjacencyIntMap -> Rep AdjacencyIntMap x

to :: Rep AdjacencyIntMap x -> AdjacencyIntMap

Semigroup AdjacencyIntMap

Defined via overlay.

Defined in Algebra.Graph.AdjacencyIntMap

Methods

(<>) :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

sconcat :: NonEmpty AdjacencyIntMap -> AdjacencyIntMap

stimes :: Integral b => b -> AdjacencyIntMap -> AdjacencyIntMap

Monoid AdjacencyIntMap

Defined via overlay and empty.

Defined in Algebra.Graph.AdjacencyIntMap

Methods

mempty :: AdjacencyIntMap

mappend :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

mconcat :: [AdjacencyIntMap] -> AdjacencyIntMap

NFData AdjacencyIntMap

Defined in Algebra.Graph.AdjacencyIntMap

Methods

rnf :: AdjacencyIntMap -> ()

ToGraph AdjacencyIntMap

Defined in Algebra.Graph.ToGraph

Associated Types

type ToVertex AdjacencyIntMap

Methods

toGraph :: AdjacencyIntMap -> Graph (ToVertex AdjacencyIntMap)

foldg :: r -> (ToVertex AdjacencyIntMap -> r) -> (r -> r -> r) -> (r -> r -> r) -> AdjacencyIntMap -> r

isEmpty :: AdjacencyIntMap -> Bool

hasVertex :: ToVertex AdjacencyIntMap -> AdjacencyIntMap -> Bool

hasEdge :: ToVertex AdjacencyIntMap -> ToVertex AdjacencyIntMap -> AdjacencyIntMap -> Bool

vertexCount :: AdjacencyIntMap -> Int

edgeCount :: AdjacencyIntMap -> Int

vertexList :: AdjacencyIntMap -> [ToVertex AdjacencyIntMap]

edgeList :: AdjacencyIntMap -> [(ToVertex AdjacencyIntMap, ToVertex AdjacencyIntMap)]

vertexSet :: AdjacencyIntMap -> Set (ToVertex AdjacencyIntMap)

vertexIntSet :: AdjacencyIntMap -> IntSet

edgeSet :: AdjacencyIntMap -> Set (ToVertex AdjacencyIntMap, ToVertex AdjacencyIntMap)

preSet :: ToVertex AdjacencyIntMap -> AdjacencyIntMap -> Set (ToVertex AdjacencyIntMap)

preIntSet :: Int -> AdjacencyIntMap -> IntSet

postSet :: ToVertex AdjacencyIntMap -> AdjacencyIntMap -> Set (ToVertex AdjacencyIntMap)

postIntSet :: Int -> AdjacencyIntMap -> IntSet

adjacencyList :: AdjacencyIntMap -> [(ToVertex AdjacencyIntMap, [ToVertex AdjacencyIntMap])]

dfsForest :: AdjacencyIntMap -> Forest (ToVertex AdjacencyIntMap)

dfsForestFrom :: [ToVertex AdjacencyIntMap] -> AdjacencyIntMap -> Forest (ToVertex AdjacencyIntMap)

dfs :: [ToVertex AdjacencyIntMap] -> AdjacencyIntMap -> [ToVertex AdjacencyIntMap]

reachable :: ToVertex AdjacencyIntMap -> AdjacencyIntMap -> [ToVertex AdjacencyIntMap]

topSort :: AdjacencyIntMap -> Either (Cycle (ToVertex AdjacencyIntMap)) [ToVertex AdjacencyIntMap]

isAcyclic :: AdjacencyIntMap -> Bool

toAdjacencyMap :: AdjacencyIntMap -> AdjacencyMap (ToVertex AdjacencyIntMap)

toAdjacencyMapTranspose :: AdjacencyIntMap -> AdjacencyMap (ToVertex AdjacencyIntMap)

toAdjacencyIntMap :: AdjacencyIntMap -> AdjacencyIntMap

toAdjacencyIntMapTranspose :: AdjacencyIntMap -> AdjacencyIntMap

isDfsForestOf :: Forest (ToVertex AdjacencyIntMap) -> AdjacencyIntMap -> Bool

isTopSortOf :: [ToVertex AdjacencyIntMap] -> AdjacencyIntMap -> Bool

Graph AdjacencyIntMap

Defined in Algebra.Graph.Class

Associated Types

type Vertex AdjacencyIntMap

Methods

empty :: AdjacencyIntMap

vertex :: Vertex AdjacencyIntMap -> AdjacencyIntMap

overlay :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

connect :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

type Rep AdjacencyIntMap

Defined in Algebra.Graph.AdjacencyIntMap

type Rep AdjacencyIntMap = D1 ('MetaData "AdjacencyIntMap" "Algebra.Graph.AdjacencyIntMap" "algebraic-graphs-0.6-31pq0b3ITnUItSsb9kEuq8" 'True) (C1 ('MetaCons "AM" 'PrefixI 'True) (S1 ('MetaSel ('Just "adjacencyIntMap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (IntMap IntSet))))

type ToVertex AdjacencyIntMap

Defined in Algebra.Graph.ToGraph

type ToVertex AdjacencyIntMap = Int

type Vertex AdjacencyIntMap

Defined in Algebra.Graph.Class

type Vertex AdjacencyIntMap = Int

adjacencyIntMap :: AdjacencyIntMap -> IntMap IntSet

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

adjacencyIntMap empty      == IntMap.empty
adjacencyIntMap (vertex x) == IntMap.singleton x IntSet.empty
adjacencyIntMap (edge 1 1) == IntMap.singleton 1 (IntSet.singleton 1)
adjacencyIntMap (edge 1 2) == IntMap.fromList [(1,IntSet.singleton 2), (2,IntSet.empty)]

fromAdjacencyMap :: AdjacencyMap Int -> AdjacencyIntMap

Construct an AdjacencyIntMap from an AdjacencyMap with vertices of type Int. Complexity: O(n + m) time and memory.

fromAdjacencyMap == stars . AdjacencyMap.adjacencyList

Basic graph construction primitives

empty :: AdjacencyIntMap

Construct the empty graph.

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

vertex :: Int -> AdjacencyIntMap

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

edge :: Int -> Int -> AdjacencyIntMap

Construct the graph comprising a single edge.

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 :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

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 :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

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 :: [Int] -> AdjacencyIntMap

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
vertices                == overlays . map vertex
hasVertex x  . vertices == elem x
vertexCount  . vertices == length . nub
vertexIntSet . vertices == IntSet.fromList

edges :: [(Int, Int)] -> AdjacencyIntMap

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 :: [AdjacencyIntMap] -> AdjacencyIntMap

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 :: [AdjacencyIntMap] -> AdjacencyIntMap

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 :: AdjacencyIntMap -> AdjacencyIntMap -> 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 :: AdjacencyIntMap -> 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 :: Int -> AdjacencyIntMap -> 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 :: Int -> Int -> AdjacencyIntMap -> 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 :: AdjacencyIntMap -> 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 :: AdjacencyIntMap -> 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 :: AdjacencyIntMap -> [Int]

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 :: AdjacencyIntMap -> [(Int, Int)]

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 :: AdjacencyIntMap -> [(Int, [Int])]

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, [])]
adjacencyList (star 2 [3,1]) == [(1, []), (2, [1,3]), (3, [])]
stars . adjacencyList        == id

vertexIntSet :: AdjacencyIntMap -> IntSet

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

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

edgeSet :: AdjacencyIntMap -> Set (Int, Int)

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

preIntSet :: Int -> AdjacencyIntMap -> IntSet

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

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

postIntSet :: Int -> AdjacencyIntMap -> IntSet

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

postIntSet x empty      == IntSet.empty
postIntSet x (vertex x) == IntSet.empty
postIntSet x (edge x y) == IntSet.fromList [y]
postIntSet 2 (edge 1 2) == IntSet.empty

Standard families of graphs

path :: [Int] -> AdjacencyIntMap

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 :: [Int] -> AdjacencyIntMap

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 :: [Int] -> AdjacencyIntMap

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 :: [Int] -> [Int] -> AdjacencyIntMap

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 :: Int -> [Int] -> AdjacencyIntMap

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 :: [(Int, [Int])] -> AdjacencyIntMap

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)

fromAdjacencyIntSets :: [(Int, IntSet)] -> AdjacencyIntMap

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.

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

tree :: Tree Int -> AdjacencyIntMap

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 :: Forest Int -> AdjacencyIntMap

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 :: Int -> AdjacencyIntMap -> AdjacencyIntMap

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 :: Int -> Int -> AdjacencyIntMap -> AdjacencyIntMap

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 :: Int -> Int -> AdjacencyIntMap -> AdjacencyIntMap

The function replaceVertex x y replaces vertex x with vertex y in a given AdjacencyIntMap. 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 :: (Int -> Bool) -> Int -> AdjacencyIntMap -> AdjacencyIntMap

Merge vertices satisfying a given predicate into a given vertex. Complexity: O((n + m) * log(n)) time, 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

transpose :: AdjacencyIntMap -> AdjacencyIntMap

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 :: (Int -> Int) -> AdjacencyIntMap -> AdjacencyIntMap

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 AdjacencyIntMap. 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 :: (Int -> Bool) -> AdjacencyIntMap -> AdjacencyIntMap

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 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

Relational operations

compose :: AdjacencyIntMap -> AdjacencyIntMap -> AdjacencyIntMap

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]

closure :: AdjacencyIntMap -> AdjacencyIntMap

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
postIntSet x (closure y) == IntSet.fromList (reachable x y)

reflexiveClosure :: AdjacencyIntMap -> AdjacencyIntMap

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 :: AdjacencyIntMap -> AdjacencyIntMap

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 :: AdjacencyIntMap -> AdjacencyIntMap

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 :: AdjacencyIntMap -> 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