algebraic-graphs-0.7: A library for algebraic graph construction and transformation

Copyright | (c) Andrey Mokhov 2016-2022
License | MIT (see the file LICENSE)
Maintainer | andrey.mokhov@gmail.com
Stability | experimental
Safe Haskell | None
Language | Haskell2010

Algebra.Graph.AdjacencyMap

Contents

- Data structure
- Basic graph construction primitives
- Relations on graphs
- Graph properties
- Standard families of graphs
- Graph transformation
- Graph composition
- Relational operations
- Miscellaneous

Description

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

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

Synopsis

- data AdjacencyMap a
- adjacencyMap :: AdjacencyMap a -> Map a (Set a)
- empty :: AdjacencyMap a
- vertex :: a -> AdjacencyMap a
- edge :: Ord a => a -> a -> AdjacencyMap a
- overlay :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a
- connect :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a
- vertices :: Ord a => [a] -> AdjacencyMap a
- edges :: Ord a => [(a, a)] -> AdjacencyMap a
- overlays :: Ord a => [AdjacencyMap a] -> AdjacencyMap a
- connects :: Ord a => [AdjacencyMap a] -> AdjacencyMap a
- isSubgraphOf :: Ord a => AdjacencyMap a -> AdjacencyMap a -> Bool
- isEmpty :: AdjacencyMap a -> Bool
- hasVertex :: Ord a => a -> AdjacencyMap a -> Bool
- hasEdge :: Ord a => a -> a -> AdjacencyMap a -> Bool
- vertexCount :: AdjacencyMap a -> Int
- edgeCount :: AdjacencyMap a -> Int
- vertexList :: AdjacencyMap a -> [a]
- edgeList :: AdjacencyMap a -> [(a, a)]
- adjacencyList :: AdjacencyMap a -> [(a, [a])]
- vertexSet :: AdjacencyMap a -> Set a
- edgeSet :: Eq a => AdjacencyMap a -> Set (a, a)
- preSet :: Ord a => a -> AdjacencyMap a -> Set a
- postSet :: Ord a => a -> AdjacencyMap a -> Set a
- path :: Ord a => [a] -> AdjacencyMap a
- circuit :: Ord a => [a] -> AdjacencyMap a
- clique :: Ord a => [a] -> AdjacencyMap a
- biclique :: Ord a => [a] -> [a] -> AdjacencyMap a
- star :: Ord a => a -> [a] -> AdjacencyMap a
- stars :: Ord a => [(a, [a])] -> AdjacencyMap a
- fromAdjacencySets :: Ord a => [(a, Set a)] -> AdjacencyMap a
- tree :: Ord a => Tree a -> AdjacencyMap a
- forest :: Ord a => Forest a -> AdjacencyMap a
- removeVertex :: Ord a => a -> AdjacencyMap a -> AdjacencyMap a
- removeEdge :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a
- replaceVertex :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a
- mergeVertices :: Ord a => (a -> Bool) -> a -> AdjacencyMap a -> AdjacencyMap a
- transpose :: Ord a => AdjacencyMap a -> AdjacencyMap a
- gmap :: (Ord a, Ord b) => (a -> b) -> AdjacencyMap a -> AdjacencyMap b
- induce :: (a -> Bool) -> AdjacencyMap a -> AdjacencyMap a
- induceJust :: Ord a => AdjacencyMap (Maybe a) -> AdjacencyMap a
- compose :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a
- box :: (Ord a, Ord b) => AdjacencyMap a -> AdjacencyMap b -> AdjacencyMap (a, b)
- closure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- reflexiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- symmetricClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- transitiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- consistent :: Ord a => AdjacencyMap a -> Bool

# Data structure

data AdjacencyMap a #

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

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

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

The `Show` instance is defined using basic graph construction primitives:

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

The `Eq` instance satisfies all axioms of algebraic graphs:

- `overlay` is commutative and associative:

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

- `connect` is associative and has `empty` as the identity:

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

- `connect` distributes over `overlay`:

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

- `connect` can be decomposed:

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

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

- `overlay` has `empty` as the identity and is idempotent:

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

- Absorption and saturation of `connect`:

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

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

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

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

Here are a few examples:

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

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

    isSubgraphOf x y ==> x <= y

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

#### Instances

Instances details

Ord a => Graph (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (AdjacencyMap a) |
Instance details Defined in Algebra.Graph.Class type Vertex (AdjacencyMap a) = 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 #
Ord a => ToGraph (AdjacencyMap a) # | See Algebra.Graph.AdjacencyMap.
Instance details Defined in Algebra.Graph.ToGraph Associated Types
type ToVertex (AdjacencyMap a) |
Instance details Defined in Algebra.Graph.ToGraph type ToVertex (AdjacencyMap a) = a
Methods toGraph :: AdjacencyMap a -> Graph (ToVertex (AdjacencyMap a)) # foldg :: r -> (ToVertex (AdjacencyMap a) -> r) -> (r -> r -> r) -> (r -> r -> r) -> AdjacencyMap a -> r # isEmpty :: AdjacencyMap a -> Bool # hasVertex :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Bool # hasEdge :: ToVertex (AdjacencyMap a) -> ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Bool # vertexCount :: AdjacencyMap a -> Int # edgeCount :: AdjacencyMap a -> Int # vertexList :: AdjacencyMap a -> [ToVertex (AdjacencyMap a)] # edgeList :: AdjacencyMap a -> [(ToVertex (AdjacencyMap a), ToVertex (AdjacencyMap a))] # vertexSet :: AdjacencyMap a -> Set (ToVertex (AdjacencyMap a)) # vertexIntSet :: AdjacencyMap a -> IntSet # edgeSet :: AdjacencyMap a -> Set (ToVertex (AdjacencyMap a), ToVertex (AdjacencyMap a)) # preSet :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Set (ToVertex (AdjacencyMap a)) # preIntSet :: Int -> AdjacencyMap a -> IntSet # postSet :: ToVertex (AdjacencyMap a) -> AdjacencyMap a -> Set (ToVertex (AdjacencyMap a)) # postIntSet :: Int -> AdjacencyMap a -> IntSet # adjacencyList :: AdjacencyMap a -> [(ToVertex (AdjacencyMap a), [ToVertex (AdjacencyMap a)])] # dfsForest :: AdjacencyMap a -> Forest (ToVertex (AdjacencyMap a)) # dfsForestFrom :: AdjacencyMap a -> [ToVertex (AdjacencyMap a)] -> Forest (ToVertex (AdjacencyMap a)) # dfs :: AdjacencyMap a -> [ToVertex (AdjacencyMap a)] -> [ToVertex (AdjacencyMap a)] # reachable :: AdjacencyMap a -> ToVertex (AdjacencyMap a) -> [ToVertex (AdjacencyMap a)] # topSort :: AdjacencyMap a -> Either (Cycle (ToVertex (AdjacencyMap a))) [ToVertex (AdjacencyMap a)] # isAcyclic :: AdjacencyMap a -> Bool # toAdjacencyMap :: AdjacencyMap a -> AdjacencyMap (ToVertex (AdjacencyMap a)) # toAdjacencyMapTranspose :: AdjacencyMap a -> AdjacencyMap (ToVertex (AdjacencyMap a)) # toAdjacencyIntMap :: AdjacencyMap a -> AdjacencyIntMap # toAdjacencyIntMapTranspose :: AdjacencyMap a -> AdjacencyIntMap # isDfsForestOf :: Forest (ToVertex (AdjacencyMap a)) -> AdjacencyMap a -> Bool # isTopSortOf :: [ToVertex (AdjacencyMap a)] -> AdjacencyMap a -> Bool #
NFData a => NFData (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.AdjacencyMap Methods rnf :: AdjacencyMap a -> () #
Ord a => Monoid (AdjacencyMap a) # | Defined via `overlay` and `empty`.
Instance details Defined in Algebra.Graph.AdjacencyMap Methods mempty :: AdjacencyMap a # mappend :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a # mconcat :: [AdjacencyMap a] -> AdjacencyMap a #
Ord a => Semigroup (AdjacencyMap a) # | Defined via `overlay`.
Instance details Defined in Algebra.Graph.AdjacencyMap Methods (<>) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a # sconcat :: NonEmpty (AdjacencyMap a) -> AdjacencyMap a # stimes :: Integral b => b -> AdjacencyMap a -> AdjacencyMap a #
IsString a => IsString (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.AdjacencyMap Methods fromString :: String -> AdjacencyMap a #
Generic (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.AdjacencyMap Associated Types
type Rep (AdjacencyMap a) |
Instance details Defined in Algebra.Graph.AdjacencyMap type Rep (AdjacencyMap a) = D1 ('MetaData "AdjacencyMap" "Algebra.Graph.AdjacencyMap" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'True) (C1 ('MetaCons "AM" 'PrefixI 'True) (S1 ('MetaSel ('Just "adjacencyMap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map a (Set a)))))
Methods from :: AdjacencyMap a -> Rep (AdjacencyMap a) x # to :: Rep (AdjacencyMap a) x -> AdjacencyMap a #
(Ord a, Num a) => Num (AdjacencyMap a) # | Note: this does not satisfy the usual ring laws; see `AdjacencyMap` for more details.
Instance details Defined in Algebra.Graph.AdjacencyMap Methods (+) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a # (-) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a # (*) :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a # negate :: AdjacencyMap a -> AdjacencyMap a # abs :: AdjacencyMap a -> AdjacencyMap a # signum :: AdjacencyMap a -> AdjacencyMap a # fromInteger :: Integer -> AdjacencyMap a #
(Ord a, Show a) => Show (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.AdjacencyMap Methods showsPrec :: Int -> AdjacencyMap a -> ShowS # show :: AdjacencyMap a -> String # showList :: [AdjacencyMap a] -> ShowS #
Eq a => Eq (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.AdjacencyMap Methods (==) :: AdjacencyMap a -> AdjacencyMap a -> Bool # (/=) :: AdjacencyMap a -> AdjacencyMap a -> Bool #
Ord a => Ord (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.AdjacencyMap Methods compare :: AdjacencyMap a -> AdjacencyMap a -> Ordering # (<) :: AdjacencyMap a -> AdjacencyMap a -> Bool # (<=) :: AdjacencyMap a -> AdjacencyMap a -> Bool # (>) :: AdjacencyMap a -> AdjacencyMap a -> Bool # (>=) :: AdjacencyMap a -> AdjacencyMap a -> Bool # max :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a # min :: AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a #
type Vertex (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.Class type Vertex (AdjacencyMap a) = a
type ToVertex (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.ToGraph type ToVertex (AdjacencyMap a) = a
type Rep (AdjacencyMap a) # |
Instance details Defined in Algebra.Graph.AdjacencyMap type Rep (AdjacencyMap a) = D1 ('MetaData "AdjacencyMap" "Algebra.Graph.AdjacencyMap" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'True) (C1 ('MetaCons "AM" 'PrefixI 'True) (S1 ('MetaSel ('Just "adjacencyMap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map a (Set a)))))

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

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

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

# Basic graph construction primitives

empty :: AdjacencyMap a #

Construct the empty graph.

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

vertex :: a -> AdjacencyMap a #

Construct the graph comprising a single isolated vertex.

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

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

Construct the graph comprising a single edge.

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

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

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

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

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

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

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

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

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

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

# Relations on graphs

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

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

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

# Graph properties

isEmpty :: AdjacencyMap a -> Bool #

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

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

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

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

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

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

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

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

vertexCount :: AdjacencyMap a -> Int #

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

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

edgeCount :: AdjacencyMap a -> Int #

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

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

vertexList :: AdjacencyMap a -> [a] #

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

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

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

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

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

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

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

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

vertexSet :: AdjacencyMap a -> Set a #

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

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

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

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

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

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

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

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

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

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

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

# Standard families of graphs

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

# Graph transformation

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

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

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

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

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

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

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

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

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

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

Merge vertices satisfying a given predicate into a given vertex. Complexity: O((n + m) * log(n)) time, assuming that the predicate takes 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 :: Ord a => AdjacencyMap a -> AdjacencyMap a #

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

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

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

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

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

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

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Complexity: O(n + m) time, assuming that the predicate takes 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 :: Ord a => AdjacencyMap (Maybe a) -> AdjacencyMap a #

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

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

# Graph composition

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

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

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

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

Compute the Cartesian product of graphs. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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

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

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

# Relational operations

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

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

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

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

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

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

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

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

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

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

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

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

# Miscellaneous

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

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

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