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.NonEmpty

Contents

- Non-empty algebraic graphs
- Basic graph construction primitives
- Graph folding
- Relations on graphs
- Graph properties
- Standard families of graphs
- Graph transformation
- Graph composition

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 data type `Graph` for algebraic graphs that are known to be non-empty at compile time. To avoid name clashes with Algebra.Graph, this module can be imported qualified:

    import qualified Algebra.Graph.NonEmpty as NonEmpty

The naming convention generally follows that of Data.List.NonEmpty: we use suffix `1` to indicate the functions whose interface must be changed compared to Algebra.Graph, e.g. `vertices1`.

Synopsis

- data Graph a
  - = Vertex a
  - | Overlay (Graph a) (Graph a)
  - | Connect (Graph a) (Graph a)
- toNonEmpty :: Graph a -> Maybe (Graph a)
- vertex :: a -> Graph a
- edge :: a -> a -> Graph a
- overlay :: Graph a -> Graph a -> Graph a
- overlay1 :: Graph a -> Graph a -> Graph a
- connect :: Graph a -> Graph a -> Graph a
- vertices1 :: NonEmpty a -> Graph a
- edges1 :: NonEmpty (a, a) -> Graph a
- overlays1 :: NonEmpty (Graph a) -> Graph a
- connects1 :: NonEmpty (Graph a) -> Graph a
- foldg1 :: (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
- isSubgraphOf :: Ord a => Graph a -> Graph a -> Bool
- (===) :: Eq a => Graph a -> Graph a -> Bool
- size :: Graph a -> Int
- hasVertex :: Eq a => a -> Graph a -> Bool
- hasEdge :: Eq a => a -> a -> Graph a -> Bool
- vertexCount :: Ord a => Graph a -> Int
- edgeCount :: Ord a => Graph a -> Int
- vertexList1 :: Ord a => Graph a -> NonEmpty a
- edgeList :: Ord a => Graph a -> [(a, a)]
- vertexSet :: Ord a => Graph a -> Set a
- edgeSet :: Ord a => Graph a -> Set (a, a)
- path1 :: NonEmpty a -> Graph a
- circuit1 :: NonEmpty a -> Graph a
- clique1 :: NonEmpty a -> Graph a
- biclique1 :: NonEmpty a -> NonEmpty a -> Graph a
- star :: a -> [a] -> Graph a
- stars1 :: NonEmpty (a, [a]) -> Graph a
- tree :: Tree a -> Graph a
- mesh1 :: NonEmpty a -> NonEmpty b -> Graph (a, b)
- torus1 :: NonEmpty a -> NonEmpty b -> Graph (a, b)
- removeVertex1 :: Eq a => a -> Graph a -> Maybe (Graph a)
- removeEdge :: Eq a => a -> a -> Graph a -> Graph a
- replaceVertex :: Eq a => a -> a -> Graph a -> Graph a
- mergeVertices :: (a -> Bool) -> a -> Graph a -> Graph a
- splitVertex1 :: Eq a => a -> NonEmpty a -> Graph a -> Graph a
- transpose :: Graph a -> Graph a
- induce1 :: (a -> Bool) -> Graph a -> Maybe (Graph a)
- induceJust1 :: Graph (Maybe a) -> Maybe (Graph a)
- simplify :: Ord a => Graph a -> Graph a
- sparsify :: Graph a -> Graph (Either Int a)
- sparsifyKL :: Int -> Graph Int -> Graph
- box :: Graph a -> Graph b -> Graph (a, b)

# Non-empty algebraic graphs

data Graph a #

Non-empty algebraic graphs, which are constructed using three primitives: `vertex`, `overlay` and `connect`. See module Algebra.Graph for algebraic graphs that can be empty.

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 `signum` method of the type class `Num` cannot be implemented and will throw an error. Furthermore, 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 satisfies the following laws of non-empty algebraic graphs.

- `overlay` is commutative, associative and idempotent:

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

- `connect` is associative:

      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

- `connect` satisfies absorption and saturation:

      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, defined as the number of vertex leaves (note that n <= s). If `g` is a `Graph`, the corresponding n, m and s can be computed as follows:

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

Converting a `Graph` to the corresponding `AdjacencyMap` 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 `Ord` 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

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

Constructors

Vertex a |
Overlay (Graph a) (Graph a) |
Connect (Graph a) (Graph a) |

#### Instances

Instances details

Applicative Graph # |
Instance details Defined in Algebra.Graph.NonEmpty 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 #
Functor Graph # |
Instance details Defined in Algebra.Graph.NonEmpty Methods fmap :: (a -> b) -> Graph a -> Graph b # (<$) :: a -> Graph b -> Graph a #
Monad Graph # |
Instance details Defined in Algebra.Graph.NonEmpty Methods (>>=) :: Graph a -> (a -> Graph b) -> Graph b # (>>) :: Graph a -> Graph b -> Graph b # return :: a -> Graph a #
ToGraph (Graph a) # |

Instance details

Defined in Algebra.Graph.NonEmpty

Associated Types

type ToVertex (Graph a) |
Instance details Defined in Algebra.Graph.NonEmpty type ToVertex (Graph a) = a

Methods

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

foldg :: r -> (ToVertex (Graph a) -> r) -> (r -> r -> r) -> (r -> r -> r) -> Graph a -> r #

isEmpty :: Graph a -> Bool #

hasVertex :: ToVertex (Graph a) -> Graph a -> Bool #

hasEdge :: ToVertex (Graph a) -> ToVertex (Graph a) -> Graph a -> Bool #

vertexCount :: Graph a -> Int #

edgeCount :: Graph a -> Int #

vertexList :: Graph a -> [ToVertex (Graph a)] #

edgeList :: Graph a -> [(ToVertex (Graph a), ToVertex (Graph a))] #

vertexSet :: Graph a -> Set (ToVertex (Graph a)) #

vertexIntSet :: Graph a -> IntSet #

edgeSet :: Graph a -> Set (ToVertex (Graph a), ToVertex (Graph a)) #

preSet :: ToVertex (Graph a) -> Graph a -> Set (ToVertex (Graph a)) #

preIntSet :: Int -> Graph a -> IntSet #

postSet :: ToVertex (Graph a) -> Graph a -> Set (ToVertex (Graph a)) #

postIntSet :: Int -> Graph a -> IntSet #

adjacencyList :: Graph a -> [(ToVertex (Graph a), [ToVertex (Graph a)])] #

dfsForest :: Graph a -> Forest (ToVertex (Graph a)) #

dfsForestFrom :: Graph a -> [ToVertex (Graph a)] -> Forest (ToVertex (Graph a)) #

dfs :: Graph a -> [ToVertex (Graph a)] -> [ToVertex (Graph a)] #

reachable :: Graph a -> ToVertex (Graph a) -> [ToVertex (Graph a)] #

topSort :: Graph a -> Either (Cycle (ToVertex (Graph a))) [ToVertex (Graph a)] #

isAcyclic :: Graph a -> Bool #

toAdjacencyMap :: Graph a -> AdjacencyMap (ToVertex (Graph a)) #

toAdjacencyMapTranspose :: Graph a -> AdjacencyMap (ToVertex (Graph a)) #

toAdjacencyIntMap :: Graph a -> AdjacencyIntMap #

toAdjacencyIntMapTranspose :: Graph a -> AdjacencyIntMap #

isDfsForestOf :: Forest (ToVertex (Graph a)) -> Graph a -> Bool #

isTopSortOf :: [ToVertex (Graph a)] -> Graph a -> Bool #

NFData a => NFData (Graph a) # |
Instance details Defined in Algebra.Graph.NonEmpty Methods rnf :: Graph a -> () #
Semigroup (Graph a) # | Defined via `overlay`.
Instance details Defined in Algebra.Graph.NonEmpty Methods (<>) :: Graph a -> Graph a -> Graph a # sconcat :: NonEmpty (Graph a) -> Graph a # stimes :: Integral b => b -> Graph a -> Graph a #
IsString a => IsString (Graph a) # |
Instance details Defined in Algebra.Graph.NonEmpty Methods fromString :: String -> Graph a #
Num a => Num (Graph a) # | Note: this does not satisfy the usual ring laws; see `Graph` for more details.
Instance details Defined in Algebra.Graph.NonEmpty 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 #
Show a => Show (Graph a) # |
Instance details Defined in Algebra.Graph.NonEmpty Methods showsPrec :: Int -> Graph a -> ShowS # show :: Graph a -> String # showList :: [Graph a] -> ShowS #
Ord a => Eq (Graph a) # |
Instance details Defined in Algebra.Graph.NonEmpty Methods (==) :: Graph a -> Graph a -> Bool # (/=) :: Graph a -> Graph a -> Bool #
Ord a => Ord (Graph a) # |
Instance details Defined in Algebra.Graph.NonEmpty 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 #
type ToVertex (Graph a) # |
Instance details Defined in Algebra.Graph.NonEmpty type ToVertex (Graph a) = a

toNonEmpty :: Graph a -> Maybe (Graph a) #

Convert an algebraic graph (from Algebra.Graph) into a non-empty algebraic graph. Returns `Nothing` if the argument is `empty`. Complexity: O(s) time, memory and size.

    toNonEmpty empty       == Nothing
    toNonEmpty (toGraph x) == Just (x :: Graph a)

# Basic graph construction primitives

vertex :: a -> Graph a #

Construct the graph comprising a single isolated vertex. An alias for the constructor `Vertex`.

    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)
    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. An alias for the constructor `Overlay`. This is a commutative, associative and idempotent operation. Complexity: O(1) time and memory, O(s1 + s2) size.

    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

overlay1 :: Graph a -> Graph a -> Graph a #

Overlay a possibly empty graph (from Algebra.Graph) with a non-empty graph. If the first argument is `empty`, the function returns the second argument; otherwise it is semantically the same as `overlay`. Complexity: O(s1) time and memory, and O(s1 + s2) size.

                   overlay1 empty x == x
    x /= empty ==> overlay1 x     y == overlay (fromJust $ toNonEmpty x) y

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

Connect two graphs. An alias for the constructor `Connect`. This is an associative operation, 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).

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

vertices1 :: NonEmpty 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.

    vertices1 [x]           == vertex x
    hasVertex x . vertices1 == elem x
    vertexCount . vertices1 == length . nub
    vertexSet   . vertices1 == Set.fromList . toList

edges1 :: NonEmpty (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.

    edges1 [(x,y)]     == edge x y
    edges1             == overlays1 . fmap (uncurry edge)
    edgeCount . edges1 == length . nub

overlays1 :: NonEmpty (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.

    overlays1 [x]   == x
    overlays1 [x,y] == overlay x y

connects1 :: NonEmpty (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.

    connects1 [x]   == x
    connects1 [x,y] == connect x y

# Graph folding

foldg1 :: (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: 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.

    foldg1 vertex    overlay connect        == id
    foldg1 vertex    overlay (flip connect) == transpose
    foldg1 (const 1) (+)     (+)            == size
    foldg1 (== 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 x             (overlay x y) ==  True
    isSubgraphOf (overlay x y) (connect x y) ==  True
    isSubgraphOf (path1 xs)    (circuit1 xs) ==  True
    isSubgraphOf x y                         ==> x <= y

(===) :: Eq a => Graph a -> Graph a -> Bool infix 4 #

Structural equality on graph expressions. Complexity: O(s) time.

        x === x     == True
    x + y === x + y == True
    1 + 2 === 2 + 1 == False
    x + y === x * y == False

# Graph properties

size :: Graph a -> Int #

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

    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 (vertex y) == (x == y)

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

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

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

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

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

vertexList1 :: Ord a => Graph a -> NonEmpty a #

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

    vertexList1 (vertex x)  == [x]
    vertexList1 . vertices1 == 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 (vertex x)     == []
    edgeList (edge x y)     == [(x,y)]
    edgeList (star 2 [3,1]) == [(2,1), (2,3)]
    edgeList . edges1       == nub . sort . toList
    edgeList . transpose    == sort . map swap . edgeList

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 . vertex    == Set.singleton
    vertexSet . vertices1 == Set.fromList . toList
    vertexSet . clique1   == Set.fromList . toList

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 (vertex x) == Set.empty
    edgeSet (edge x y) == Set.singleton (x,y)
    edgeSet . edges1   == Set.fromList . toList

# Standard families of graphs

path1 :: NonEmpty 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.

    path1 [x]       == vertex x
    path1 [x,y]     == edge x y
    path1 . reverse == transpose . path1

circuit1 :: NonEmpty 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.

    circuit1 [x]       == edge x x
    circuit1 [x,y]     == edges1 [(x,y), (y,x)]
    circuit1 . reverse == transpose . circuit1

clique1 :: NonEmpty 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.

    clique1 [x]        == vertex x
    clique1 [x,y]      == edge x y
    clique1 [x,y,z]    == edges1 [(x,y), (x,z), (y,z)]
    clique1 (xs <> ys) == connect (clique1 xs) (clique1 ys)
    clique1 . reverse  == transpose . clique1

biclique1 :: NonEmpty a -> NonEmpty 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.

    biclique1 [x1,x2] [y1,y2] == edges1 [(x1,y1), (x1,y2), (x2,y1), (x2,y2)]
    biclique1 xs      ys      == connect (vertices1 xs) (vertices1 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] == edges1 [(x,y), (x,z)]

stars1 :: NonEmpty (a, [a]) -> Graph a #

The stars formed by overlaying a non-empty list of `star`s. Complexity: O(L) time, memory and size, where L is the total size of the input.

    stars1 [(x, [] )]               == vertex x
    stars1 [(x, [y])]               == edge x y
    stars1 [(x, ys )]               == star x ys
    stars1                          == overlays1 . fmap (uncurry star)
    overlay (stars1 xs) (stars1 ys) == stars1 (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 []]])                       == path1 [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 []]]) == edges1 [(1,2), (1,3), (3,4), (3,5)]

mesh1 :: NonEmpty a -> NonEmpty b -> Graph (a, b) #

Construct a mesh graph from two lists of vertices. Complexity: O(L1 * L2) time, memory and size, where L1 and L2 are the lengths of the given lists.

    mesh1 [x]     [y]        == vertex (x, y)
    mesh1 xs      ys         == box (path1 xs) (path1 ys)
    mesh1 [1,2,3] ['a', 'b'] == edges1 [ ((1,'a'),(1,'b')), ((1,'a'),(2,'a'))
                                       , ((1,'b'),(2,'b')), ((2,'a'),(2,'b'))
                                       , ((2,'a'),(3,'a')), ((2,'b'),(3,'b'))
                                       , ((3,'a'),(3,'b')) ]

torus1 :: NonEmpty a -> NonEmpty b -> Graph (a, b) #

Construct a torus graph from two lists of vertices. Complexity: O(L1 * L2) time, memory and size, where L1 and L2 are the lengths of the given lists.

    torus1 [x]   [y]        == edge (x,y) (x,y)
    torus1 xs    ys         == box (circuit1 xs) (circuit1 ys)
    torus1 [1,2] ['a', 'b'] == edges1 [ ((1,'a'),(1,'b')), ((1,'a'),(2,'a'))
                                      , ((1,'b'),(1,'a')), ((1,'b'),(2,'b'))
                                      , ((2,'a'),(1,'a')), ((2,'a'),(2,'b'))
                                      , ((2,'b'),(1,'b')), ((2,'b'),(2,'a')) ]

# Graph transformation

removeVertex1 :: Eq a => a -> Graph a -> Maybe (Graph a) #

Remove a vertex from a given graph. Returns `Nothing` if the resulting graph is empty. Complexity: O(s) time, memory and size.

    removeVertex1 x (vertex x)          == Nothing
    removeVertex1 1 (vertex 2)          == Just (vertex 2)
    removeVertex1 x (edge x x)          == Nothing
    removeVertex1 1 (edge 1 2)          == Just (vertex 2)
    removeVertex1 x >=> removeVertex1 x == removeVertex1 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)       == vertices1 [x,y]
    removeEdge x y . removeEdge x y == removeEdge x y
    removeEdge 1 1 (1 * 1 * 2 * 2)  == 1 * 2 * 2
    removeEdge 1 2 (1 * 1 * 2 * 2)  == 1 * 1 + 2 * 2
    size (removeEdge x y z)         <= 3 * size z

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

splitVertex1 :: Eq a => a -> NonEmpty a -> Graph a -> Graph a #

Split a vertex into a list of vertices with the same connectivity. Complexity: O(s + k * L) time, memory and size, where k is the number of occurrences of the vertex in the expression and L is the length of the given list.

    splitVertex1 x [x]                 == id
    splitVertex1 x [y]                 == replaceVertex x y
    splitVertex1 1 [0,1] $ 1 * (2 + 3) == (0 + 1) * (2 + 3)

transpose :: Graph a -> Graph a #

Transpose a given graph. Complexity: O(s) time, memory and size.

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

induce1 :: (a -> Bool) -> Graph a -> Maybe (Graph a) #

Construct the induced subgraph of a given graph by removing the vertices that do not satisfy a given predicate. Returns `Nothing` if the resulting graph is empty. Complexity: O(s) time, memory and size, assuming that the predicate takes constant time.

    induce1 (const True ) x == Just x
    induce1 (const False) x == Nothing
    induce1 (/= x)          == removeVertex1 x
    induce1 p >=> induce1 q == induce1 (\x -> p x && q x)

induceJust1 :: Graph (Maybe a) -> Maybe (Graph a) #

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

    induceJust1 (vertex Nothing)                               == Nothing
    induceJust1 (edge (Just x) Nothing)                        == Just (vertex x)
    induceJust1 . fmap Just                                    == Just
    induceJust1 . fmap (\x -> if p x then Just x else Nothing) == induce1 p

simplify :: Ord a => Graph a -> Graph a #

Simplify a graph expression. Semantically, this is the identity function, but it simplifies a given expression according to the laws of the algebra. The function does not compute the simplest possible expression, but uses heuristics to obtain useful simplifications in reasonable time. Complexity: the function performs O(s) graph comparisons. It is guaranteed that the size of the result does not exceed the size of the given expression.

    simplify             ==  id
    size (simplify x)    <=  size x
    simplify 1           === 1
    simplify (1 + 1)     === 1
    simplify (1 + 2 + 1) === 1 + 2
    simplify (1 * 1 * 1) === 1 * 1

sparsify :: Graph a -> Graph (Either Int a) #

Sparsify a graph by adding intermediate `Left` `Int` vertices between the original vertices (wrapping the latter in `Right`) such that the resulting graph is sparse, i.e. contains only O(s) edges, but preserves the reachability relation between the original vertices. Sparsification is useful when working with dense graphs, as it can reduce the number of edges from O(n^2) down to O(n) by replacing cliques, bicliques and similar densely connected structures by sparse subgraphs built out of intermediate vertices. Complexity: O(s) time, memory and size.

    sort . reachable x       == sort . rights . reachable (sparsify x) . Right
    vertexCount (sparsify x) <= vertexCount x + size x + 1
    edgeCount   (sparsify x) <= 3 * size x
    size        (sparsify x) <= 3 * size x

sparsifyKL :: Int -> Graph Int -> Graph #

Sparsify a graph whose vertices are integers in the range `[1..n]`, where `n` is the first argument of the function, producing an array-based graph representation from Data.Graph (introduced by King and Launchbury, hence the name of the function). In the resulting graph, vertices `[1..n]` correspond to the original vertices, and all vertices greater than `n` are introduced by the sparsification procedure.

Complexity: O(s) time and memory. Note that thanks to sparsification, the resulting graph has a linear number of edges with respect to the size of the original algebraic representation even though the latter can potentially contain a quadratic O(s^2) number of edges.

    sort . reachable x                 == sort . filter (<= n) . reachable (sparsifyKL n x)
    length (vertices $ sparsifyKL n x) <= vertexCount x + size x + 1
    length (edges    $ sparsifyKL n x) <= 3 * size x

# Graph composition

box :: Graph a -> Graph b -> Graph (a, b) #

Compute the Cartesian product of graphs. Complexity: O(s1 * s2) time, memory and size, where s1 and s2 are the sizes of the given graphs.

    box (path1 [0,1]) (path1 ['a','b']) == edges1 [ ((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`, and has singleton graphs as identities. 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
    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