Algebra.Graph

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

Contents

- Algebraic data type for graphs
- Basic graph construction primitives
- Graph folding
- Relations on graphs
- Graph properties
- Standard families of graphs
- Graph transformation
- Graph composition
- Context

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 core data type `Graph` and associated algorithms. For graphs that are known to be non-empty at compile time, see Algebra.Graph.NonEmpty. `Graph` is an instance of type classes defined in modules Algebra.Graph.Class and Algebra.Graph.HigherKinded.Class, which can be used for polymorphic graph construction and manipulation.

Synopsis

- data Graph a
  - = Empty
  - | Vertex a
  - | Overlay (Graph a) (Graph a)
  - | Connect (Graph a) (Graph a)
- empty :: Graph a
- vertex :: a -> Graph a
- edge :: a -> a -> Graph a
- overlay :: Graph a -> Graph a -> Graph a
- connect :: Graph a -> Graph a -> Graph a
- vertices :: [a] -> Graph a
- edges :: [(a, a)] -> Graph a
- overlays :: [Graph a] -> Graph a
- connects :: [Graph a] -> Graph a
- foldg :: b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
- buildg :: (forall r. r -> (a -> r) -> (r -> r -> r) -> (r -> r -> r) -> r) -> Graph a
- isSubgraphOf :: Ord a => Graph a -> Graph a -> Bool
- (===) :: Eq a => Graph a -> Graph a -> Bool
- isEmpty :: 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
- vertexList :: Ord a => Graph a -> [a]
- edgeList :: Ord a => Graph a -> [(a, a)]
- vertexSet :: Ord a => Graph a -> Set a
- edgeSet :: Ord a => Graph a -> Set (a, a)
- adjacencyList :: Ord a => Graph a -> [(a, [a])]
- path :: [a] -> Graph a
- circuit :: [a] -> Graph a
- clique :: [a] -> Graph a
- biclique :: [a] -> [a] -> Graph a
- star :: a -> [a] -> Graph a
- stars :: [(a, [a])] -> Graph a
- tree :: Tree a -> Graph a
- forest :: Forest a -> Graph a
- mesh :: [a] -> [b] -> Graph (a, b)
- torus :: [a] -> [b] -> Graph (a, b)
- deBruijn :: Int -> [a] -> Graph [a]
- removeVertex :: Eq a => a -> Graph a -> 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
- splitVertex :: Eq a => a -> [a] -> Graph a -> Graph a
- transpose :: Graph a -> Graph a
- induce :: (a -> Bool) -> Graph a -> Graph a
- induceJust :: Graph (Maybe a) -> Graph a
- simplify :: Ord a => Graph a -> Graph a
- sparsify :: Graph a -> Graph (Either Int a)
- sparsifyKL :: Int -> Graph Int -> Graph
- compose :: Ord a => Graph a -> Graph a -> Graph a
- box :: Graph a -> Graph b -> Graph (a, b)
- data Context a = Context {
  - inputs :: [a]
  - outputs :: [a]
  }
- context :: (a -> Bool) -> Graph a -> Maybe (Context a)

# Algebraic data type for graphs

data Graph a #

The `Graph` data type is a deep embedding of the core graph construction primitives `empty`, `vertex`, `overlay` and `connect`. 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 `Eq` instance is currently implemented using the `AdjacencyMap` as the canonical graph representation and satisfies all axioms of algebraic graphs:

- `overlay` is commutative and associative:

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

- `connect` is associative 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 will denote the number of vertices in the graph, m will denote the number of edges in the graph, and s will denote the size of the corresponding `Graph` expression. For example, if `g` is a `Graph` then n, m and s can be computed as follows:

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

Note that `size` counts all leaves of the expression:

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

Converting 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 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

Deforestation (fusion) is implemented for some functions in this module. This means that when a function tagged as a "good producer" is composed with a function tagged as a "good consumer", the intermediate structure will not be built.

Constructors

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

#### Instances

Instances details

Graph Graph # |
Instance details Defined in Algebra.Graph.HigherKinded.Class Methods connect :: Graph a -> Graph a -> Graph a #
Alternative Graph # |
Instance details Defined in Algebra.Graph Methods empty :: Graph a # (<|>) :: Graph a -> Graph a -> Graph a # some :: Graph a -> Graph [a] # many :: Graph a -> Graph [a] #
Applicative Graph # | `<*>` is a good consumer of its first argument and a good producer.
Instance details Defined in Algebra.Graph 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 # | `fmap` is a good consumer and producer.
Instance details Defined in Algebra.Graph Methods fmap :: (a -> b) -> Graph a -> Graph b # (<$) :: a -> Graph b -> Graph a #
Monad Graph # | `>>=` is a good consumer and producer.
Instance details Defined in Algebra.Graph Methods (>>=) :: Graph a -> (a -> Graph b) -> Graph b # (>>) :: Graph a -> Graph b -> Graph b # return :: a -> Graph a #
MonadPlus Graph # |
Instance details Defined in Algebra.Graph Methods mzero :: Graph a # mplus :: Graph a -> Graph a -> Graph a #
Graph (Graph a) # |
Instance details Defined in Algebra.Graph.Class Associated Types
type Vertex (Graph a) |
Instance details Defined in Algebra.Graph.Class type Vertex (Graph a) = a
Methods empty :: Graph a # vertex :: Vertex (Graph a) -> Graph a # overlay :: Graph a -> Graph a -> Graph a # connect :: Graph a -> Graph a -> Graph a #
Ord a => ToGraph (Graph a) # | See Algebra.Graph.
Instance details Defined in Algebra.Graph.ToGraph Associated Types
type ToVertex (Graph a) |
Instance details Defined in Algebra.Graph.ToGraph 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 Methods rnf :: Graph a -> () #
Monoid (Graph a) # | Defined via `overlay` and `empty`.
Instance details Defined in Algebra.Graph Methods mempty :: Graph a # mappend :: Graph a -> Graph a -> Graph a # mconcat :: [Graph a] -> Graph a #
Semigroup (Graph a) # | Defined via `overlay`.
Instance details Defined in Algebra.Graph 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 Methods fromString :: String -> Graph a #
Generic (Graph a) # |
Instance details Defined in Algebra.Graph Associated Types
type Rep (Graph a) |
Instance details Defined in Algebra.Graph type Rep (Graph a) = D1 ('MetaData "Graph" "Algebra.Graph" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'False) ((C1 ('MetaCons "Empty" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Vertex" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a))) :+: (C1 ('MetaCons "Overlay" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a)) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a))) :+: C1 ('MetaCons "Connect" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a)) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a)))))
Methods from :: Graph a -> Rep (Graph a) x # to :: Rep (Graph a) x -> 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 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 Methods showsPrec :: Int -> Graph a -> ShowS # show :: Graph a -> String # showList :: [Graph a] -> ShowS #
Ord a => Eq (Graph a) # | `==` is a good consumer of both arguments.
Instance details Defined in Algebra.Graph Methods (==) :: Graph a -> Graph a -> Bool # (/=) :: Graph a -> Graph a -> Bool #
Ord a => Ord (Graph a) # | `compare` is a good consumer of both arguments.
Instance details Defined in Algebra.Graph 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 Vertex (Graph a) # |
Instance details Defined in Algebra.Graph.Class type Vertex (Graph a) = a
type ToVertex (Graph a) # |
Instance details Defined in Algebra.Graph.ToGraph type ToVertex (Graph a) = a
type Rep (Graph a) # |
Instance details Defined in Algebra.Graph type Rep (Graph a) = D1 ('MetaData "Graph" "Algebra.Graph" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'False) ((C1 ('MetaCons "Empty" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Vertex" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a))) :+: (C1 ('MetaCons "Overlay" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a)) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a))) :+: C1 ('MetaCons "Connect" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a)) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Graph a)))))

# Basic graph construction primitives

empty :: Graph a #

Construct the empty graph. An alias for the constructor `Empty`.

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

vertex :: a -> Graph a #

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

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

edge :: a -> a -> Graph a #

Construct the graph comprising a single edge.

    edge x y               == connect (vertex x) (vertex y)
    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 with the identity `empty`. Complexity: O(1) time and memory, O(s1 + s2) size.

    isEmpty     (overlay x y) == isEmpty   x   && isEmpty   y
    hasVertex z (overlay x y) == hasVertex z x || hasVertex z y
    vertexCount (overlay x y) >= vertexCount x
    vertexCount (overlay x y) <= vertexCount x + vertexCount y
    edgeCount   (overlay x y) >= edgeCount x
    edgeCount   (overlay x y) <= edgeCount x   + edgeCount y
    size        (overlay x y) == size x        + size y
    vertexCount (overlay 1 2) == 2
    edgeCount   (overlay 1 2) == 0

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

Connect two graphs. An alias for the constructor `Connect`. This is an associative operation with the identity `empty`, which distributes over `overlay` and obeys the decomposition axiom. Complexity: O(1) time and memory, O(s1 + s2) size. Note that the number of edges in the resulting graph is quadratic with respect to the number of vertices of the arguments: m = O(m1 + m2 + n1 * n2).

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

vertices :: [a] -> Graph a #

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

Good consumer of lists and producer of graphs.

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

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

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

Good consumer of lists and producer of graphs.

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

overlays :: [Graph a] -> Graph a #

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

Good consumer of lists and producer of graphs.

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

connects :: [Graph a] -> Graph a #

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

Good consumer of lists and producer of graphs.

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

# Graph folding

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

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

Good consumer.

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

buildg :: (forall r. r -> (a -> r) -> (r -> r -> r) -> (r -> r -> r) -> r) -> Graph a #

Build a graph given an interpretation of the four graph construction primitives `empty`, `vertex`, `overlay` and `connect`, in this order. See examples for further clarification.

Functions expressed with `buildg` are good producers.

    buildg f                                                   == f empty vertex overlay connect
    buildg (\e _ _ _ -> e)                                     == empty
    buildg (\_ v _ _ -> v x)                                   == vertex x
    buildg (\e v o c -> o (foldg e v o c x) (foldg e v o c y)) == overlay x y
    buildg (\e v o c -> c (foldg e v o c x) (foldg e v o c y)) == connect x y
    buildg (\e v o _ -> foldr o e (map v xs))                  == vertices xs
    buildg (\e v o c -> foldg e v o (flip c) g)                == transpose g
    foldg e v o c (buildg f)                                   == f e v o c

# 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.

Good consumer of both arguments.

    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

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

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

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

# Graph properties

isEmpty :: Graph a -> Bool #

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

Good consumer.

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

size :: Graph a -> Int #

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

Good consumer.

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

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

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

Good consumer.

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

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

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

Good consumer.

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

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

Good consumer.

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

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

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

Good consumer.

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

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

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

Good consumer of graphs and producer of lists.

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

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

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

Good consumer of graphs and producer of lists.

    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

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.

Good consumer.

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

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

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

Good consumer.

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

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

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

Good consumer.

    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

# Standard families of graphs

path :: [a] -> Graph a #

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

Good producer.

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

circuit :: [a] -> Graph a #

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

Good producer.

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

clique :: [a] -> Graph a #

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

Good consumer of lists and producer of graphs.

    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 :: [a] -> [a] -> Graph a #

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

Good consumer of both arguments and producer of graphs.

    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 :: 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.

Good consumer of lists and good producer of graphs.

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

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

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

Good consumer of lists and producer of graphs.

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

tree :: Tree a -> Graph a #

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

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

forest :: Forest a -> Graph a #

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

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

mesh :: [a] -> [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.

    mesh xs     []   == empty
    mesh []     ys   == empty
    mesh [x]    [y]  == vertex (x, y)
    mesh xs     ys   == box (path xs) (path ys)
    mesh [1..3] "ab" == edges [ ((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')) ]

torus :: [a] -> [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.

    torus xs    []   == empty
    torus []    ys   == empty
    torus [x]   [y]  == edge (x,y) (x,y)
    torus xs    ys   == box (circuit xs) (circuit ys)
    torus [1,2] "ab" == edges [ ((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')) ]

deBruijn :: Int -> [a] -> Graph [a] #

Construct a De Bruijn graph of a given non-negative dimension using symbols from a given alphabet. Complexity: O(A^(D + 1)) time, memory and size, where A is the size of the alphabet and D is the dimension of the graph.

              deBruijn 0 xs               == edge [] []
    n > 0 ==> deBruijn n []               == empty
              deBruijn 1 [0,1]            == edges [ ([0],[0]), ([0],[1]), ([1],[0]), ([1],[1]) ]
              deBruijn 2 "0"              == edge "00" "00"
              deBruijn 2 "01"             == edges [ ("00","00"), ("00","01"), ("01","10"), ("01","11")
                                                   , ("10","00"), ("10","01"), ("11","10"), ("11","11") ]
              transpose   (deBruijn n xs) == fmap reverse $ deBruijn n xs
              vertexCount (deBruijn n xs) == (length $ nub xs)^n
    n > 0 ==> edgeCount   (deBruijn n xs) == (length $ nub xs)^(n + 1)

# Graph transformation

removeVertex :: Eq a => a -> Graph a -> Graph a #

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

Good consumer and producer.

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

removeEdge :: Eq a => a -> a -> Graph a -> Graph a #

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

    removeEdge x y (edge x y)       == vertices [x,y]
    removeEdge x y . removeEdge x y == removeEdge x y
    removeEdge x y . 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
    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.

Good consumer and producer.

    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.

Good consumer and producer.

    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

splitVertex :: Eq a => a -> [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.

Good consumer of lists and producer of graphs.

    splitVertex x []                  == removeVertex x
    splitVertex x [x]                 == id
    splitVertex x [y]                 == replaceVertex x y
    splitVertex 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.

Good consumer and producer.

    transpose empty       == empty
    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

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

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

Good consumer and producer.

    induce (const True ) x      == x
    induce (const False) x      == empty
    induce (/= x)               == removeVertex x
    induce p . induce q         == induce (\x -> p x && q x)
    isSubgraphOf (induce p x) x == True

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

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

Good consumer and producer.

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

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.

Good consumer.

    simplify              == id
    size (simplify x)     <= size x
    simplify empty       === empty
    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

compose :: Ord a => Graph a -> Graph a -> Graph 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, O(n + m) memory, and O(m1 + m2) size, where n and m stand for the number of vertices and edges in the resulting graph, while m1 and m2 are the number of edges in the original graphs. Note that the number of edges in the resulting graph may be quadratic, i.e. m = O(m1 * m2), but the algebraic representation requires only O(m1 + m2) operations to list them.

Good consumer of both arguments and good producer.

    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]
    size (compose x y)                        <= edgeCount x + edgeCount y + 1

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 (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

# Context

data Context a #

The `Context` of a subgraph comprises its `inputs` and `outputs`, i.e. all the vertices that are connected to the subgraph's vertices. Note that inputs and outputs can belong to the subgraph itself. In general, there are no guarantees on the order of vertices in `inputs` and `outputs`; furthermore, there may be repetitions.

Constructors

Context |
Fields inputs :: [a] outputs :: [a]

#### Instances

Instances details

Show a => Show (Context a) # |
Instance details Defined in Algebra.Graph Methods showsPrec :: Int -> Context a -> ShowS # show :: Context a -> String # showList :: [Context a] -> ShowS #
Eq a => Eq (Context a) # |
Instance details Defined in Algebra.Graph Methods (==) :: Context a -> Context a -> Bool # (/=) :: Context a -> Context a -> Bool #

context :: (a -> Bool) -> Graph a -> Maybe (Context a) #

Extract the `Context` of a subgraph specified by a given predicate. Returns `Nothing` if the specified subgraph is empty.

Good consumer.

    context (const False) x                   == Nothing
    context (== 1)        (edge 1 2)          == Just (Context [   ] [2  ])
    context (== 2)        (edge 1 2)          == Just (Context [1  ] [   ])
    context (const True ) (edge 1 2)          == Just (Context [1  ] [2  ])
    context (== 4)        (3 * 1 * 4 * 1 * 5) == Just (Context [3,1] [1,5])