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.Bipartite.AdjacencyMap

Contents

- Data structure
- Basic graph construction primitives
- Conversion functions
- Graph properties
- Standard families of graphs
- Graph transformation
- Graph composition
- 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 for undirected bipartite graphs and associated functions. See Algebra.Graph.Bipartite.AdjacencyMap.Algorithm for basic bipartite graph algorithms.

To avoid name clashes with Algebra.Graph.AdjacencyMap, this module can be imported qualified:

    import qualified Algebra.Graph.Bipartite.AdjacencyMap as Bipartite

Synopsis

- data AdjacencyMap a b
- leftAdjacencyMap :: AdjacencyMap a b -> Map a (Set b)
- rightAdjacencyMap :: AdjacencyMap a b -> Map b (Set a)
- empty :: AdjacencyMap a b
- leftVertex :: a -> AdjacencyMap a b
- rightVertex :: b -> AdjacencyMap a b
- vertex :: Either a b -> AdjacencyMap a b
- edge :: a -> b -> AdjacencyMap a b
- overlay :: (Ord a, Ord b) => AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b
- connect :: (Ord a, Ord b) => AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b
- vertices :: (Ord a, Ord b) => [a] -> [b] -> AdjacencyMap a b
- edges :: (Ord a, Ord b) => [(a, b)] -> AdjacencyMap a b
- overlays :: (Ord a, Ord b) => [AdjacencyMap a b] -> AdjacencyMap a b
- connects :: (Ord a, Ord b) => [AdjacencyMap a b] -> AdjacencyMap a b
- swap :: AdjacencyMap a b -> AdjacencyMap b a
- toBipartite :: (Ord a, Ord b) => AdjacencyMap (Either a b) -> AdjacencyMap a b
- toBipartiteWith :: (Ord a, Ord b, Ord c) => (a -> Either b c) -> AdjacencyMap a -> AdjacencyMap b c
- fromBipartite :: (Ord a, Ord b) => AdjacencyMap a b -> AdjacencyMap (Either a b)
- fromBipartiteWith :: Ord c => (a -> c) -> (b -> c) -> AdjacencyMap a b -> AdjacencyMap c
- isEmpty :: AdjacencyMap a b -> Bool
- hasLeftVertex :: Ord a => a -> AdjacencyMap a b -> Bool
- hasRightVertex :: Ord b => b -> AdjacencyMap a b -> Bool
- hasVertex :: (Ord a, Ord b) => Either a b -> AdjacencyMap a b -> Bool
- hasEdge :: (Ord a, Ord b) => a -> b -> AdjacencyMap a b -> Bool
- leftVertexCount :: AdjacencyMap a b -> Int
- rightVertexCount :: AdjacencyMap a b -> Int
- vertexCount :: AdjacencyMap a b -> Int
- edgeCount :: AdjacencyMap a b -> Int
- leftVertexList :: AdjacencyMap a b -> [a]
- rightVertexList :: AdjacencyMap a b -> [b]
- vertexList :: AdjacencyMap a b -> [Either a b]
- edgeList :: AdjacencyMap a b -> [(a, b)]
- leftVertexSet :: AdjacencyMap a b -> Set a
- rightVertexSet :: AdjacencyMap a b -> Set b
- vertexSet :: (Ord a, Ord b) => AdjacencyMap a b -> Set (Either a b)
- edgeSet :: (Ord a, Ord b) => AdjacencyMap a b -> Set (a, b)
- leftAdjacencyList :: AdjacencyMap a b -> [(a, [b])]
- rightAdjacencyList :: AdjacencyMap a b -> [(b, [a])]
- data List a b
  - = Nil
  - | Cons a (List b a)
- evenList :: [(a, b)] -> List a b
- oddList :: a -> [(b, a)] -> List a b
- path :: (Ord a, Ord b) => List a b -> AdjacencyMap a b
- circuit :: (Ord a, Ord b) => [(a, b)] -> AdjacencyMap a b
- biclique :: (Ord a, Ord b) => [a] -> [b] -> AdjacencyMap a b
- star :: (Ord a, Ord b) => a -> [b] -> AdjacencyMap a b
- stars :: (Ord a, Ord b) => [(a, [b])] -> AdjacencyMap a b
- mesh :: (Ord a, Ord b) => [a] -> [b] -> AdjacencyMap (a, b) (a, b)
- removeLeftVertex :: Ord a => a -> AdjacencyMap a b -> AdjacencyMap a b
- removeRightVertex :: Ord b => b -> AdjacencyMap a b -> AdjacencyMap a b
- removeEdge :: (Ord a, Ord b) => a -> b -> AdjacencyMap a b -> AdjacencyMap a b
- bimap :: (Ord a, Ord b, Ord c, Ord d) => (a -> c) -> (b -> d) -> AdjacencyMap a b -> AdjacencyMap c d
- box :: (Ord a, Ord b) => AdjacencyMap a a -> AdjacencyMap b b -> AdjacencyMap (a, b) (a, b)
- boxWith :: (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f) => (a -> c -> e) -> (b -> d -> e) -> (a -> d -> f) -> (b -> c -> f) -> AdjacencyMap a b -> AdjacencyMap c d -> AdjacencyMap e f
- consistent :: (Ord a, Ord b) => AdjacencyMap a b -> Bool

# Data structure

data AdjacencyMap a b #

The `AdjacencyMap` data type represents an undirected bipartite graph. The two type parameters determine the types of vertices of each part. If the types coincide, the vertices of the left part are still treated as disjoint from the vertices of the right part. See examples for more details.

We define a `Num` instance as a convenient notation for working with bipartite graphs:

    0                     == rightVertex 0
    swap 1                == leftVertex 1
    swap 1 + 2            == vertices [1] [2]
    swap 1 * 2            == edge 1 2
    swap 1 + 2 * swap 3   == overlay (leftVertex 1) (edge 3 2)
    swap 1 * (2 + swap 3) == connect (leftVertex 1) (vertices [3] [2])

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                 == "empty"
    show 1                     == "rightVertex 1"
    show (swap 2)              == "leftVertex 2"
    show (1 + 2)               == "vertices [] [1,2]"
    show (swap (1 + 2))        == "vertices [1,2] []"
    show (swap 1 * 2)          == "edge 1 2"
    show (swap 1 * 2 * swap 3) == "edges [(1,2),(3,2)]"
    show (swap 1 * 2 + swap 3) == "overlay (leftVertex 3) (edge 1 2)"

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

- `overlay` is commutative and associative:

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

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

        x * empty == x
        empty * x == x
            x * y == y * 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

- `connect` has the same effect as `overlay` on vertices of the same part:

       leftVertex x * leftVertex y  ==  leftVertex x + leftVertex y
      rightVertex x * rightVertex y == rightVertex x + rightVertex y

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 of the graph, respectively. In addition, l and r will denote the number of vertices in the left and right parts of the graph, respectively.

#### Instances

Instances details

(Ord a, Ord b) => Monoid (AdjacencyMap a b) # | Defined via `overlay` and `empty`.
Instance details Defined in Algebra.Graph.Bipartite.AdjacencyMap Methods mempty :: AdjacencyMap a b # mappend :: AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b # mconcat :: [AdjacencyMap a b] -> AdjacencyMap a b #
(Ord a, Ord b) => Semigroup (AdjacencyMap a b) # | Defined via `overlay`.
Instance details Defined in Algebra.Graph.Bipartite.AdjacencyMap Methods (<>) :: AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b # sconcat :: NonEmpty (AdjacencyMap a b) -> AdjacencyMap a b # stimes :: Integral b0 => b0 -> AdjacencyMap a b -> AdjacencyMap a b #
Generic (AdjacencyMap a b) # |

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

Associated Types

type Rep (AdjacencyMap a b) |

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

type Rep (AdjacencyMap a b) = D1 ('MetaData "AdjacencyMap" "Algebra.Graph.Bipartite.AdjacencyMap" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'False) (C1 ('MetaCons "BAM" 'PrefixI 'True) (S1 ('MetaSel ('Just "leftAdjacencyMap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map a (Set b))) :*: S1 ('MetaSel ('Just "rightAdjacencyMap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map b (Set a)))))

Methods

from :: AdjacencyMap a b -> Rep (AdjacencyMap a b) x #

to :: Rep (AdjacencyMap a b) x -> AdjacencyMap a b #

(Ord a, Ord b, Num b) => Num (AdjacencyMap a b) # | Note: this does not satisfy the usual ring laws; see `AdjacencyMap` for more details.

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

Methods

(+) :: AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b #

(-) :: AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b #

(*) :: AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b #

negate :: AdjacencyMap a b -> AdjacencyMap a b #

abs :: AdjacencyMap a b -> AdjacencyMap a b #

signum :: AdjacencyMap a b -> AdjacencyMap a b #

fromInteger :: Integer -> AdjacencyMap a b #

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

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

Methods

compare :: AdjacencyMap a b -> AdjacencyMap a b -> Ordering #

(<) :: AdjacencyMap a b -> AdjacencyMap a b -> Bool #

(<=) :: AdjacencyMap a b -> AdjacencyMap a b -> Bool #

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

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

max :: AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b #

min :: AdjacencyMap a b -> AdjacencyMap a b -> AdjacencyMap a b #

type Rep (AdjacencyMap a b) # |

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

type Rep (AdjacencyMap a b) = D1 ('MetaData "AdjacencyMap" "Algebra.Graph.Bipartite.AdjacencyMap" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'False) (C1 ('MetaCons "BAM" 'PrefixI 'True) (S1 ('MetaSel ('Just "leftAdjacencyMap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map a (Set b))) :*: S1 ('MetaSel ('Just "rightAdjacencyMap") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Map b (Set a)))))

leftAdjacencyMap :: AdjacencyMap a b -> Map a (Set b) #

The adjacency map of the left part of the graph: each left vertex is associated with a set of its right neighbours. Complexity: O(1) time and memory.

    leftAdjacencyMap empty           == Map.empty
    leftAdjacencyMap (leftVertex x)  == Map.singleton x Set.empty
    leftAdjacencyMap (rightVertex x) == Map.empty
    leftAdjacencyMap (edge x y)      == Map.singleton x (Set.singleton y)

rightAdjacencyMap :: AdjacencyMap a b -> Map b (Set a) #

The adjacency map of the right part of the graph: each right vertex is associated with a set of its left neighbours. Complexity: O(1) time and memory.

    rightAdjacencyMap empty           == Map.empty
    rightAdjacencyMap (leftVertex x)  == Map.empty
    rightAdjacencyMap (rightVertex x) == Map.singleton x Set.empty
    rightAdjacencyMap (edge x y)      == Map.singleton y (Set.singleton x)

# Basic graph construction primitives

empty :: AdjacencyMap a b #

Construct the empty graph.

    isEmpty empty           == True
    leftAdjacencyMap empty  == Map.empty
    rightAdjacencyMap empty == Map.empty
    hasVertex x empty       == False

leftVertex :: a -> AdjacencyMap a b #

Construct the graph comprising a single isolated vertex in the left part.

    leftAdjacencyMap (leftVertex x)  == Map.singleton x Set.empty
    rightAdjacencyMap (leftVertex x) == Map.empty
    hasLeftVertex x (leftVertex y)   == (x == y)
    hasRightVertex x (leftVertex y)  == False
    hasEdge x y (leftVertex z)       == False

rightVertex :: b -> AdjacencyMap a b #

Construct the graph comprising a single isolated vertex in the right part.

    leftAdjacencyMap (rightVertex x)  == Map.empty
    rightAdjacencyMap (rightVertex x) == Map.singleton x Set.empty
    hasLeftVertex x (rightVertex y)   == False
    hasRightVertex x (rightVertex y)  == (x == y)
    hasEdge x y (rightVertex z)       == False

vertex :: Either a b -> AdjacencyMap a b #

Construct the graph comprising a single isolated vertex.

    vertex . Left  == leftVertex
    vertex . Right == rightVertex

edge :: a -> b -> AdjacencyMap a b #

Construct the graph comprising a single edge.

    edge x y                     == connect (leftVertex x) (rightVertex y)
    leftAdjacencyMap (edge x y)  == Map.singleton x (Set.singleton y)
    rightAdjacencyMap (edge x y) == Map.singleton y (Set.singleton x)
    hasEdge x y (edge x y)       == True
    hasEdge 1 2 (edge 2 1)       == False

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

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

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

Connect two graphs, filtering out the edges between vertices of the same part. This is a commutative and 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 in the arguments: O(m1 + m2 + l1 * r2 + l2 * r1).

    connect (leftVertex x)     (leftVertex y)     == vertices [x,y] []
    connect (leftVertex x)     (rightVertex y)    == edge x y
    connect (rightVertex x)    (leftVertex y)     == edge y x
    connect (rightVertex x)    (rightVertex y)    == vertices [] [x,y]
    connect (vertices xs1 ys1) (vertices xs2 ys2) == overlay (biclique xs1 ys2) (biclique xs2 ys1)
    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)                     >= leftVertexCount x * rightVertexCount y
    edgeCount   (connect x y)                     <= leftVertexCount x * rightVertexCount y + rightVertexCount x * leftVertexCount y + edgeCount x + edgeCount y

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

Construct the graph comprising given lists of isolated vertices in each part. Complexity: O(L * log(L)) time and O(L) memory, where L is the total length of two lists.

    vertices [] []                    == empty
    vertices [x] []                   == leftVertex x
    vertices [] [x]                   == rightVertex x
    vertices xs ys                    == overlays (map leftVertex xs ++ map rightVertex ys)
    hasLeftVertex  x (vertices xs ys) == elem x xs
    hasRightVertex y (vertices xs ys) == elem y ys

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

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)
    hasEdge x y . edges == elem (x,y)
    edgeCount   . edges == length . nub

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

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, Ord b) => [AdjacencyMap a b] -> AdjacencyMap a b #

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

swap :: AdjacencyMap a b -> AdjacencyMap b a #

Swap the parts of a given graph. Complexity: O(1) time and memory.

    swap empty            == empty
    swap . leftVertex     == rightVertex
    swap (vertices xs ys) == vertices ys xs
    swap (edge x y)       == edge y x
    swap . edges          == edges . map Data.Tuple.swap
    swap . swap           == id

# Conversion functions

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

Construct a bipartite `AdjacencyMap` from an Algebra.Graph.AdjacencyMap, adding any missing edges to make the graph undirected and filtering out the edges within the same parts. Complexity: O(m * log(n)).

    toBipartite empty                      == empty
    toBipartite (vertex (Left x))          == leftVertex x
    toBipartite (vertex (Right x))         == rightVertex x
    toBipartite (edge (Left x) (Left y))   == vertices [x,y] []
    toBipartite (edge (Left x) (Right y))  == edge x y
    toBipartite (edge (Right x) (Left y))  == edge y x
    toBipartite (edge (Right x) (Right y)) == vertices [] [x,y]
    toBipartite . clique                   == uncurry biclique . partitionEithers
    toBipartite . fromBipartite            == id

toBipartiteWith :: (Ord a, Ord b, Ord c) => (a -> Either b c) -> AdjacencyMap a -> AdjacencyMap b c #

Construct a bipartite `AdjacencyMap` from an Algebra.Graph.AdjacencyMap, where the two parts are identified by a separate function, adding any missing edges to make the graph undirected and filtering out the edges within the same parts. Complexity: O(m * log(n)).

    toBipartiteWith f empty == empty
    toBipartiteWith Left x  == vertices (vertexList x) []
    toBipartiteWith Right x == vertices [] (vertexList x)
    toBipartiteWith f       == toBipartite . gmap f
    toBipartiteWith id      == toBipartite

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

Construct an Algebra.Graph.AdjacencyMap from a bipartite `AdjacencyMap`. Complexity: O(m * log(n)).

    fromBipartite empty          == empty
    fromBipartite (leftVertex x) == vertex (Left x)
    fromBipartite (edge x y)     == edges [(Left x, Right y), (Right y, Left x)]
    toBipartite . fromBipartite  == id

fromBipartiteWith :: Ord c => (a -> c) -> (b -> c) -> AdjacencyMap a b -> AdjacencyMap c #

Construct an Algebra.Graph.AdjacencyMap from a bipartite `AdjacencyMap` given a way to inject vertices of the two parts into the resulting vertex type. Complexity: O(m * log(n)).

    fromBipartiteWith Left Right             == fromBipartite
    fromBipartiteWith id id (vertices xs ys) == vertices (xs ++ ys)
    fromBipartiteWith id id . edges          == symmetricClosure . edges

# Graph properties

isEmpty :: AdjacencyMap a b -> Bool #

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

    isEmpty empty                 == True
    isEmpty (overlay empty empty) == True
    isEmpty (vertex x)            == False
    isEmpty                       == (==) empty

hasLeftVertex :: Ord a => a -> AdjacencyMap a b -> Bool #

Check if a graph contains a given vertex in the left part. Complexity: O(log(l)) time.

    hasLeftVertex x empty           == False
    hasLeftVertex x (leftVertex y)  == (x == y)
    hasLeftVertex x (rightVertex y) == False

hasRightVertex :: Ord b => b -> AdjacencyMap a b -> Bool #

Check if a graph contains a given vertex in the right part. Complexity: O(log(r)) time.

    hasRightVertex x empty           == False
    hasRightVertex x (leftVertex y)  == False
    hasRightVertex x (rightVertex y) == (x == y)

hasVertex :: (Ord a, Ord b) => Either a b -> AdjacencyMap a b -> Bool #

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

    hasVertex . Left  == hasLeftVertex
    hasVertex . Right == hasRightVertex

hasEdge :: (Ord a, Ord b) => a -> b -> AdjacencyMap a b -> 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            == elem (x,y) . edgeList

leftVertexCount :: AdjacencyMap a b -> Int #

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

    leftVertexCount empty           == 0
    leftVertexCount (leftVertex x)  == 1
    leftVertexCount (rightVertex x) == 0
    leftVertexCount (edge x y)      == 1
    leftVertexCount . edges         == length . nub . map fst

rightVertexCount :: AdjacencyMap a b -> Int #

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

    rightVertexCount empty           == 0
    rightVertexCount (leftVertex x)  == 0
    rightVertexCount (rightVertex x) == 1
    rightVertexCount (edge x y)      == 1
    rightVertexCount . edges         == length . nub . map snd

vertexCount :: AdjacencyMap a b -> Int #

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

    vertexCount empty      == 0
    vertexCount (vertex x) == 1
    vertexCount (edge x y) == 2
    vertexCount x          == leftVertexCount x + rightVertexCount x

edgeCount :: AdjacencyMap a b -> Int #

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

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

leftVertexList :: AdjacencyMap a b -> [a] #

The sorted list of vertices of the left part of a graph. Complexity: O(l) time and memory.

    leftVertexList empty              == []
    leftVertexList (leftVertex x)     == [x]
    leftVertexList (rightVertex x)    == []
    leftVertexList . flip vertices [] == nub . sort

rightVertexList :: AdjacencyMap a b -> [b] #

The sorted list of vertices of the right part of a graph. Complexity: O(r) time and memory.

    rightVertexList empty           == []
    rightVertexList (leftVertex x)  == []
    rightVertexList (rightVertex x) == [x]
    rightVertexList . vertices []   == nub . sort

vertexList :: AdjacencyMap a b -> [Either a b] #

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

    vertexList empty                             == []
    vertexList (vertex x)                        == [x]
    vertexList (edge x y)                        == [Left x, Right y]
    vertexList (vertices (lefts xs) (rights xs)) == nub (sort xs)

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

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 . edges    == nub . sort

leftVertexSet :: AdjacencyMap a b -> Set a #

The set of vertices of the left part of a graph. Complexity: O(l) time and memory.

    leftVertexSet empty              == Set.empty
    leftVertexSet . leftVertex       == Set.singleton
    leftVertexSet . rightVertex      == const Set.empty
    leftVertexSet . flip vertices [] == Set.fromList

rightVertexSet :: AdjacencyMap a b -> Set b #

The set of vertices of the right part of a graph. Complexity: O(r) time and memory.

    rightVertexSet empty         == Set.empty
    rightVertexSet . leftVertex  == const Set.empty
    rightVertexSet . rightVertex == Set.singleton
    rightVertexSet . vertices [] == Set.fromList

vertexSet :: (Ord a, Ord b) => AdjacencyMap a b -> Set (Either a b) #

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

    vertexSet empty                             == Set.empty
    vertexSet . vertex                          == Set.singleton
    vertexSet (edge x y)                        == Set.fromList [Left x, Right y]
    vertexSet (vertices (lefts xs) (rights xs)) == Set.fromList xs

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

The set of edges of a graph. Complexity: O(n + 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

leftAdjacencyList :: AdjacencyMap a b -> [(a, [b])] #

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

    leftAdjacencyList empty            == []
    leftAdjacencyList (vertices [] xs) == []
    leftAdjacencyList (vertices xs []) == [(x, []) | x <- nub (sort xs)]
    leftAdjacencyList (edge x y)       == [(x, [y])]
    leftAdjacencyList (star x ys)      == [(x, nub (sort ys))]

rightAdjacencyList :: AdjacencyMap a b -> [(b, [a])] #

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

    rightAdjacencyList empty            == []
    rightAdjacencyList (vertices [] xs) == [(x, []) | x <- nub (sort xs)]
    rightAdjacencyList (vertices xs []) == []
    rightAdjacencyList (edge x y)       == [(y, [x])]
    rightAdjacencyList (star x ys)      == [(y, [x])  | y <- nub (sort ys)]

# Standard families of graphs

data List a b #

A list of values of two alternating types. The first type argument denotes the type of the value at the head.

With the `OverloadedLists` extension it is possible to use the standard list notation to construct a `List` where the two types coincide, for example:

    [1, 2, 3, 4, 5] :: List Int Int

We make use of this shorthand notation in the examples below.

Constructors

Nil |
Cons a (List b a) |

#### Instances

Instances details

Generic (List a b) # |

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

Associated Types

type Rep (List a b) |

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

type Rep (List a b) = D1 ('MetaData "List" "Algebra.Graph.Bipartite.AdjacencyMap" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'False) (C1 ('MetaCons "Nil" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Cons" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (List b a))))

Methods

from :: List a b -> Rep (List a b) x #

to :: Rep (List a b) x -> List a b #

IsList (List a a) # |

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

Associated Types

type Item (List a a) |
Instance details Defined in Algebra.Graph.Bipartite.AdjacencyMap type Item (List a a) = a

Methods

fromList :: [Item (List a a)] -> List a a #

fromListN :: Int -> [Item (List a a)] -> List a a #

toList :: List a a -> [Item (List a a)] #

(Show a, Show b) => Show (List a b) # |
Instance details Defined in Algebra.Graph.Bipartite.AdjacencyMap Methods showsPrec :: Int -> List a b -> ShowS # show :: List a b -> String # showList :: [List a b] -> ShowS #
(Eq a, Eq b) => Eq (List a b) # |
Instance details Defined in Algebra.Graph.Bipartite.AdjacencyMap Methods (==) :: List a b -> List a b -> Bool # (/=) :: List a b -> List a b -> Bool #
(Ord a, Ord b) => Ord (List a b) # |
Instance details Defined in Algebra.Graph.Bipartite.AdjacencyMap Methods compare :: List a b -> List a b -> Ordering # (<) :: List a b -> List a b -> Bool # (<=) :: List a b -> List a b -> Bool # (>) :: List a b -> List a b -> Bool # (>=) :: List a b -> List a b -> Bool # max :: List a b -> List a b -> List a b # min :: List a b -> List a b -> List a b #
type Rep (List a b) # |

Instance details

Defined in Algebra.Graph.Bipartite.AdjacencyMap

type Rep (List a b) = D1 ('MetaData "List" "Algebra.Graph.Bipartite.AdjacencyMap" "algebraic-graphs-0.7-3cBpMvBpglnBK3x0Rk38no" 'False) (C1 ('MetaCons "Nil" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "Cons" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (List b a))))

type Item (List a a) # |
Instance details Defined in Algebra.Graph.Bipartite.AdjacencyMap type Item (List a a) = a

evenList :: [(a, b)] -> List a b #

Construct a `List` of even length from a list of pairs.

    evenList []                 == Nil
    evenList [(1,2), (3,4)]     == [1, 2, 3, 4] :: List Int Int
    evenList [(1,'a'), (2,'b')] == Cons 1 (Cons 'a' (Cons 2 (Cons 'b' Nil)))

oddList :: a -> [(b, a)] -> List a b #

Construct a `List` of odd length given the first element and a list of pairs.

    oddList 1 []                 == Cons 1 Nil
    oddList 1 [(2,3), (4,5)]     == [1, 2, 3, 4, 5] :: List Int Int
    oddList 1 [('a',2), ('b',3)] == Cons 1 (Cons 'a' (Cons 2 (Cons 'b' (Cons 3 Nil))))

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

The path on a `List` of vertices. Complexity: O(L * log(L)) time, where L is the length of the given list.

    path Nil                   == empty
    path (Cons x Nil)          == leftVertex x
    path (Cons x (Cons y Nil)) == edge x y
    path [1, 2, 3, 4, 5]       == edges [(1,2), (3,2), (3,4), (5,4)]

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

The circuit on a list of pairs of vertices. Complexity: O(L * log(L)) time, where L is the length of the given list.

    circuit []                    == empty
    circuit [(x,y)]               == edge x y
    circuit [(1,2), (3,4), (5,6)] == edges [(1,2), (3,2), (3,4), (5,4), (5,6), (1,6)]
    circuit . reverse             == swap . circuit . map Data.Tuple.swap

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

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

    biclique [] [] == empty
    biclique xs [] == vertices xs []
    biclique [] ys == vertices [] ys
    biclique xs ys == connect (vertices xs []) (vertices [] ys)

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

The star formed by a center vertex connected to a list of leaves. Complexity: O(L * log(L)) time, where L is the length of the given list.

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

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

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

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

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

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

    mesh xs []           == empty
    mesh [] ys           == empty
    mesh [x] [y]         == leftVertex (x,y)
    mesh [1,1] ['a','b'] == biclique [(1,'a'), (1,'b')] [(1,'a'), (1,'b')]
    mesh [1,2] ['a','b'] == biclique [(1,'a'), (2,'b')] [(1,'b'), (2,'a')]

# Graph transformation

removeLeftVertex :: Ord a => a -> AdjacencyMap a b -> AdjacencyMap a b #

Remove a vertex from the left part of a given graph. Complexity: O(r * log(l)) time.

    removeLeftVertex x (leftVertex x)       == empty
    removeLeftVertex 1 (leftVertex 2)       == leftVertex 2
    removeLeftVertex x (rightVertex y)      == rightVertex y
    removeLeftVertex x (edge x y)           == rightVertex y
    removeLeftVertex x . removeLeftVertex x == removeLeftVertex x

removeRightVertex :: Ord b => b -> AdjacencyMap a b -> AdjacencyMap a b #

Remove a vertex from the right part of a given graph. Complexity: O(l * log(r)) time.

    removeRightVertex x (rightVertex x)       == empty
    removeRightVertex 1 (rightVertex 2)       == rightVertex 2
    removeRightVertex x (leftVertex y)        == leftVertex y
    removeRightVertex y (edge x y)            == leftVertex x
    removeRightVertex x . removeRightVertex x == removeRightVertex x

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

Remove an edge from a given graph. Complexity: O(log(l) + log(r)) time.

    removeEdge x y (edge x y)            == vertices [x] [y]
    removeEdge x y . removeEdge x y      == removeEdge x y
    removeEdge x y . removeLeftVertex x  == removeLeftVertex x
    removeEdge x y . removeRightVertex y == removeRightVertex y

bimap :: (Ord a, Ord b, Ord c, Ord d) => (a -> c) -> (b -> d) -> AdjacencyMap a b -> AdjacencyMap c d #

Transform a graph by applying given functions to the vertices of each part. Complexity: O((n + m) * log(n)) time.

    bimap f g empty           == empty
    bimap f g . vertex        == vertex . Data.Bifunctor.bimap f g
    bimap f g (edge x y)      == edge (f x) (g y)
    bimap id id               == id
    bimap f1 g1 . bimap f2 g2 == bimap (f1 . f2) (g1 . g2)

# Graph composition

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

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

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

Up to isomorphism between the resulting vertex types, this operation is commutative, associative, distributes over `overlay`, has singleton graphs as identities and swapping 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 (leftVertex ())  ~~ x
    box x (rightVertex ()) ~~ swap x
    box x empty            ~~ empty
    vertexCount (box x y)  == vertexCount x * vertexCount y
    edgeCount   (box x y)  == vertexCount x * edgeCount y + edgeCount x * vertexCount y

boxWith :: (Ord a, Ord b, Ord c, Ord d, Ord e, Ord f) => (a -> c -> e) -> (b -> d -> e) -> (a -> d -> f) -> (b -> c -> f) -> AdjacencyMap a b -> AdjacencyMap c d -> AdjacencyMap e f #

Compute the generalised Cartesian product of two graphs. The resulting vertices are obtained using the given vertex combinators. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

See `box` for some examples.

    box == boxWith (,) (,) (,) (,)

# Miscellaneous

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

Check that the internal graph representation is consistent, i.e. that all edges that are present in the `leftAdjacencyMap` are also present in the `rightAdjacencyMap` map. 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 (edge x y)      == True
    consistent (edges x)       == True
    consistent (toBipartite x) == True
    consistent (swap x)        == True
    consistent (circuit x)     == True
    consistent (biclique x y)  == True