Algebra.Graph.NonEmpty.AdjacencyMap

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.AdjacencyMap

Contents

- Data structure
- Basic graph construction primitives
- Relations on graphs
- Graph properties
- Standard families of graphs
- Graph transformation
- Graph closure
- 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 data type `AdjacencyMap` for graphs that are known to be non-empty at compile time. To avoid name clashes with Algebra.Graph.AdjacencyMap, this module can be imported qualified:

    import qualified Algebra.Graph.NonEmpty.AdjacencyMap 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.AdjacencyMap, e.g. `vertices1`.

Synopsis

- data AdjacencyMap a
- toNonEmpty :: AdjacencyMap a -> Maybe (AdjacencyMap a)
- fromNonEmpty :: AdjacencyMap a -> AdjacencyMap a
- vertex :: a -> AdjacencyMap a
- edge :: Ord a => a -> a -> AdjacencyMap a
- overlay :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a
- connect :: Ord a => AdjacencyMap a -> AdjacencyMap a -> AdjacencyMap a
- vertices1 :: Ord a => NonEmpty a -> AdjacencyMap a
- edges1 :: Ord a => NonEmpty (a, a) -> AdjacencyMap a
- overlays1 :: Ord a => NonEmpty (AdjacencyMap a) -> AdjacencyMap a
- connects1 :: Ord a => NonEmpty (AdjacencyMap a) -> AdjacencyMap a
- isSubgraphOf :: Ord a => AdjacencyMap a -> AdjacencyMap a -> Bool
- hasVertex :: Ord a => a -> AdjacencyMap a -> Bool
- hasEdge :: Ord a => a -> a -> AdjacencyMap a -> Bool
- vertexCount :: AdjacencyMap a -> Int
- edgeCount :: AdjacencyMap a -> Int
- vertexList1 :: AdjacencyMap a -> NonEmpty a
- edgeList :: AdjacencyMap a -> [(a, a)]
- vertexSet :: AdjacencyMap a -> Set a
- edgeSet :: Ord a => AdjacencyMap a -> Set (a, a)
- preSet :: Ord a => a -> AdjacencyMap a -> Set a
- postSet :: Ord a => a -> AdjacencyMap a -> Set a
- path1 :: Ord a => NonEmpty a -> AdjacencyMap a
- circuit1 :: Ord a => NonEmpty a -> AdjacencyMap a
- clique1 :: Ord a => NonEmpty a -> AdjacencyMap a
- biclique1 :: Ord a => NonEmpty a -> NonEmpty a -> AdjacencyMap a
- star :: Ord a => a -> [a] -> AdjacencyMap a
- stars1 :: Ord a => NonEmpty (a, [a]) -> AdjacencyMap a
- tree :: Ord a => Tree a -> AdjacencyMap a
- removeVertex1 :: Ord a => a -> AdjacencyMap a -> Maybe (AdjacencyMap a)
- removeEdge :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a
- replaceVertex :: Ord a => a -> a -> AdjacencyMap a -> AdjacencyMap a
- mergeVertices :: Ord a => (a -> Bool) -> a -> AdjacencyMap a -> AdjacencyMap a
- transpose :: Ord a => AdjacencyMap a -> AdjacencyMap a
- gmap :: (Ord a, Ord b) => (a -> b) -> AdjacencyMap a -> AdjacencyMap b
- induce1 :: (a -> Bool) -> AdjacencyMap a -> Maybe (AdjacencyMap a)
- induceJust1 :: Ord a => AdjacencyMap (Maybe a) -> Maybe (AdjacencyMap a)
- closure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- reflexiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- symmetricClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- transitiveClosure :: Ord a => AdjacencyMap a -> AdjacencyMap a
- consistent :: Ord a => AdjacencyMap a -> Bool

# Data structure

data AdjacencyMap a #

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

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

Note: the `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 `Show` instance is defined using basic graph construction primitives:

    show (1         :: AdjacencyMap Int) == "vertex 1"
    show (1 + 2     :: AdjacencyMap Int) == "vertices1 [1,2]"
    show (1 * 2     :: AdjacencyMap Int) == "edge 1 2"
    show (1 * 2 * 3 :: AdjacencyMap Int) == "edges1 [(1,2),(1,3),(2,3)]"
    show (1 * 2 + 3 :: AdjacencyMap Int) == "overlay (vertex 3) (edge 1 2)"

The `Eq` instance satisfies the following laws of 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 and m will denote the number of vertices and edges in the graph, respectively.

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

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

Here are a few examples:

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

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

    isSubgraphOf x y ==> x <= y

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

#### Instances

Instances details

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

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

Convert a possibly empty `AdjacencyMap` into NonEmpty.`AdjacencyMap`. Returns `Nothing` if the argument is `empty`. Complexity: O(1) time, memory and size.

    toNonEmpty empty          == Nothing
    toNonEmpty . fromNonEmpty == Just

fromNonEmpty :: AdjacencyMap a -> AdjacencyMap a #

Convert a NonEmpty.`AdjacencyMap` into an `AdjacencyMap`. The resulting graph is guaranteed to be non-empty. Complexity: O(1) time, memory and size.

    isEmpty . fromNonEmpty    == const False
    toNonEmpty . fromNonEmpty == Just

# Basic graph construction primitives

vertex :: a -> AdjacencyMap a #

Construct the graph comprising a single isolated vertex.

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

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

Construct the graph comprising a single edge.

    edge x y               == connect (vertex x) (vertex y)
    hasEdge x y (edge x y) == True
    edgeCount   (edge x y) == 1
    vertexCount (edge 1 1) == 1
    vertexCount (edge 1 2) == 2

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

Overlay two graphs. This is a commutative, associative and idempotent operation with the identity `empty`. Complexity: O((n + m) * log(n)) time and O(n + m) memory.

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

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

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

    hasVertex z (connect x y) == hasVertex z x || hasVertex z y
    vertexCount (connect x y) >= vertexCount x
    vertexCount (connect x y) <= vertexCount x + vertexCount y
    edgeCount   (connect x y) >= edgeCount x
    edgeCount   (connect x y) >= edgeCount y
    edgeCount   (connect x y) >= vertexCount x * vertexCount y
    edgeCount   (connect x y) <= vertexCount x * vertexCount y + edgeCount x + edgeCount y
    vertexCount (connect 1 2) == 2
    edgeCount   (connect 1 2) == 1

vertices1 :: Ord a => NonEmpty a -> AdjacencyMap a #

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

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

edges1 :: Ord a => NonEmpty (a, a) -> AdjacencyMap a #

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

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

overlays1 :: Ord a => NonEmpty (AdjacencyMap a) -> AdjacencyMap a #

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

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

connects1 :: Ord a => NonEmpty (AdjacencyMap a) -> AdjacencyMap a #

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

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

# Relations on graphs

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

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

    isSubgraphOf x             (overlay x y) ==  True
    isSubgraphOf (overlay x y) (connect x y) ==  True
    isSubgraphOf (path1 xs)    (circuit1 xs) ==  True
    isSubgraphOf x y                         ==> x <= y

# Graph properties

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

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

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

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

Check if a graph contains a given edge. Complexity: O(log(n)) 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 :: AdjacencyMap a -> Int #

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

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

edgeCount :: AdjacencyMap a -> Int #

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

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

vertexList1 :: AdjacencyMap a -> NonEmpty a #

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

    vertexList1 (vertex x)  == [x]
    vertexList1 . vertices1 == nub . sort

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

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

    edgeList (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 :: AdjacencyMap a -> Set a #

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

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

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

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

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

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

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

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

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

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

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

# Standard families of graphs

path1 :: Ord a => NonEmpty a -> AdjacencyMap a #

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

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

circuit1 :: Ord a => NonEmpty a -> AdjacencyMap a #

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

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

clique1 :: Ord a => NonEmpty a -> AdjacencyMap a #

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

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

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

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

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

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

    star x []    == vertex x
    star x [y]   == edge x y
    star x [y,z] == edges1 [(x,y), (x,z)]

stars1 :: Ord a => NonEmpty (a, [a]) -> AdjacencyMap a #

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

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

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

    tree (Node x [])                                         == vertex x
    tree (Node x [Node y [Node z []]])                       == 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)]

# Graph transformation

removeVertex1 :: Ord a => a -> AdjacencyMap a -> Maybe (AdjacencyMap a) #

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

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

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

    removeEdge x y (edge x y)       == 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

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

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

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

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

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

    mergeVertices (const False) x    == id
    mergeVertices (== x) y           == replaceVertex x y
    mergeVertices even 1 (0 * 2)     == 1 * 1
    mergeVertices odd  1 (3 + 4 * 5) == 4 * 1

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

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

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

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

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

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

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

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

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

# Graph closure

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

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

    closure (vertex x)       == edge x x
    closure (edge x x)       == edge x x
    closure (edge x y)       == edges1 [(x,x), (x,y), (y,y)]
    closure (path1 $ nub xs) == reflexiveClosure (clique1 $ nub xs)
    closure                  == reflexiveClosure . transitiveClosure
    closure                  == transitiveClosure . reflexiveClosure
    closure . closure        == closure
    postSet x (closure y)    == Set.fromList (reachable y x)

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

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

    reflexiveClosure (vertex x)         == edge x x
    reflexiveClosure (edge x x)         == edge x x
    reflexiveClosure (edge x y)         == edges1 [(x,x), (x,y), (y,y)]
    reflexiveClosure . reflexiveClosure == reflexiveClosure

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

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

    symmetricClosure (vertex x)         == vertex x
    symmetricClosure (edge x y)         == edges1 [(x,y), (y,x)]
    symmetricClosure x                  == overlay x (transpose x)
    symmetricClosure . symmetricClosure == symmetricClosure

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

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

    transitiveClosure (vertex x)          == vertex x
    transitiveClosure (edge x y)          == edge x y
    transitiveClosure (path1 $ nub xs)    == clique1 (nub xs)
    transitiveClosure . transitiveClosure == transitiveClosure

# Miscellaneous

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

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

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