-----------------------------------------------------------------------------
-- |
-- Module     : Algebra.Graph.Undirected
-- Copyright  : (c) Andrey Mokhov 2016-2022
-- License    : MIT (see the file LICENSE)
-- Maintainer : [email protected]
-- Stability  : experimental
--
-- __Alga__ is a library for algebraic construction and manipulation of graphs
-- in Haskell. See <https://github.com/snowleopard/alga-paper this paper> for the
-- motivation behind the library, the underlying theory, and implementation details.
--
-- This module defines an undirected version of algebraic graphs. Undirected
-- graphs satisfy all laws of the 'Algebra.Graph.Class.Undirected' type class,
-- including the commutativity of 'connect'.
--
-- To avoid name clashes with "Algebra.Graph", this module can be imported
-- qualified:
--
-- @
-- import qualified Algebra.Graph.Undirected as Undirected
-- @

-----------------------------------------------------------------------------
module Algebra.Graph.Undirected (
    -- * Algebraic data type for graphs
    Graph, fromUndirected, toUndirected,

    -- * Basic graph construction primitives
    empty, vertex, edge, overlay, connect, vertices, edges, overlays, connects,

    -- * Graph folding
    foldg,

    -- * Relations on graphs
    isSubgraphOf, toRelation,

    -- * Graph properties
    isEmpty, size, hasVertex, hasEdge, vertexCount, edgeCount, vertexList,
    edgeList, vertexSet, edgeSet, adjacencyList, neighbours,

    -- * Standard families of graphs
    path, circuit, clique, biclique, star, stars, tree, forest,

    -- * Graph transformation
    removeVertex, removeEdge, replaceVertex, mergeVertices, induce, induceJust,
    complement
    ) where

import Algebra.Graph.Internal
import Algebra.Graph.ToGraph (toGraph)
import Control.Applicative (Alternative)
import Control.DeepSeq
import Control.Monad
import Data.Coerce
import Data.List (tails)
import GHC.Generics
import Data.Set (Set)
import Data.Tree (Tree, Forest)
import Data.String

import qualified Algebra.Graph                    as G
import qualified Algebra.Graph.Relation.Symmetric as SR
import qualified Data.Set                         as Set

-- TODO: Specialise the API for graphs with vertices of type 'Int'.

{-| The 'Graph' data type provides the four algebraic graph construction
primitives 'empty', 'vertex', 'overlay' and 'connect', as well as various
derived functions. The only difference compared to the 'Algebra.Graph.Graph'
data type defined in "Algebra.Graph" is that the 'connect' operation is
/commutative/. We define a 'Num' instance as a convenient notation for working
with undirected 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 'SR.Relation' 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, commutative 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

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 an undirected 'Graph' to the corresponding 'SR.Relation' 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
'edge' 1 2 == 'edge' 2 1@

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@
-}
newtype Graph a = UG (G.Graph a)
    deriving ( Applicative Graph
forall a. Graph a
forall a. Graph a -> Graph [a]
forall a. Graph a -> Graph a -> Graph a
forall (f :: * -> *).
Applicative f
-> (forall a. f a)
-> (forall a. f a -> f a -> f a)
-> (forall a. f a -> f [a])
-> (forall a. f a -> f [a])
-> Alternative f
many :: forall a. Graph a -> Graph [a]
$cmany :: forall a. Graph a -> Graph [a]
some :: forall a. Graph a -> Graph [a]
$csome :: forall a. Graph a -> Graph [a]
<|> :: forall a. Graph a -> Graph a -> Graph a
$c<|> :: forall a. Graph a -> Graph a -> Graph a
empty :: forall a. Graph a
$cempty :: forall a. Graph a
Alternative, Functor Graph
forall a. a -> Graph a
forall a b. Graph a -> Graph b -> Graph a
forall a b. Graph a -> Graph b -> Graph b
forall a b. Graph (a -> b) -> Graph a -> Graph b
forall a b c. (a -> b -> c) -> Graph a -> Graph b -> Graph c
forall (f :: * -> *).
Functor f
-> (forall a. a -> f a)
-> (forall a b. f (a -> b) -> f a -> f b)
-> (forall a b c. (a -> b -> c) -> f a -> f b -> f c)
-> (forall a b. f a -> f b -> f b)
-> (forall a b. f a -> f b -> f a)
-> Applicative f
<* :: forall a b. Graph a -> Graph b -> Graph a
$c<* :: forall a b. Graph a -> Graph b -> Graph a
*> :: forall a b. Graph a -> Graph b -> Graph b
$c*> :: forall a b. Graph a -> Graph b -> Graph b
liftA2 :: forall a b c. (a -> b -> c) -> Graph a -> Graph b -> Graph c
$cliftA2 :: forall a b c. (a -> b -> c) -> Graph a -> Graph b -> Graph c
<*> :: forall a b. Graph (a -> b) -> Graph a -> Graph b
$c<*> :: forall a b. Graph (a -> b) -> Graph a -> Graph b
pure :: forall a. a -> Graph a
$cpure :: forall a. a -> Graph a
Applicative, forall a b. a -> Graph b -> Graph a
forall a b. (a -> b) -> Graph a -> Graph b
forall (f :: * -> *).
(forall a b. (a -> b) -> f a -> f b)
-> (forall a b. a -> f b -> f a) -> Functor f
<$ :: forall a b. a -> Graph b -> Graph a
$c<$ :: forall a b. a -> Graph b -> Graph a
fmap :: forall a b. (a -> b) -> Graph a -> Graph b
$cfmap :: forall a b. (a -> b) -> Graph a -> Graph b
Functor, forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
forall a x. Rep (Graph a) x -> Graph a
forall a x. Graph a -> Rep (Graph a) x
$cto :: forall a x. Rep (Graph a) x -> Graph a
$cfrom :: forall a x. Graph a -> Rep (Graph a) x
Generic, String -> Graph a
forall a. IsString a => String -> Graph a
forall a. (String -> a) -> IsString a
fromString :: String -> Graph a
$cfromString :: forall a. IsString a => String -> Graph a
IsString, Applicative Graph
forall a. a -> Graph a
forall a b. Graph a -> Graph b -> Graph b
forall a b. Graph a -> (a -> Graph b) -> Graph b
forall (m :: * -> *).
Applicative m
-> (forall a b. m a -> (a -> m b) -> m b)
-> (forall a b. m a -> m b -> m b)
-> (forall a. a -> m a)
-> Monad m
return :: forall a. a -> Graph a
$creturn :: forall a. a -> Graph a
>> :: forall a b. Graph a -> Graph b -> Graph b
$c>> :: forall a b. Graph a -> Graph b -> Graph b
>>= :: forall a b. Graph a -> (a -> Graph b) -> Graph b
$c>>= :: forall a b. Graph a -> (a -> Graph b) -> Graph b
Monad
             , Monad Graph
Alternative Graph
forall a. Graph a
forall a. Graph a -> Graph a -> Graph a
forall (m :: * -> *).
Alternative m
-> Monad m
-> (forall a. m a)
-> (forall a. m a -> m a -> m a)
-> MonadPlus m
mplus :: forall a. Graph a -> Graph a -> Graph a
$cmplus :: forall a. Graph a -> Graph a -> Graph a
mzero :: forall a. Graph a
$cmzero :: forall a. Graph a
MonadPlus, Graph a -> ()
forall a. NFData a => Graph a -> ()
forall a. (a -> ()) -> NFData a
rnf :: Graph a -> ()
$crnf :: forall a. NFData a => Graph a -> ()
NFData )

instance (Show a, Ord a) => Show (Graph a) where
    show :: Graph a -> String
show = forall a. Show a => a -> String
show forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => Graph a -> Relation a
toRelation

-- | __Note:__ this does not satisfy the usual ring laws; see 'Graph' for more
-- details.
instance Num a => Num (Graph a) where
    fromInteger :: Integer -> Graph a
fromInteger = forall a. a -> Graph a
vertex forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Num a => Integer -> a
fromInteger
    + :: Graph a -> Graph a -> Graph a
(+)         = forall a. Graph a -> Graph a -> Graph a
overlay
    * :: Graph a -> Graph a -> Graph a
(*)         = forall a. Graph a -> Graph a -> Graph a
connect
    signum :: Graph a -> Graph a
signum      = forall a b. a -> b -> a
const forall a. Graph a
empty
    abs :: Graph a -> Graph a
abs         = forall a. a -> a
id
    negate :: Graph a -> Graph a
negate      = forall a. a -> a
id

instance Ord a => Eq (Graph a) where
    == :: Graph a -> Graph a -> Bool
(==) = forall a. Ord a => Graph a -> Graph a -> Bool
eqR

instance Ord a => Ord (Graph a) where
    compare :: Graph a -> Graph a -> Ordering
compare = forall a. Ord a => Graph a -> Graph a -> Ordering
ordR

-- | Defined via 'overlay'.
instance Semigroup (Graph a) where
    <> :: Graph a -> Graph a -> Graph a
(<>) = forall a. Graph a -> Graph a -> Graph a
overlay

-- | Defined via 'overlay' and 'empty'.
instance Monoid (Graph a) where
    mempty :: Graph a
mempty = forall a. Graph a
empty

-- TODO: Find a more efficient equality check.
-- Check if two graphs are equal by converting them to symmetric relations.
eqR :: Ord a => Graph a -> Graph a -> Bool
eqR :: forall a. Ord a => Graph a -> Graph a -> Bool
eqR Graph a
x Graph a
y = forall a. Ord a => Graph a -> Relation a
toRelation Graph a
x forall a. Eq a => a -> a -> Bool
== forall a. Ord a => Graph a -> Relation a
toRelation Graph a
y

-- TODO: Find a more efficient comparison.
-- Compare two graphs by converting them to their symmetric relations.
ordR :: Ord a => Graph a -> Graph a -> Ordering
ordR :: forall a. Ord a => Graph a -> Graph a -> Ordering
ordR Graph a
x Graph a
y = forall a. Ord a => a -> a -> Ordering
compare (forall a. Ord a => Graph a -> Relation a
toRelation Graph a
x) (forall a. Ord a => Graph a -> Relation a
toRelation Graph a
y)

-- | Construct an undirected graph from a given "Algebra.Graph".
-- Complexity: /O(1)/ time.
--
-- @
-- toUndirected ('Algebra.Graph.edge' 1 2)         == 'edge' 1 2
-- toUndirected . 'fromUndirected'   == id
-- 'vertexCount' . toUndirected      == 'Algebra.Graph.vertexCount'
-- (*2) . 'edgeCount' . toUndirected >= 'Algebra.Graph.edgeCount'
-- @
toUndirected :: G.Graph a -> Graph a
toUndirected :: forall a. Graph a -> Graph a
toUndirected = coerce :: forall a b. Coercible a b => a -> b
coerce

-- | Extract the underlying "Algebra.Graph".
-- Complexity: /O(n + m)/ time.
--
-- @
-- fromUndirected ('Algebra.Graph.edge' 1 2)     == 'Algebra.Graph.edges' [(1,2),(2,1)]
-- 'toUndirected' . 'fromUndirected' == id
-- 'Algebra.Graph.vertexCount' . fromUndirected  == 'vertexCount'
-- 'Algebra.Graph.edgeCount' . fromUndirected    <= (*2) . 'edgeCount'
-- @
fromUndirected :: Ord a => Graph a -> G.Graph a
fromUndirected :: forall a. Ord a => Graph a -> Graph a
fromUndirected = forall t. ToGraph t => t -> Graph (ToVertex t)
toGraph forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Relation a -> Relation a
SR.fromSymmetric forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => Graph a -> Relation a
toRelation

-- | Construct the /empty graph/.
--
-- @
-- 'isEmpty'     empty == True
-- 'hasVertex' x empty == False
-- 'vertexCount' empty == 0
-- 'edgeCount'   empty == 0
-- 'size'        empty == 1
-- @
empty :: Graph a
empty :: forall a. Graph a
empty = forall (f :: * -> *) (g :: * -> *) x. Coercible f g => f x -> g x
coerce00 forall a. Graph a
G.empty
{-# INLINE empty #-}

-- | Construct the graph comprising /a single isolated vertex/.
--
-- @
-- 'isEmpty'     (vertex x) == False
-- 'hasVertex' x (vertex y) == (x == y)
-- 'vertexCount' (vertex x) == 1
-- 'edgeCount'   (vertex x) == 0
-- 'size'        (vertex x) == 1
-- @
vertex :: a -> Graph a
vertex :: forall a. a -> Graph a
vertex = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. a -> Graph a
G.vertex
{-# INLINE vertex #-}

-- | Construct the graph comprising /a single edge/.
--
-- @
-- edge x y               == 'connect' ('vertex' x) ('vertex' y)
-- edge x y               == 'edge' y x
-- edge x y               == 'edges' [(x,y), (y,x)]
-- 'hasEdge' x y (edge x y) == True
-- 'edgeCount'   (edge x y) == 1
-- 'vertexCount' (edge 1 1) == 1
-- 'vertexCount' (edge 1 2) == 2
-- @
edge :: a -> a -> Graph a
edge :: forall a. a -> a -> Graph a
edge = forall a b c d (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible f g) =>
(a -> c -> f x) -> b -> d -> g x
coerce20 forall a. a -> a -> Graph a
G.edge
{-# INLINE edge #-}

-- | /Overlay/ two graphs. 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
-- @
overlay :: Graph a -> Graph a -> Graph a
overlay :: forall a. Graph a -> Graph a -> Graph a
overlay = forall a b c d (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible f g) =>
(a -> c -> f x) -> b -> d -> g x
coerce20 forall a. Graph a -> Graph a -> Graph a
G.overlay
{-# INLINE overlay #-}

-- | /Connect/ two graphs. This is a commutative and 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)/.
--
-- @
-- 'connect' x y               == 'connect' y x
-- '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 `div` 2
-- 'size'        (connect x y) == 'size' x        + 'size' y
-- 'vertexCount' (connect 1 2) == 2
-- 'edgeCount'   (connect 1 2) == 1
-- @
connect :: Graph a -> Graph a -> Graph a
connect :: forall a. Graph a -> Graph a -> Graph a
connect = forall a b c d (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible f g) =>
(a -> c -> f x) -> b -> d -> g x
coerce20 forall a. Graph a -> Graph a -> Graph a
G.connect
{-# INLINE connect #-}

-- | 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.
--
-- @
-- vertices []            == 'empty'
-- vertices [x]           == 'vertex' x
-- vertices               == 'overlays' . map 'vertex'
-- 'hasVertex' x . vertices == 'elem' x
-- 'vertexCount' . vertices == 'length' . 'Data.List.nub'
-- 'vertexSet'   . vertices == Set . 'Set.fromList'
-- @
vertices :: [a] -> Graph a
vertices :: forall a. [a] -> Graph a
vertices = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. [a] -> Graph a
G.vertices
{-# INLINE vertices #-}

-- | Construct the graph from a list of edges.
-- Complexity: /O(L)/ time, memory and size, where /L/ is the length of the
-- given list.
--
-- @
-- edges []             == 'empty'
-- edges [(x,y)]        == 'edge' x y
-- edges [(x,y), (y,x)] == 'edge' x y
-- @
edges :: [(a, a)] -> Graph a
edges :: forall a. [(a, a)] -> Graph a
edges = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. [(a, a)] -> Graph a
G.edges
{-# INLINE edges #-}

-- | 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.
--
-- @
-- overlays []        == 'empty'
-- overlays [x]       == x
-- overlays [x,y]     == 'overlay' x y
-- overlays           == 'foldr' 'overlay' 'empty'
-- 'isEmpty' . overlays == 'all' 'isEmpty'
-- @
overlays :: [Graph a] -> Graph a
overlays :: forall a. [Graph a] -> Graph a
overlays = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. [Graph a] -> Graph a
G.overlays
{-# INLINE overlays #-}

-- | 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.
--
-- @
-- connects []        == 'empty'
-- connects [x]       == x
-- connects [x,y]     == 'connect' x y
-- connects           == 'foldr' 'connect' 'empty'
-- 'isEmpty' . connects == 'all' 'isEmpty'
-- connects           == connects . 'reverse'
-- @
connects :: [Graph a] -> Graph a
connects :: forall a. [Graph a] -> Graph a
connects = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. [Graph a] -> Graph a
G.connects
{-# INLINE connects #-}

-- | 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.
--
-- @
-- foldg 'empty' 'vertex'        'overlay' 'connect'        == id
-- foldg 'empty' 'vertex'        'overlay' ('flip' 'connect') == id
-- foldg 1     ('const' 1)     (+)     (+)            == 'size'
-- foldg True  ('const' False) (&&)    (&&)           == 'isEmpty'
-- foldg False (== x)        (||)    (||)           == 'hasVertex' x
-- @
foldg :: b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
foldg :: forall b a.
b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
foldg = forall b a.
(b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b)
-> b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
coerce forall b a.
b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
G.foldg
  where
    coerce :: (b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> G.Graph a -> b)
           -> (b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) ->   Graph a -> b)
    coerce :: forall b a.
(b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b)
-> b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
coerce = coerce :: forall a b. Coercible a b => a -> b
Data.Coerce.coerce
{-# INLINE foldg #-}

-- | The 'isSubgraphOf' function takes two graphs and returns 'True' if the
-- first graph is a /subgraph/ of the second.
-- Complexity: /O(s + m * log(m))/ time. Note that the number of edges /m/ of a
-- graph can be quadratic with respect to the expression size /s/.
--
-- @
-- isSubgraphOf '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 ('edge' x y)    ('edge' y x)    ==  True
-- isSubgraphOf x y                         ==> x <= y
-- @
isSubgraphOf :: Ord a => Graph a -> Graph a -> Bool
isSubgraphOf :: forall a. Ord a => Graph a -> Graph a -> Bool
isSubgraphOf Graph a
x Graph a
y = forall a. Ord a => Relation a -> Relation a -> Bool
SR.isSubgraphOf (forall a. Ord a => Graph a -> Relation a
toRelation Graph a
x) (forall a. Ord a => Graph a -> Relation a
toRelation Graph a
y)
{-# NOINLINE [1] isSubgraphOf #-}

-- TODO: This is a very inefficient implementation. Find a way to construct a
-- symmetric relation directly, without building intermediate representations
-- for all subgraphs.
-- | Convert an undirected graph to a symmetric 'SR.Relation'.
toRelation :: Ord a => Graph a -> SR.Relation a
toRelation :: forall a. Ord a => Graph a -> Relation a
toRelation = forall b a.
b -> (a -> b) -> (b -> b -> b) -> (b -> b -> b) -> Graph a -> b
foldg forall a. Relation a
SR.empty forall a. a -> Relation a
SR.vertex forall a. Ord a => Relation a -> Relation a -> Relation a
SR.overlay forall a. Ord a => Relation a -> Relation a -> Relation a
SR.connect
{-# INLINE toRelation #-}

-- | Check if a graph is empty.
-- Complexity: /O(s)/ time.
--
-- @
-- isEmpty 'empty'                       == True
-- isEmpty ('overlay' 'empty' 'empty')       == True
-- isEmpty ('vertex' x)                  == False
-- isEmpty ('removeVertex' x $ 'vertex' x) == True
-- isEmpty ('removeEdge' x y $ 'edge' x y) == False
-- @
isEmpty :: Graph a -> Bool
isEmpty :: forall a. Graph a -> Bool
isEmpty = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(f x -> a) -> g x -> b
coerce01 forall a. Graph a -> Bool
G.isEmpty
{-# INLINE isEmpty #-}

-- | The /size/ of a graph, i.e. the number of leaves of the expression
-- including 'empty' leaves.
-- Complexity: /O(s)/ time.
--
-- @
-- 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
-- @
size :: Graph a -> Int
size :: forall a. Graph a -> Int
size = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(f x -> a) -> g x -> b
coerce01 forall a. Graph a -> Int
G.size
{-# INLINE size #-}

-- | Check if a graph contains a given vertex.
-- Complexity: /O(s)/ time.
--
-- @
-- hasVertex x 'empty'            == False
-- hasVertex x ('vertex' y)       == (x == y)
-- hasVertex x . 'removeVertex' x == 'const' False
-- @
hasVertex :: Eq a => a -> Graph a -> Bool
hasVertex :: forall a. Eq a => a -> Graph a -> Bool
hasVertex = forall a b c d (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible f g) =>
(a -> f x -> c) -> b -> g x -> d
coerce11 forall a. Eq a => a -> Graph a -> Bool
G.hasVertex
{-# INLINE hasVertex #-}
{-# SPECIALISE hasVertex :: Int -> Graph Int -> Bool #-}

-- TODO: Optimise this further.
-- | Check if a graph contains a given edge.
-- Complexity: /O(s)/ time.
--
-- @
-- hasEdge x y 'empty'            == False
-- hasEdge x y ('vertex' z)       == False
-- hasEdge x y ('edge' x y)       == True
-- hasEdge x y ('edge' y x)       == True
-- hasEdge x y . 'removeEdge' x y == 'const' False
-- hasEdge x y                  == 'elem' (min x y, max x y) . 'edgeList'
-- @
hasEdge :: Eq a => a -> a -> Graph a -> Bool
hasEdge :: forall a. Eq a => a -> a -> Graph a -> Bool
hasEdge a
s a
t (UG Graph a
g) = forall a. Eq a => a -> a -> Graph a -> Bool
G.hasEdge a
s a
t Graph a
g Bool -> Bool -> Bool
|| forall a. Eq a => a -> a -> Graph a -> Bool
G.hasEdge a
t a
s Graph a
g
{-# INLINE hasEdge #-}
{-# SPECIALISE hasEdge :: Int -> Int -> Graph Int -> Bool #-}

-- | The number of vertices in a graph.
-- Complexity: /O(s * log(n))/ time.
--
-- @
-- vertexCount 'empty'             ==  0
-- vertexCount ('vertex' x)        ==  1
-- vertexCount                   ==  'length' . 'vertexList'
-- vertexCount x \< vertexCount y ==> x \< y
-- @
vertexCount :: Ord a => Graph a -> Int
vertexCount :: forall a. Ord a => Graph a -> Int
vertexCount = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(f x -> a) -> g x -> b
coerce01 forall a. Ord a => Graph a -> Int
G.vertexCount
{-# INLINE [1] vertexCount #-}

-- | The number of edges in a graph.
-- Complexity: /O(s + m * log(m))/ time. Note that the number of edges /m/ of a
-- graph can be quadratic with respect to the expression size /s/.
--
-- @
-- edgeCount 'empty'      == 0
-- edgeCount ('vertex' x) == 0
-- edgeCount ('edge' x y) == 1
-- edgeCount            == 'length' . 'edgeList'
-- @
edgeCount :: Ord a => Graph a -> Int
edgeCount :: forall a. Ord a => Graph a -> Int
edgeCount = forall a. Ord a => Relation a -> Int
SR.edgeCount forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => Graph a -> Relation a
toRelation
{-# INLINE [1] edgeCount #-}

-- | The sorted list of vertices of a given graph.
-- Complexity: /O(s * log(n))/ time and /O(n)/ memory.
--
-- @
-- vertexList 'empty'      == []
-- vertexList ('vertex' x) == [x]
-- vertexList . 'vertices' == 'Data.List.nub' . 'Data.List.sort'
-- @
vertexList :: Ord a => Graph a -> [a]
vertexList :: forall a. Ord a => Graph a -> [a]
vertexList = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(f x -> a) -> g x -> b
coerce01 forall a. Ord a => Graph a -> [a]
G.vertexList
{-# INLINE [1] vertexList #-}

-- | The sorted list of edges of a graph.
-- Complexity: /O(s + m * log(m))/ time and /O(m)/ memory. Note that the number of
-- edges /m/ of a graph can be quadratic with respect to the expression size /s/.
--
-- @
-- edgeList 'empty'          == []
-- edgeList ('vertex' x)     == []
-- edgeList ('edge' x y)     == [(min x y, max y x)]
-- edgeList ('star' 2 [3,1]) == [(1,2), (2,3)]
-- @
edgeList :: Ord a => Graph a -> [(a, a)]
edgeList :: forall a. Ord a => Graph a -> [(a, a)]
edgeList = forall a. Ord a => Relation a -> [(a, a)]
SR.edgeList forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => Graph a -> Relation a
toRelation
{-# INLINE [1] edgeList #-}

-- | The set of vertices of a given graph.
-- Complexity: /O(s * log(n))/ time and /O(n)/ memory.
--
-- @
-- vertexSet 'empty'      == Set.'Set.empty'
-- vertexSet . 'vertex'   == Set.'Set.singleton'
-- vertexSet . 'vertices' == Set.'Set.fromList'
-- @
vertexSet :: Ord a => Graph a -> Set a
vertexSet :: forall a. Ord a => Graph a -> Set a
vertexSet = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(f x -> a) -> g x -> b
coerce01 forall a. Ord a => Graph a -> Set a
G.vertexSet
{-# INLINE vertexSet #-}

-- | The set of edges of a given graph.
-- Complexity: /O(s * log(m))/ time and /O(m)/ memory.
--
-- @
-- edgeSet 'empty'      == Set.'Set.empty'
-- edgeSet ('vertex' x) == Set.'Set.empty'
-- edgeSet ('edge' x y) == Set.'Set.singleton' ('min' x y, 'max' x y)
-- @
edgeSet :: Ord a => Graph a -> Set (a, a)
edgeSet :: forall a. Ord a => Graph a -> Set (a, a)
edgeSet = forall a. Ord a => Relation a -> Set (a, a)
SR.edgeSet forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => Graph a -> Relation a
toRelation
{-# INLINE [1] edgeSet #-}

-- | The sorted /adjacency list/ of a graph.
-- Complexity: /O(n + m)/ time and memory.
--
-- @
-- adjacencyList 'empty'          == []
-- adjacencyList ('vertex' x)     == [(x, [])]
-- adjacencyList ('edge' 1 2)     == [(1, [2]), (2, [1])]
-- adjacencyList ('star' 2 [3,1]) == [(1, [2]), (2, [1,3]), (3, [2])]
-- 'stars' . adjacencyList        == id
-- @
adjacencyList :: Ord a => Graph a -> [(a, [a])]
adjacencyList :: forall a. Ord a => Graph a -> [(a, [a])]
adjacencyList = forall a. Eq a => Relation a -> [(a, [a])]
SR.adjacencyList forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => Graph a -> Relation a
toRelation
{-# INLINE adjacencyList #-}
{-# SPECIALISE adjacencyList :: Graph Int -> [(Int, [Int])] #-}

-- | The set of vertices /adjacent/ to a given vertex.
--
-- @
-- neighbours x 'empty'      == Set.'Set.empty'
-- neighbours x ('vertex' x) == Set.'Set.empty'
-- neighbours x ('edge' x y) == Set.'Set.fromList' [y]
-- neighbours y ('edge' x y) == Set.'Set.fromList' [x]
-- @
neighbours :: Ord a => a -> Graph a -> Set a
neighbours :: forall a. Ord a => a -> Graph a -> Set a
neighbours a
x = forall a. Ord a => a -> Relation a -> Set a
SR.neighbours a
x forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Ord a => Graph a -> Relation a
toRelation
{-# INLINE neighbours #-}

-- | The /path/ on a list of vertices.
-- Complexity: /O(L)/ time, memory and size, where /L/ is the length of the
-- given list.
--
-- @
-- path []        == 'empty'
-- path [x]       == 'vertex' x
-- path [x,y]     == 'edge' x y
-- path . 'reverse' == path
-- @
path :: [a] -> Graph a
path :: forall a. [a] -> Graph a
path = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. [a] -> Graph a
G.path
{-# INLINE path #-}

-- | The /circuit/ on a list of vertices.
-- Complexity: /O(L)/ time, memory and size, where /L/ is the length of the
-- given list.
--
-- @
-- circuit []        == 'empty'
-- circuit [x]       == 'edge' x x
-- circuit [x,y]     == 'edge' (x,y)
-- circuit . 'reverse' == circuit
-- @
circuit :: [a] -> Graph a
circuit :: forall a. [a] -> Graph a
circuit = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. [a] -> Graph a
G.circuit
{-# INLINE circuit #-}

-- | The /clique/ on a list of vertices.
-- Complexity: /O(L)/ time, memory and size, where /L/ is the length of the
-- given list.
--
-- @
-- 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'  == clique
-- @
clique :: [a] -> Graph a
clique :: forall a. [a] -> Graph a
clique = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. [a] -> Graph a
G.clique
{-# INLINE clique #-}

-- | 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.
--
-- @
-- biclique []      []      == 'empty'
-- biclique [x]     []      == 'vertex' x
-- biclique []      [y]     == 'vertex' y
-- biclique [x1,x2] [y1,y2] == 'edges' [(x1,y1), (x1,y2), (x2,x2), (x2,y2)]
-- biclique xs      ys      == 'connect' ('vertices' xs) ('vertices' ys)
-- @
biclique :: [a] -> [a] -> Graph a
biclique :: forall a. [a] -> [a] -> Graph a
biclique = forall a b c d (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible f g) =>
(a -> c -> f x) -> b -> d -> g x
coerce20 forall a. [a] -> [a] -> Graph a
G.biclique
{-# INLINE biclique #-}

-- | The /star/ formed by a centre vertex connected to a list of leaves.
-- Complexity: /O(L)/ time, memory and size, where /L/ is the length of the
-- given list.
--
-- @
-- star x []    == 'vertex' x
-- star x [y]   == 'edge' x y
-- star x [y,z] == 'edges' [(x,y), (x,z)]
-- star x ys    == 'connect' ('vertex' x) ('vertices' ys)
-- @
star :: a -> [a] -> Graph a
star :: forall a. a -> [a] -> Graph a
star = forall a b c d (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible f g) =>
(a -> c -> f x) -> b -> d -> g x
coerce20 forall a. a -> [a] -> Graph a
G.star
{-# INLINE star #-}

-- | 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.
--
-- @
-- 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)
-- @
stars :: [(a, [a])] -> Graph a
stars :: forall a. [(a, [a])] -> Graph a
stars = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. [(a, [a])] -> Graph a
G.stars
{-# INLINE stars #-}

-- | 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)]
-- @
tree :: Tree a -> Graph a
tree :: forall a. Tree a -> Graph a
tree = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. Tree a -> Graph a
G.tree
{-# INLINE tree #-}

-- | 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'
-- @
forest :: Forest a -> Graph a
forest :: forall a. Forest a -> Graph a
forest = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. Forest a -> Graph a
G.forest
{-# INLINE forest #-}

-- | Remove a vertex from a given graph.
-- Complexity: /O(s)/ time, memory and size.
--
-- @
-- 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
-- @
removeVertex :: Eq a => a -> Graph a -> Graph a
removeVertex :: forall a. Eq a => a -> Graph a -> Graph a
removeVertex = forall a b c d (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible f g) =>
(a -> f x -> c) -> b -> g x -> d
coerce11 forall a. Eq a => a -> Graph a -> Graph a
G.removeVertex
{-# INLINE removeVertex #-}
{-# SPECIALISE removeVertex :: Int -> Graph Int -> Graph Int #-}

-- TODO: Optimise by doing a single graph traversal.
-- | 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                  == removeEdge y x
-- 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
-- @
removeEdge :: Eq a => a -> a -> Graph a -> Graph a
removeEdge :: forall a. Eq a => a -> a -> Graph a -> Graph a
removeEdge a
s a
t = coerce :: forall a b. Coercible a b => a -> b
Data.Coerce.coerce forall a b. (a -> b) -> a -> b
$ forall a. Eq a => a -> a -> Graph a -> Graph a
G.removeEdge a
s a
t forall b c a. (b -> c) -> (a -> b) -> a -> c
. forall a. Eq a => a -> a -> Graph a -> Graph a
G.removeEdge a
t a
s
{-# INLINE removeEdge #-}
{-# SPECIALISE removeEdge :: Int -> Int -> Graph Int -> Graph Int #-}

-- | The function @'replaceVertex' x y@ replaces vertex @x@ with vertex @y@ in a
-- given 'Graph'. If @y@ already exists, @x@ and @y@ will be merged.
-- Complexity: /O(s)/ time, memory and size.
--
-- @
-- replaceVertex x x            == id
-- replaceVertex x y ('vertex' x) == 'vertex' y
-- replaceVertex x y            == 'mergeVertices' (== x) y
-- @
replaceVertex :: Eq a => a -> a -> Graph a -> Graph a
replaceVertex :: forall a. Eq a => a -> a -> Graph a -> Graph a
replaceVertex = forall a b c d p q (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible p q, Coercible f g) =>
(a -> c -> f x -> p) -> b -> d -> g x -> q
coerce21 forall a. Eq a => a -> a -> Graph a -> Graph a
G.replaceVertex
{-# INLINE replaceVertex #-}
{-# SPECIALISE replaceVertex :: Int -> Int -> Graph Int -> Graph Int #-}

-- | Merge vertices satisfying a given predicate into a given vertex.
-- Complexity: /O(s)/ time, memory and size, assuming that the predicate takes
-- constant time.
--
-- @
-- mergeVertices ('const' False) x    == id
-- mergeVertices (== x) y           == 'replaceVertex' x y
-- mergeVertices 'even' 1 (0 * 2)     == 1 * 1
-- mergeVertices 'odd'  1 (3 + 4 * 5) == 4 * 1
-- @
mergeVertices :: (a -> Bool) -> a -> Graph a -> Graph a
mergeVertices :: forall a. (a -> Bool) -> a -> Graph a -> Graph a
mergeVertices = forall a b c d p q (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible p q, Coercible f g) =>
(a -> c -> f x -> p) -> b -> d -> g x -> q
coerce21 forall a. (a -> Bool) -> a -> Graph a -> Graph a
G.mergeVertices
{-# INLINE mergeVertices #-}

-- | 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.
--
-- @
-- 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
-- @
induce :: (a -> Bool) -> Graph a -> Graph a
induce :: forall a. (a -> Bool) -> Graph a -> Graph a
induce = forall a b c d (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible c d, Coercible f g) =>
(a -> c -> f x) -> b -> d -> g x
coerce20 forall a. (a -> Bool) -> Graph a -> Graph a
G.induce
{-# INLINE induce #-}

-- | Construct the /induced subgraph/ of a given graph by removing the vertices
-- that are 'Nothing'.
-- Complexity: /O(s)/ time, memory and size.
--
-- @
-- 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
-- @
induceJust :: Graph (Maybe a) -> Graph a
induceJust :: forall a. Graph (Maybe a) -> Graph a
induceJust = forall a b (f :: * -> *) (g :: * -> *) x.
(Coercible a b, Coercible f g) =>
(a -> f x) -> b -> g x
coerce10 forall a. Graph (Maybe a) -> Graph a
G.induceJust
{-# INLINE induceJust #-}

-- | The edge complement of a graph. Note that, as can be seen from the examples
-- below, this operation ignores self-loops.
-- Complexity: /O(n^2 * log n)/ time, /O(n^2)/ memory.
--
-- @
-- complement 'empty'           == 'empty'
-- complement ('vertex' x)      == ('vertex' x)
-- complement ('edge' 1 2)      == ('vertices' [1, 2])
-- complement ('edge' 0 0)      == ('edge' 0 0)
-- complement ('star' 1 [2, 3]) == ('overlay' ('vertex' 1) ('edge' 2 3))
-- complement . complement    == id
-- @
complement :: Ord a => Graph a -> Graph a
complement :: forall a. Ord a => Graph a -> Graph a
complement Graph a
g = forall a. Graph a -> Graph a -> Graph a
overlay (forall a. [a] -> Graph a
vertices [a]
vsOld) (forall a. [(a, a)] -> Graph a
edges forall a b. (a -> b) -> a -> b
$ forall a. Set a -> [a]
Set.toAscList Set (a, a)
esNew)
  where
    vsOld :: [a]
vsOld = forall a. Ord a => Graph a -> [a]
vertexList Graph a
g
    esOld :: Set (a, a)
esOld = forall a. Ord a => Graph a -> Set (a, a)
edgeSet Graph a
g
    loops :: Set (a, a)
loops = forall a. (a -> Bool) -> Set a -> Set a
Set.filter (forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry forall a. Eq a => a -> a -> Bool
(==)) Set (a, a)
esOld
    esAll :: Set (a, a)
esAll = forall a. Eq a => [a] -> Set a
Set.fromAscList [ (a
x, a
y) | a
x:[a]
ys <- forall a. [a] -> [[a]]
tails [a]
vsOld, a
y <- [a]
ys ]
    esNew :: Set (a, a)
esNew = forall a. Ord a => Set a -> Set a -> Set a
Set.union Set (a, a)
loops (forall a. Ord a => Set a -> Set a -> Set a
Set.difference Set (a, a)
esAll Set (a, a)
esOld)