content: -----------------------------------------------------------------------------
  -- |
  -- Module     : Algebra.Graph.HigherKinded.Class
  -- Copyright  : (c) Andrey Mokhov 2016-2025
  -- License    : MIT (see the file LICENSE)
  -- Maintainer : andrey.mokhov@gmail.com
  -- 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 the core type class 'Graph', a few graph subclasses, and
  -- basic polymorphic graph construction primitives. Functions that cannot be
  -- implemented fully polymorphically and require the use of an intermediate data
  -- type are not included. For example, to compute the size of a 'Graph'
  -- expression you will need to use a concrete data type, such as "Algebra.Graph".
  --
  -- See "Algebra.Graph.Class" for alternative definitions where the core type
  -- class is not higher-kinded and permits more instances.
  -----------------------------------------------------------------------------
  module Algebra.Graph.HigherKinded.Class (
      -- * The core type class
      Graph (..), empty, vertex, overlay,
  
      -- * Undirected graphs
      Undirected,
  
      -- * Reflexive graphs
      Reflexive,
  
      -- * Transitive graphs
      Transitive,
  
      -- * Preorders
      Preorder,
  
      -- * Basic graph construction primitives
      edge, vertices, edges, overlays, connects,
  
      -- * Relations on graphs
      isSubgraphOf,
  
      -- * Graph properties
      hasEdge,
  
      -- * Standard families of graphs
      path, circuit, clique, biclique, star, stars, tree, forest, mesh, torus,
      deBruijn,
  
      -- * Graph transformation
      removeVertex, replaceVertex, mergeVertices, splitVertex, induce
      ) where
  
  import Control.Applicative (Alternative(empty, (<|>)))
  import Control.Monad (MonadPlus, mfilter)
  import Data.Tree (Forest, Tree (..))
  
  import qualified Algebra.Graph as G
  
  {-|
  The core type class for constructing algebraic graphs is defined by introducing
  the 'connect' method to the standard 'MonadPlus' class and reusing the following
  existing methods:
  
  * The 'empty' method comes from the 'Control.Applicative.Alternative' class and
  corresponds to the /empty graph/. This module simply re-exports it.
  
  * The 'vertex' graph construction primitive is an alias for 'pure' of the
  'Applicative' type class.
  
  * Graph 'overlay' is an alias for 'mplus' of the 'MonadPlus' type class.
  
  The 'Graph' type class is characterised by the following minimal set of axioms.
  In equations we use @+@ and @*@ as convenient shortcuts for 'overlay' and
  'connect', respectively.
  
      * 'overlay' is commutative and associative:
  
          >       x + y == y + x
          > x + (y + z) == (x + y) + z
  
      * 'connect' is associative and has 'empty' as the identity:
  
          >   x * empty == x
          >   empty * x == x
          > x * (y * z) == (x * y) * z
  
      * 'connect' distributes over 'overlay':
  
          > x * (y + z) == x * y + x * z
          > (x + y) * z == x * z + y * z
  
      * 'connect' can be decomposed:
  
          > x * y * z == x * y + x * z + y * z
  
  The following useful theorems can be proved from the above set of axioms.
  
      * 'overlay' has 'empty' as the identity and is idempotent:
  
          >   x + empty == x
          >   empty + x == x
          >       x + x == x
  
      * Absorption and saturation of 'connect':
  
          > x * y + x + y == x * y
          >     x * x * x == x * x
  
  The core type class 'Graph' corresponds to unlabelled directed graphs.
  'Undirected', 'Reflexive', 'Transitive' and 'Preorder' graphs can be obtained
  by extending the minimal set of axioms.
  
  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.
  -}
  class MonadPlus g => Graph g where
      -- | Connect two graphs.
      connect :: g a -> g a -> g a
  
  instance Graph G.Graph where
      connect = G.connect
  
  -- | Construct the graph comprising a single isolated vertex. An alias for 'pure'.
  vertex :: Graph g => a -> g a
  vertex = pure
  
  -- | Overlay two graphs. An alias for '<|>'.
  overlay :: Graph g => g a -> g a -> g a
  overlay = (<|>)
  
  {-|
  The class of /undirected graphs/ that satisfy the following additional axiom.
  
      * 'connect' is commutative:
  
          > x * y == y * x
  -}
  class Graph g => Undirected g
  
  {-|
  The class of /reflexive graphs/ that satisfy the following additional axiom.
  
      * Each vertex has a /self-loop/:
  
          > vertex x == vertex x * vertex x
  
      Or, alternatively, if we remember that 'vertex' is an alias for 'pure':
  
          > pure x == pure x * pure x
  
  Note that by applying the axiom in the reverse direction, one can always remove
  all self-loops resulting in an /irreflexive graph/. This type class can
  therefore be also used in the context of irreflexive graphs.
  -}
  class Graph g => Reflexive g
  
  {-|
  The class of /transitive graphs/ that satisfy the following additional axiom.
  
      * The /closure/ axiom: graphs with equal transitive closures are equal.
  
          > y /= empty ==> x * y + x * z + y * z == x * y + y * z
  
  By repeated application of the axiom one can turn any graph into its transitive
  closure or transitive reduction.
  -}
  class Graph g => Transitive g
  
  {-|
  The class of /preorder graphs/ that are both reflexive and transitive.
  -}
  class (Reflexive g, Transitive g) => Preorder g
  
  -- | Construct the graph comprising a single edge.
  --
  -- @
  -- edge x y               == 'connect' ('vertex' x) ('vertex' y)
  -- 'vertexCount' (edge 1 1) == 1
  -- 'vertexCount' (edge 1 2) == 2
  -- @
  edge :: Graph g => a -> a -> g a
  edge x y = connect (vertex x) (vertex y)
  
  -- | 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 :: Graph g => [a] -> g a
  vertices []     = empty
  vertices [x]    = vertex x
  vertices (x:xs) = vertex x `overlay` vertices xs
  
  -- | 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 :: Graph g => [(a, a)] -> g a
  edges = overlays . map (uncurry edge)
  
  -- | 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 g => [g a] -> g a
  overlays []     = empty
  overlays [x]    = x
  overlays (x:xs) = x `overlay` overlays xs
  
  -- | 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 :: Graph g => [g a] -> g a
  connects []     = empty
  connects [x]    = x
  connects (x:xs) = x `connect` connects xs
  
  -- | The 'isSubgraphOf' function takes two graphs and returns 'True' if the
  -- first graph is a /subgraph/ of the second. Here is the current implementation:
  --
  -- @
  -- isSubgraphOf x y = 'overlay' x y == y
  -- @
  -- The complexity therefore depends on the complexity of equality testing of
  -- the specific graph instance.
  --
  -- @
  -- 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 :: (Graph g, Eq (g a)) => g a -> g a -> Bool
  isSubgraphOf x y = overlay x y == y
  
  -- | 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                  == 'elem' (x,y) . 'edgeList'
  -- @
  hasEdge :: (Eq (g a), Graph g, Ord a) => a -> a -> g a -> Bool
  hasEdge u v = (edge u v `isSubgraphOf`) . induce (\x -> x == u || x == v)
  
  -- | 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 :: Graph g => [a] -> g a
  path xs = case xs of []     -> empty
                       [x]    -> vertex x
                       (_:ys) -> edges (zip xs ys)
  
  -- | 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] == 'edges' [(x,y), (y,x)]
  -- @
  circuit :: Graph g => [a] -> g a
  circuit []     = empty
  circuit (x:xs) = path $ [x] ++ xs ++ [x]
  
  -- | 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 :: Graph g => [a] -> g a
  clique = connects . map vertex
  
  -- | 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,y1), (x2,y2)]
  -- biclique xs      ys      == 'connect' ('vertices' xs) ('vertices' ys)
  -- @
  biclique :: Graph g => [a] -> [a] -> g a
  biclique xs [] = vertices xs
  biclique [] ys = vertices ys
  biclique xs ys = connect (vertices xs) (vertices ys)
  
  -- | 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 :: Graph g => a -> [a] -> g a
  star x [] = vertex x
  star x ys = connect (vertex x) (vertices ys)
  
  -- | 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 :: Graph g => [(a, [a])] -> g a
  stars = overlays . map (uncurry star)
  
  -- | 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 :: Graph g => Tree a -> g a
  tree (Node x []) = vertex x
  tree (Node x f ) = star x (map rootLabel f)
           `overlay` forest (filter (not . null . subForest) f)
  
  -- | 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 :: Graph g => Forest a -> g a
  forest = overlays . map tree
  
  -- | Construct a /mesh graph/ from two lists of vertices.
  -- Complexity: /O(L1 * L2)/ time, memory and size, where /L1/ and /L2/ are the
  -- lengths of the given lists.
  --
  -- @
  -- mesh xs     []   == 'empty'
  -- mesh []     ys   == 'empty'
  -- mesh [x]    [y]  == 'vertex' (x, y)
  -- mesh xs     ys   == 'box' ('path' xs) ('path' ys)
  -- mesh [1..3] "ab" == 'edges' [ ((1,\'a\'),(1,\'b\')), ((1,\'a\'),(2,\'a\')), ((1,\'b\'),(2,\'b\')), ((2,\'a\'),(2,\'b\'))
  --                           , ((2,\'a\'),(3,\'a\')), ((2,\'b\'),(3,\'b\')), ((3,\'a\'),(3,\'b\')) ]
  -- @
  mesh :: Graph g => [a] -> [b] -> g (a, b)
  mesh []  _   = empty
  mesh _   []  = empty
  mesh [x] [y] = vertex (x, y)
  mesh xs  ys  = stars $  [ ((a1, b1), [(a1, b2), (a2, b1)]) | (a1, a2) <- ipxs, (b1, b2) <- ipys ]
                       ++ [ ((lx,y1), [(lx,y2)]) | (y1,y2) <- ipys]
                       ++ [ ((x1,ly), [(x2,ly)]) | (x1,x2) <- ipxs]
    where
      lx = last xs
      ly = last ys
      ipxs = init (pairs xs)
      ipys = init (pairs ys)
  
  -- | Construct a /torus graph/ from two lists of vertices.
  -- Complexity: /O(L1 * L2)/ time, memory and size, where /L1/ and /L2/ are the
  -- lengths of the given lists.
  --
  -- @
  -- torus xs    []   == 'empty'
  -- torus []    ys   == 'empty'
  -- torus [x]   [y]  == 'edge' (x,y) (x,y)
  -- torus xs    ys   == 'box' ('circuit' xs) ('circuit' ys)
  -- torus [1,2] "ab" == 'edges' [ ((1,\'a\'),(1,\'b\')), ((1,\'a\'),(2,\'a\')), ((1,\'b\'),(1,\'a\')), ((1,\'b\'),(2,\'b\'))
  --                           , ((2,\'a\'),(1,\'a\')), ((2,\'a\'),(2,\'b\')), ((2,\'b\'),(1,\'b\')), ((2,\'b\'),(2,\'a\')) ]
  -- @
  torus :: Graph g => [a] -> [b] -> g (a, b)
  torus xs ys = stars [ ((a1, b1), [(a1, b2), (a2, b1)]) | (a1, a2) <- pairs xs, (b1, b2) <- pairs ys ]
  
  -- | Auxiliary function for 'mesh' and 'torus'
  pairs :: [a] -> [(a, a)]
  pairs [] = []
  pairs as@(x:xs) = zip as (xs ++ [x])
  
  -- | Construct a /De Bruijn graph/ of a given non-negative dimension using symbols
  -- from a given alphabet.
  -- Complexity: /O(A^(D + 1))/ time, memory and size, where /A/ is the size of the
  -- alphabet and /D/ is the dimension of the graph.
  --
  -- @
  --           deBruijn 0 xs               == 'edge' [] []
  -- n > 0 ==> deBruijn n []               == 'empty'
  --           deBruijn 1 [0,1]            == 'edges' [ ([0],[0]), ([0],[1]), ([1],[0]), ([1],[1]) ]
  --           deBruijn 2 "0"              == 'edge' "00" "00"
  --           deBruijn 2 "01"             == 'edges' [ ("00","00"), ("00","01"), ("01","10"), ("01","11")
  --                                                , ("10","00"), ("10","01"), ("11","10"), ("11","11") ]
  --           transpose   (deBruijn n xs) == 'fmap' 'reverse' $ deBruijn n xs
  --           'vertexCount' (deBruijn n xs) == ('length' $ 'Data.List.nub' xs)^n
  -- n > 0 ==> 'edgeCount'   (deBruijn n xs) == ('length' $ 'Data.List.nub' xs)^(n + 1)
  -- @
  deBruijn :: Graph g => Int -> [a] -> g [a]
  deBruijn 0   _        = edge [] []
  deBruijn len alphabet = skeleton >>= expand
    where
      overlaps = mapM (const alphabet) [2..len]
      skeleton = edges    [        (Left s, Right s)   | s <- overlaps ]
      expand v = vertices [ either ([a] ++) (++ [a]) v | a <- alphabet ]
  
  -- | 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 :: Graph g => (a -> Bool) -> g a -> g a
  induce = mfilter
  
  -- | 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, Graph g) => a -> g a -> g a
  removeVertex v = induce (/= v)
  
  -- | 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, Graph g) => a -> a -> g a -> g a
  replaceVertex u v = fmap $ \w -> if w == u then v else w
  
  -- | 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 :: Graph g => (a -> Bool) -> a -> g a -> g a
  mergeVertices p v = fmap $ \w -> if p w then v else w
  
  -- | Split a vertex into a list of vertices with the same connectivity.
  -- Complexity: /O(s + k * L)/ time, memory and size, where /k/ is the number of
  -- occurrences of the vertex in the expression and /L/ is the length of the
  -- given list.
  --
  -- @
  -- splitVertex x []                  == 'removeVertex' x
  -- splitVertex x [x]                 == id
  -- splitVertex x [y]                 == 'replaceVertex' x y
  -- splitVertex 1 [0,1] $ 1 * (2 + 3) == (0 + 1) * (2 + 3)
  -- @
  splitVertex :: (Eq a, Graph g) => a -> [a] -> g a -> g a
  splitVertex v us g = g >>= \w -> if w == v then vertices us else vertex w
  