content: {-# LANGUAGE LambdaCase #-}
  -----------------------------------------------------------------------------
  -- |
  -- Module     : Algebra.Graph.AdjacencyIntMap.Algorithm
  -- Copyright  : (c) Andrey Mokhov 2016-2025
  -- License    : MIT (see the file LICENSE)
  -- Maintainer : andrey.mokhov@gmail.com
  -- Stability  : unstable
  --
  -- __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 provides basic graph algorithms, such as /depth-first search/,
  -- implemented for the "Algebra.Graph.AdjacencyIntMap" data type.
  --
  -- Some of the worst-case complexities include the term /min(n,W)/.
  -- Following 'IntSet.IntSet' and 'IntMap.IntMap', the /W/ stands for
  -- word size (usually 32 or 64 bits).
  -----------------------------------------------------------------------------
  module Algebra.Graph.AdjacencyIntMap.Algorithm (
      -- * Algorithms
      bfsForest, bfs, dfsForest, dfsForestFrom, dfs, reachable,
      topSort, isAcyclic,
  
      -- * Correctness properties
      isDfsForestOf, isTopSortOf,
  
      -- * Type synonyms
      Cycle
      ) where
  
  import Control.Monad
  import Control.Monad.Trans.Cont
  import Control.Monad.Trans.State.Strict
  import Data.Either
  import Data.List.NonEmpty (NonEmpty(..), (<|))
  import Data.Tree
  
  import Algebra.Graph.AdjacencyIntMap
  
  import qualified Data.List          as List
  import qualified Data.IntMap.Strict as IntMap
  import qualified Data.IntSet        as IntSet
  
  -- | Compute the /breadth-first search/ forest of a graph, such that adjacent
  -- vertices are explored in the increasing order. The search is seeded by a list
  -- of vertices that will become the roots of the resulting forest. Duplicates in
  -- the list will have their first occurrence explored and subsequent ones
  -- ignored. The seed vertices that do not belong to the graph are also ignored.
  --
  -- Complexity: /O((L + m) * log n)/ time and /O(n)/ space, where /L/ is the
  -- number of seed vertices.
  --
  -- @
  -- 'forest' $ bfsForest ('edge' 1 2) [0]        == 'empty'
  -- 'forest' $ bfsForest ('edge' 1 2) [1]        == 'edge' 1 2
  -- 'forest' $ bfsForest ('edge' 1 2) [2]        == 'vertex' 2
  -- 'forest' $ bfsForest ('edge' 1 2) [0,1,2]    == 'vertices' [1,2]
  -- 'forest' $ bfsForest ('edge' 1 2) [2,1,0]    == 'vertices' [1,2]
  -- 'forest' $ bfsForest ('edge' 1 1) [1]        == 'vertex' 1
  -- 'isSubgraphOf' ('forest' $ bfsForest x vs) x == True
  -- bfsForest x ('vertexList' x)               == 'map' (\\v -> Node v []) ('Data.List.nub' $ 'vertexList' x)
  -- bfsForest x []                           == []
  -- bfsForest 'empty' vs                       == []
  -- bfsForest (3 * (1 + 4) * (1 + 5)) [1,4]  == [ Node { rootLabel = 1
  --                                                    , subForest = [ Node { rootLabel = 5
  --                                                                         , subForest = [] }]}
  --                                             , Node { rootLabel = 4
  --                                                    , subForest = [] }]
  -- 'forest' $ bfsForest ('circuit' [1..5] + 'circuit' [5,4..1]) [3] == 'path' [3,2,1] + 'path' [3,4,5]
  --
  -- @
  bfsForest :: AdjacencyIntMap -> [Int] -> Forest Int
  bfsForest g vs= evalState (explore [ v | v <- vs, hasVertex v g ]) IntSet.empty
    where
      explore = filterM discovered >=> unfoldForestM_BF walk
      walk v = (v,) <$> adjacentM v
      adjacentM v = filterM discovered $ IntSet.toList (postIntSet v g)
      discovered v = do new <- gets (not . IntSet.member v)
                        when new $ modify' (IntSet.insert v)
                        return new
  
  -- | A version of 'bfsForest' where the resulting forest is converted to a level
  -- structure. Adjacent vertices are explored in the increasing order. Flattening
  -- the result via @'concat'@ @.@ @'bfs'@ @x@ gives an enumeration of reachable
  -- vertices in the breadth-first search order.
  --
  -- Complexity: /O((L + m) * min(n,W))/ time and /O(n)/ space, where /L/ is the
  -- number of seed vertices.
  --
  -- @
  -- bfs ('edge' 1 2) [0]                == []
  -- bfs ('edge' 1 2) [1]                == [[1], [2]]
  -- bfs ('edge' 1 2) [2]                == [[2]]
  -- bfs ('edge' 1 2) [1,2]              == [[1,2]]
  -- bfs ('edge' 1 2) [2,1]              == [[2,1]]
  -- bfs ('edge' 1 1) [1]                == [[1]]
  -- bfs 'empty' vs                      == []
  -- bfs x []                          == []
  -- bfs (1 * 2 + 3 * 4 + 5 * 6) [1,2] == [[1,2]]
  -- bfs (1 * 2 + 3 * 4 + 5 * 6) [1,3] == [[1,3], [2,4]]
  -- bfs (3 * (1 + 4) * (1 + 5)) [3]   == [[3], [1,4,5]]
  --
  -- bfs ('circuit' [1..5] + 'circuit' [5,4..1]) [3]          == [[2], [1,3], [5,4]]
  -- 'concat' $ bfs ('circuit' [1..5] + 'circuit' [5,4..1]) [3] == [3,2,4,1,5]
  -- 'map' 'concat' . 'List.transpose' . 'map' 'levels' . 'bfsForest' x    == bfs x
  -- @
  bfs :: AdjacencyIntMap -> [Int] -> [[Int]]
  bfs g = map concat . List.transpose . map levels . bfsForest g
  
  dfsForestFromImpl :: AdjacencyIntMap -> [Int] -> Forest Int
  dfsForestFromImpl g vs = evalState (explore vs) IntSet.empty
    where
      explore (v:vs) = discovered v >>= \case
        True -> (:) <$> walk v <*> explore vs
        False -> explore vs
      explore [] = return []
      walk v = Node v <$> explore (adjacent v)
      adjacent v = IntSet.toList (postIntSet v g)
      discovered v = do new <- gets (not . IntSet.member v)
                        when new $ modify' (IntSet.insert v)
                        return new
  
  -- | Compute the /depth-first search/ forest of a graph, where adjacent vertices
  -- are explored in the increasing order.
  --
  -- Complexity: /O((n + m) * min(n,W))/ time and /O(n)/ space.
  --
  -- @
  -- 'forest' $ dfsForest 'empty'              == 'empty'
  -- 'forest' $ dfsForest ('edge' 1 1)         == 'vertex' 1
  -- 'forest' $ dfsForest ('edge' 1 2)         == 'edge' 1 2
  -- 'forest' $ dfsForest ('edge' 2 1)         == 'vertices' [1,2]
  -- 'isSubgraphOf' ('forest' $ dfsForest x) x == True
  -- 'isDfsForestOf' (dfsForest x) x         == True
  -- dfsForest . 'forest' . dfsForest        == dfsForest
  -- dfsForest ('vertices' vs)               == 'map' (\\v -> Node v []) ('Data.List.nub' $ 'Data.List.sort' vs)
  -- dfsForest $ 3 * (1 + 4) * (1 + 5)     == [ Node { rootLabel = 1
  --                                                 , subForest = [ Node { rootLabel = 5
  --                                                                      , subForest = [] }]}
  --                                          , Node { rootLabel = 3
  --                                                 , subForest = [ Node { rootLabel = 4
  --                                                                      , subForest = [] }]}]
  -- 'forest' (dfsForest $ 'circuit' [1..5] + 'circuit' [5,4..1]) == 'path' [1,2,3,4,5]
  -- @
  dfsForest :: AdjacencyIntMap -> Forest Int
  dfsForest g = dfsForestFromImpl g (vertexList g)
  
  -- | Compute the /depth-first search/ forest of a graph starting from the given
  -- seed vertices, where adjacent vertices are explored in the increasing order.
  -- Note that the resulting forest does not necessarily span the whole graph, as
  -- some vertices may be unreachable. The seed vertices which do not belong to
  -- the graph are ignored.
  --
  -- Complexity: /O((L + m) * log n)/ time and /O(n)/ space, where /L/ is the
  -- number of seed vertices.
  --
  -- @
  -- 'forest' $ dfsForestFrom 'empty'      vs             == 'empty'
  -- 'forest' $ dfsForestFrom ('edge' 1 1) [1]            == 'vertex' 1
  -- 'forest' $ dfsForestFrom ('edge' 1 2) [0]            == 'empty'
  -- 'forest' $ dfsForestFrom ('edge' 1 2) [1]            == 'edge' 1 2
  -- 'forest' $ dfsForestFrom ('edge' 1 2) [2]            == 'vertex' 2
  -- 'forest' $ dfsForestFrom ('edge' 1 2) [1,2]          == 'edge' 1 2
  -- 'forest' $ dfsForestFrom ('edge' 1 2) [2,1]          == 'vertices' [1,2]
  -- 'isSubgraphOf' ('forest' $ dfsForestFrom x vs) x     == True
  -- 'isDfsForestOf' (dfsForestFrom x ('vertexList' x)) x == True
  -- dfsForestFrom x ('vertexList' x)                   == 'dfsForest' x
  -- dfsForestFrom x []                               == []
  -- dfsForestFrom (3 * (1 + 4) * (1 + 5)) [1,4]      == [ Node { rootLabel = 1
  --                                                            , subForest = [ Node { rootLabel = 5
  --                                                                                 , subForest = [] }
  --                                                     , Node { rootLabel = 4
  --                                                            , subForest = [] }]
  -- 'forest' $ dfsForestFrom ('circuit' [1..5] + 'circuit' [5,4..1]) [3] == 'path' [3,2,1,5,4]
  -- @
  dfsForestFrom :: AdjacencyIntMap -> [Int] -> Forest Int
  dfsForestFrom g vs = dfsForestFromImpl g [ v | v <- vs, hasVertex v g ]
  
  
  -- | Return the list vertices visited by the /depth-first search/ in a graph,
  -- starting from the given seed vertices. Adjacent vertices are explored in the
  -- increasing order.
  --
  -- Complexity: /O((L + m) * log n)/ time and /O(n)/ space, where /L/ is the
  -- number of seed vertices.
  --
  -- @
  -- dfs 'empty'      vs    == []
  -- dfs ('edge' 1 1) [1]   == [1]
  -- dfs ('edge' 1 2) [0]   == []
  -- dfs ('edge' 1 2) [1]   == [1,2]
  -- dfs ('edge' 1 2) [2]   == [2]
  -- dfs ('edge' 1 2) [1,2] == [1,2]
  -- dfs ('edge' 1 2) [2,1] == [2,1]
  -- dfs x          []    == []
  --
  -- 'Data.List.and' [ 'hasVertex' v x | v <- dfs x vs ]       == True
  -- dfs (3 * (1 + 4) * (1 + 5)) [1,4]           == [1,5,4]
  -- dfs ('circuit' [1..5] + 'circuit' [5,4..1]) [3] == [3,2,1,5,4]
  -- @
  dfs :: AdjacencyIntMap -> [Int] -> [Int]
  dfs x = concatMap flatten . dfsForestFrom x
  
  -- | Return the list of vertices /reachable/ from a source vertex in a graph.
  -- The vertices in the resulting list appear in the /depth-first search order/.
  --
  -- Complexity: /O(m * log n)/ time and /O(n)/ space.
  --
  -- @
  -- reachable 'empty'              x == []
  -- reachable ('vertex' 1)         1 == [1]
  -- reachable ('edge' 1 1)         1 == [1]
  -- reachable ('edge' 1 2)         0 == []
  -- reachable ('edge' 1 2)         1 == [1,2]
  -- reachable ('edge' 1 2)         2 == [2]
  -- reachable ('path'    [1..8]  ) 4 == [4..8]
  -- reachable ('circuit' [1..8]  ) 4 == [4..8] ++ [1..3]
  -- reachable ('clique'  [8,7..1]) 8 == [8] ++ [1..7]
  --
  -- 'Data.List.and' [ 'hasVertex' v x | v <- reachable x y ] == True
  -- @
  reachable :: AdjacencyIntMap -> Int -> [Int]
  reachable x y = dfs x [y]
  
  type Cycle = NonEmpty
  type Result = Either (Cycle Int) [Int]
  data NodeState = Entered | Exited
  data S = S { parent :: IntMap.IntMap Int
             , entry  :: IntMap.IntMap NodeState
             , order  :: [Int] }
  
  topSortImpl :: AdjacencyIntMap -> StateT S (Cont Result) Result
  topSortImpl g = liftCallCC' callCC $ \cyclic ->
    do let vertices = map fst $ IntMap.toDescList $ adjacencyIntMap g
           adjacent = IntSet.toDescList . flip postIntSet g
           dfsRoot x = nodeState x >>= \case
             Nothing -> enterRoot x >> dfs x >> exit x
             _       -> return ()
           dfs x = forM_ (adjacent x) $ \y ->
                     nodeState y >>= \case
                       Nothing      -> enter x y >> dfs y >> exit y
                       Just Exited  -> return ()
                       Just Entered -> cyclic . Left . retrace x y =<< gets parent
       forM_ vertices dfsRoot
       Right <$> gets order
    where
      nodeState v = gets (IntMap.lookup v . entry)
      enter u v = modify' (\(S m n vs) -> S (IntMap.insert v u m)
                                            (IntMap.insert v Entered n)
                                            vs)
      enterRoot v = modify' (\(S m n vs) -> S m (IntMap.insert v Entered n) vs)
      exit v = modify' (\(S m n vs) -> S m (IntMap.alter (fmap leave) v n) (v:vs))
        where leave = \case
                Entered -> Exited
                Exited  -> error "Internal error: dfs search order violated"
      retrace curr head parent = aux (curr :| []) where
        aux xs@(curr :| _)
          | head == curr = xs
          | otherwise = aux (parent IntMap.! curr <| xs)
  
  -- | Compute a topological sort of a graph or discover a cycle.
  --
  -- Vertices are explored in the decreasing order according to their 'Ord'
  -- instance. This gives the lexicographically smallest topological ordering in
  -- the case of success. In the case of failure, the cycle is characterized by
  -- being the lexicographically smallest up to rotation with respect to
  -- @Ord@ @(Dual@ @Int)@ in the first connected component of the graph containing
  -- a cycle, where the connected components are ordered by their largest vertex
  -- with respect to @Ord a@.
  --
  -- Complexity: /O((n + m) * min(n,W))/ time and /O(n)/ space.
  --
  -- @
  -- topSort (1 * 2 + 3 * 1)                    == Right [3,1,2]
  -- topSort ('path' [1..5])                      == Right [1..5]
  -- topSort (3 * (1 * 4 + 2 * 5))              == Right [3,1,2,4,5]
  -- topSort (1 * 2 + 2 * 1)                    == Left (2 ':|' [1])
  -- topSort ('path' [5,4..1] + 'edge' 2 4)         == Left (4 ':|' [3,2])
  -- topSort ('circuit' [1..3])                   == Left (3 ':|' [1,2])
  -- topSort ('circuit' [1..3] + 'circuit' [3,2,1]) == Left (3 ':|' [2])
  -- topSort (1 * 2 + (5 + 2) * 1 + 3 * 4 * 3)  == Left (1 ':|' [2])
  -- fmap ('flip' 'isTopSortOf' x) (topSort x)      /= Right False
  -- topSort . 'vertices'                         == Right . 'nub' . 'sort'
  -- @
  topSort :: AdjacencyIntMap -> Either (Cycle Int) [Int]
  topSort g = runCont (evalStateT (topSortImpl g) initialState) id
    where
      initialState = S IntMap.empty IntMap.empty []
  
  -- | Check if a given graph is /acyclic/.
  --
  -- Complexity: /O((n + m) * min(n,W))/ time and /O(n)/ space.
  --
  -- @
  -- isAcyclic (1 * 2 + 3 * 1) == True
  -- isAcyclic (1 * 2 + 2 * 1) == False
  -- isAcyclic . 'circuit'       == 'null'
  -- isAcyclic                 == 'isRight' . 'topSort'
  -- @
  isAcyclic :: AdjacencyIntMap -> Bool
  isAcyclic = isRight . topSort
  
  -- | Check if a given forest is a correct /depth-first search/ forest of a graph.
  -- The implementation is based on the paper "Depth-First Search and Strong
  -- Connectivity in Coq" by François Pottier.
  --
  -- @
  -- isDfsForestOf []                              'empty'            == True
  -- isDfsForestOf []                              ('vertex' 1)       == False
  -- isDfsForestOf [Node 1 []]                     ('vertex' 1)       == True
  -- isDfsForestOf [Node 1 []]                     ('vertex' 2)       == False
  -- isDfsForestOf [Node 1 [], Node 1 []]          ('vertex' 1)       == False
  -- isDfsForestOf [Node 1 []]                     ('edge' 1 1)       == True
  -- isDfsForestOf [Node 1 []]                     ('edge' 1 2)       == False
  -- isDfsForestOf [Node 1 [], Node 2 []]          ('edge' 1 2)       == False
  -- isDfsForestOf [Node 2 [], Node 1 []]          ('edge' 1 2)       == True
  -- isDfsForestOf [Node 1 [Node 2 []]]            ('edge' 1 2)       == True
  -- isDfsForestOf [Node 1 [], Node 2 []]          ('vertices' [1,2]) == True
  -- isDfsForestOf [Node 2 [], Node 1 []]          ('vertices' [1,2]) == True
  -- isDfsForestOf [Node 1 [Node 2 []]]            ('vertices' [1,2]) == False
  -- isDfsForestOf [Node 1 [Node 2 [Node 3 []]]]   ('path' [1,2,3])   == True
  -- isDfsForestOf [Node 1 [Node 3 [Node 2 []]]]   ('path' [1,2,3])   == False
  -- isDfsForestOf [Node 3 [], Node 1 [Node 2 []]] ('path' [1,2,3])   == True
  -- isDfsForestOf [Node 2 [Node 3 []], Node 1 []] ('path' [1,2,3])   == True
  -- isDfsForestOf [Node 1 [], Node 2 [Node 3 []]] ('path' [1,2,3])   == False
  -- @
  isDfsForestOf :: Forest Int -> AdjacencyIntMap -> Bool
  isDfsForestOf f am = case go IntSet.empty f of
      Just seen -> seen == vertexIntSet am
      Nothing   -> False
    where
      go seen []     = Just seen
      go seen (t:ts) = do
          let root = rootLabel t
          guard $ root `IntSet.notMember` seen
          guard $ and [ hasEdge root (rootLabel subTree) am | subTree <- subForest t ]
          newSeen <- go (IntSet.insert root seen) (subForest t)
          guard $ postIntSet root am `IntSet.isSubsetOf` newSeen
          go newSeen ts
  
  -- | Check if a given list of vertices is a correct /topological sort/ of a graph.
  --
  -- @
  -- isTopSortOf [3,1,2] (1 * 2 + 3 * 1) == True
  -- isTopSortOf [1,2,3] (1 * 2 + 3 * 1) == False
  -- isTopSortOf []      (1 * 2 + 3 * 1) == False
  -- isTopSortOf []      'empty'           == True
  -- isTopSortOf [x]     ('vertex' x)      == True
  -- isTopSortOf [x]     ('edge' x x)      == False
  -- @
  isTopSortOf :: [Int] -> AdjacencyIntMap -> Bool
  isTopSortOf xs m = go IntSet.empty xs
    where
      go seen []     = seen == IntMap.keysSet (adjacencyIntMap m)
      go seen (v:vs) = postIntSet v m `IntSet.intersection` newSeen == IntSet.empty
                    && go newSeen vs
        where
          newSeen = IntSet.insert v seen
  