    module Data.Graph.Inductive.Query.ArtPoint(
        ap
    ) where

    import Data.Graph.Inductive.Graph

    ------------------------------------------------------------------------------
    -- Tree for storing the DFS numbers and back edges for each node in the graph.
    -- Each node in this tree is of the form (v,n,b) where v is the vertex number,
    -- n is its DFS number and b is the list of nodes (and their DFS numbers) that
    -- lead to back back edges for that vertex v.
    ------------------------------------------------------------------------------
    data DFSTree a = B (a,a,[(a,a)]) [DFSTree a]
         deriving (Eq, Show, Read)

    ------------------------------------------------------------------------------
    -- Tree for storing the DFS and low numbers for each node in the graph.
    -- Each node in this tree is of the form (v,n,l) where v is the vertex number,
    -- n is its DFS number and l is its low number.
    ------------------------------------------------------------------------------
    data LOWTree a = Brc (a,a,a) [LOWTree a]
         deriving (Eq, Show, Read)

    ------------------------------------------------------------------------------
    -- Finds the back edges for a given node.
    ------------------------------------------------------------------------------
    getBackEdges :: Node -> [[(Node,Int)]] -> [(Node,Int)]
    getBackEdges _ [] = []
    getBackEdges v ls   = map head (filter (elem (v,0)) (tail ls))

    ------------------------------------------------------------------------------
    -- Builds a DFS tree for a given graph. Each element (v,n,b) in the tree
    -- contains: the node number v, the DFS number n, and a list of backedges b.
    ------------------------------------------------------------------------------
    dfsTree :: (Graph gr) => Int -> Node -> [Node] -> [[(Node,Int)]] ->
                           gr a b -> ([DFSTree Int],gr a b,Int)
    dfsTree n _ []      _ g             = ([],g,n)
    dfsTree n _ _       _ g | isEmpty g = ([],g,n)
    dfsTree n u (v:vs) ls g = case match v g of
                                (Nothing, g1) -> dfsTree n u vs ls g1
                                (Just c , g1) -> (B (v,n+1,bck) ts:ts', g3, k)
                                 where  bck        = getBackEdges v ls
                                        (ts, g2,m) = dfsTree (n+1) v sc ls' g1
                                        (ts',g3,k) = dfsTree m v vs ls g2
                                        ls'        = ((v,n+1):sc'):ls
                                        sc'        = map (\x->(x,0)) sc
                                        sc         = suc' c

    ------------------------------------------------------------------------------
    -- Finds the minimum between a dfs number and a list of back edges' dfs
    -- numbers.
    ------------------------------------------------------------------------------
    minbckEdge :: Int -> [(Node,Int)] -> Int
    minbckEdge n [] = n
    minbckEdge n bs = min n (minimum (map snd bs))

    ------------------------------------------------------------------------------
    -- Returns the low number for a node in a subtree.
    ------------------------------------------------------------------------------
    getLow :: LOWTree Int -> Int
    getLow (Brc (_,_,l) _) = l

    ------------------------------------------------------------------------------
    -- Builds a low tree from a DFS tree. Each element (v,n,low) in the tree
    -- contains: the node number v, the DFS number n, and the low number low.
    ------------------------------------------------------------------------------
    lowTree :: DFSTree Int -> LOWTree Int
    lowTree (B (v,n,[]  ) [] ) = Brc (v,n,n) []
    lowTree (B (v,n,bcks) [] ) = Brc (v,n,minbckEdge n bcks) []
    lowTree (B (v,n,bcks) trs) = Brc (v,n,lowv) ts
                                 where lowv     = min (minbckEdge n bcks) lowChild
                                       lowChild = minimum (map getLow ts)
                                       ts       = map lowTree trs

    ------------------------------------------------------------------------------
    -- Builds a low tree for a given graph. Each element (v,n,low) in the tree
    -- contains: the node number v, the DFS number n, and the low number low.
    ------------------------------------------------------------------------------
    getLowTree :: (Graph gr) => gr a b -> Node -> LOWTree Int
    getLowTree g v = lowTree (head dfsf)
                      where (dfsf, _, _) = dfsTree 0 0 [v] [] g

    ------------------------------------------------------------------------------
    -- Tests if a node in a subtree is an articulation point. An non-root node v
    -- is an articulation point iff there exists at least one child w of v such
    -- that lowNumber(w) >= dfsNumber(v). The root node is an articulation point
    -- iff it has two or more children.
    ------------------------------------------------------------------------------
    isap :: LOWTree Int -> Bool
    isap (Brc (_,_,_) []) = False
    isap (Brc (_,1,_) ts) = length ts > 1
    isap (Brc (_,n,_) ts) = not (null ch)
                            where ch = filter ( >=n) (map getLow ts)

    ------------------------------------------------------------------------------
    -- Finds the articulation points by traversing the low tree.
    ------------------------------------------------------------------------------
    arp :: LOWTree Int -> [Node]
    arp (Brc (v,1,_) ts) | length ts > 1         = v:concatMap arp ts
                         | otherwise             =   concatMap arp ts
    arp (Brc (v,n,l) ts) | isap (Brc (v,n,l) ts) = v:concatMap arp ts
                         | otherwise             =   concatMap arp ts

    ------------------------------------------------------------------------------
    -- Finds the articulation points of a graph starting at a given node.
    ------------------------------------------------------------------------------
    artpoints :: (Graph gr) => gr a b -> Node -> [Node]
    artpoints g v = arp (getLowTree g v)

    {-|
       Finds the articulation points for a connected undirected graph,
       by using the low numbers criteria:

       a) The root node is an articulation point iff it has two or more children.

       b) An non-root node v is an articulation point iff there exists at least
          one child w of v such that lowNumber(w) >= dfsNumber(v).
    -}
    ap :: (Graph gr) => gr a b -> [Node]
    ap g = artpoints g v where ((_,v,_,_),_) = matchAny g