content: -- Find Dominators of a graph.
  --
  -- Author: Bertram Felgenhauer <int-e@gmx.de>
  --
  -- Implementation based on
  -- Keith D. Cooper, Timothy J. Harvey, Ken Kennedy,
  -- "A Simple, Fast Dominance Algorithm",
  -- (http://citeseer.ist.psu.edu/cooper01simple.html)
  
  module Data.Graph.Inductive.Query.Dominators (
      dom,
      iDom
  ) where
  
  import           Data.Array
  import           Data.Graph.Inductive.Graph
  import           Data.Graph.Inductive.Query.DFS
  import           Data.IntMap                    (IntMap)
  import qualified Data.IntMap                    as I
  import           Data.Tree                      (Tree (..))
  import qualified Data.Tree                      as T
  
  {-# ANN iDom "HLint: ignore Use ***" #-}
  -- | return immediate dominators for each node of a graph, given a root
  iDom :: (Graph gr) => gr a b -> Node -> [(Node,Node)]
  iDom g root = let (result, toNode, _) = idomWork g root
                in  map (\(a, b) -> (toNode ! a, toNode ! b)) (assocs result)
  
  -- | return the set of dominators of the nodes of a graph, given a root
  dom :: (Graph gr) => gr a b -> Node -> [(Node,[Node])]
  dom g root = let
      (iD, toNode, fromNode) = idomWork g root
      dom' = getDom toNode iD
      nodes' = nodes g
      rest = I.keys (I.filter (-1 ==) fromNode)
    in
      [(toNode ! i, dom' ! i) | i <- range (bounds dom')] ++
      [(n, nodes') | n <- rest]
  
  -- internal node type
  type Node' = Int
  -- array containing the immediate dominator of each node, or an approximation
  -- thereof. the dominance set of a node can be found by taking the union of
  -- {node} and the dominance set of its immediate dominator.
  type IDom = Array Node' Node'
  -- array containing the list of predecessors of each node
  type Preds = Array Node' [Node']
  -- arrays for translating internal nodes back to graph nodes and back
  type ToNode = Array Node' Node
  type FromNode = IntMap Node'
  
  idomWork :: (Graph gr) => gr a b -> Node -> (IDom, ToNode, FromNode)
  idomWork g root = let
      -- use depth first tree from root do build the first approximation
      trees@(~[tree]) = dff [root] g
      -- relabel the tree so that paths from the root have increasing nodes
      (s, ntree) = numberTree 0 tree
      -- the approximation iDom0 just maps each node to its parent
      iD0 = array (1, s-1) (tail $ treeEdges (-1) ntree)
      -- fromNode translates graph nodes to relabeled (internal) nodes
      fromNode = I.unionWith const (I.fromList (zip (T.flatten tree) (T.flatten ntree))) (I.fromList (zip (nodes g) (repeat (-1))))
      -- toNode translates internal nodes to graph nodes
      toNode = array (0, s-1) (zip (T.flatten ntree) (T.flatten tree))
      preds = array (1, s-1) [(i, filter (/= -1) (map (fromNode I.!)
                              (pre g (toNode ! i)))) | i <- [1..s-1]]
      -- iteratively improve the approximation to find iDom.
      iD = fixEq (refineIDom preds) iD0
    in
      if null trees then error "Dominators.idomWork: root not in graph"
                    else (iD, toNode, fromNode)
  
  -- for each node in iDom, find the intersection of all its predecessor's
  -- dominating sets, and update iDom accordingly.
  refineIDom :: Preds -> IDom -> IDom
  refineIDom preds iD = fmap (foldl1 (intersect iD)) preds
  
  -- find the intersection of the two given dominance sets.
  intersect :: IDom -> Node' -> Node' -> Node'
  intersect iD a b = case a `compare` b of
      LT -> intersect iD a (iD ! b)
      EQ -> a
      GT -> intersect iD (iD ! a) b
  
  -- convert an IDom to dominance sets. we translate to graph nodes here
  -- because mapping later would be more expensive and lose sharing.
  getDom :: ToNode -> IDom -> Array Node' [Node]
  getDom toNode iD = let
      res = array (0, snd (bounds iD)) ((0, [toNode ! 0]) :
            [(i, toNode ! i : res ! (iD ! i)) | i <- range (bounds iD)])
    in
      res
  
  -- relabel tree, labeling vertices with consecutive numbers in depth first order
  numberTree :: Node' -> Tree a -> (Node', Tree Node')
  numberTree n (Node _ ts) = let (n', ts') = numberForest (n+1) ts
                             in  (n', Node n ts')
  
  -- same as numberTree, for forests.
  numberForest :: Node' -> [Tree a] -> (Node', [Tree Node'])
  numberForest n []     = (n, [])
  numberForest n (t:ts) = let (n', t')   = numberTree n t
                              (n'', ts') = numberForest n' ts
                          in  (n'', t':ts')
  
  -- return the edges of the tree, with an added dummy root node.
  treeEdges :: a -> Tree a -> [(a,a)]
  treeEdges a (Node b ts) = (b,a) : concatMap (treeEdges b) ts
  
  -- find a fixed point of f, iteratively
  fixEq :: (Eq a) => (a -> a) -> a -> a
  fixEq f v | v' == v   = v
            | otherwise = fixEq f v'
      where v' = f v
  
  {-
  :m +Data.Graph.Inductive
  let g0 = mkGraph [(i,()) | i <- [0..4]] [(a,b,()) | (a,b) <- [(0,1),(1,2),(0,3),(3,2),(4,0)]] :: Gr () ()
  let g1 = mkGraph [(i,()) | i <- [0..4]] [(a,b,()) | (a,b) <- [(0,1),(1,2),(2,3),(1,3),(3,4)]] :: Gr () ()
  let g2,g3,g4 :: Int -> Gr () (); g2 n = mkGraph [(i,()) | i <- [0..n-1]] ([(a,a+1,()) | a <- [0..n-2]] ++ [(a,a+2,()) | a <- [0..n-3]]); g3 n =mkGraph [(i,()) | i <- [0..n-1]] ([(a,a+2,()) | a <- [0..n-3]] ++ [(a,a+1,()) | a <- [0..n-2]]); g4 n =mkGraph [(i,()) | i <- [0..n-1]] ([(a+2,a,()) | a <- [0..n-3]] ++ [(a+1,a,()) | a <- [0..n-2]])
  :m -Data.Graph.Inductive
  -}
  