```{-# LANGUAGE BangPatterns #-}
-- | Functions for finding /lowest common ancestors/ in trees in /O(1)/ time, with /O(n)/ preprocessing.
module Data.RangeMin.LCA (Index, lowestCommonAncestor, quickLCA) where

import Data.RangeMin.LCA.IndexM
import Data.RangeMin
import Data.RangeMin.Common.Vector
import qualified Data.RangeMin.Fusion as F
import qualified Data.Vector.Unboxed as UV
import qualified Data.Vector.Primitive as PV
import qualified Data.Vector as V

import Data.Tree

-- | Labels a tree in depth-first order.
dfOrder :: Tree a -> (Tree (Index, a), Int)
dfOrder tree = execIndexM \$ indexer tree where
indexer (Node a ts) = do
i <- getIndex
ts' <- mapM indexer ts
return (Node (i, a) ts')

type Depth = Int
data Trav a = Trav {-# UNPACK #-} !Depth {-# UNPACK #-} !Index a

travel :: (a -> Index) -> Tree a -> [Trav a] -> [Trav a]
travel f = trav' 0 where
trav' !d (Node x ts) zs = let me = Trav d (f x) x in
me:foldr (\ t -> trav' (d+1) t . (me:)) zs ts

-- | Takes a tree and indexes it in depth-first order, returning the number of nodes, the indexed
-- tree, and the lowest common ancestor function.
quickLCA :: Tree a -> (Int, Tree (Index, a), Index -> Index -> (Index, a))
quickLCA tree = case dfOrder tree of
(iTree, n) -> (n, iTree, lowestCommonAncestor n fst iTree)

-- | @'lowestCommonAncestor' n ix tree@ takes a tree whose nodes are mapped by
-- @ix@ to a unique index in the range @0..n-1@, and returns a function
-- which takes two indices (corresponding to two nodes in the tree) and returns
-- the label of their /lowest common ancestor/.
--
-- This takes /O(n)/ preprocessing and answers queries in /O(1)/, as it is an
-- application of "Data.RangeMin".
--
-- For binary trees, consider using "Data.RangeMin.LCA.Binary".
lowestCommonAncestor :: Int -> (a -> Index) -> Tree a -> Index -> Index -> a
lowestCommonAncestor !n ix tree = vals `seq` lca
where	vmap :: (a -> b) -> V.Vector a -> V.Vector b
vmap = F.map
!(!depths, !ixs, !vals) =
(F.unzip3 :: V.Vector (Int, Int, a) -> (PV.Vector Int, UV.Vector Int, V.Vector a))
(vmap (\ (Trav d j a) -> (d, j, a)) \$ F.fromListN n (travel ix tree []))
rM :: Int -> Int -> Int
!rM = intRangeMin depths
{-# NOINLINE lca #-}
lca !i !j = let
iIx = ixs ! i
jIx = ixs ! j
in vals ! case compare iIx jIx of
EQ	-> iIx
LT	-> rM iIx (jIx - iIx + 1)
GT	-> rM jIx (iIx - jIx + 1)
```