```module Data.Trees.KdTree where

-- by Issac Trotts

import Data.Maybe

import qualified Data.Foldable as F
import qualified Data.List as L
import Test.QuickCheck

class Point p where
-- |dimension returns the number of coordinates of a point.
dimension :: p -> Int

-- |coord gets the k'th coordinate, starting from 0.
coord :: Int -> p -> Double

-- |dist2 returns the squared distance between two points.
dist2 :: p -> p -> Double
dist2 a b = sum . map diff2 \$ [0..dimension a - 1]
where diff2 i = (coord i a - coord i b)^2

-- |compareDistance p a b  compares the distances of a and b to p.
compareDistance :: (Point p) => p -> p -> p -> Ordering
compareDistance p a b = dist2 p a `compare` dist2 p b

data Point3d = Point3d { p3x :: Double, p3y :: Double, p3z :: Double }
deriving (Eq, Ord, Show)

instance Point Point3d where
dimension _ = 3

coord 0 p = p3x p
coord 1 p = p3y p
coord 2 p = p3z p

data KdTree point = KdNode { kdLeft :: KdTree point,
kdPoint :: point,
kdRight :: KdTree point,
kdAxis :: Int }
| KdEmpty
deriving (Eq, Ord, Show)

instance Functor KdTree where
fmap _ KdEmpty = KdEmpty
fmap f (KdNode l x r axis) = KdNode (fmap f l) (f x) (fmap f r) axis

instance F.Foldable KdTree where
foldr f init KdEmpty = init
foldr f init (KdNode l x r _) = F.foldr f init3 l
where 	init3 = f x init2
init2 = F.foldr f init r

fromList :: Point p => [p] -> KdTree p
fromList points = fromListWithDepth points 0

-- Select axis based on depth so that axis cycles through all valid values
fromListWithDepth :: Point p => [p] -> Int -> KdTree p
fromListWithDepth [] _ = KdEmpty
fromListWithDepth points depth = node
where   axis = axisFromDepth (head points) depth

-- Sort point list and choose median as pivot element
sortedPoints =
L.sortBy (\a b -> coord axis a `compare` coord axis b) points
medianIndex = length sortedPoints `div` 2

-- Create node and construct subtrees
node = KdNode { kdLeft = fromListWithDepth (take medianIndex sortedPoints) (depth+1),
kdPoint = sortedPoints !! medianIndex,
kdRight = fromListWithDepth (drop (medianIndex+1) sortedPoints) (depth+1),
kdAxis = axis }

axisFromDepth :: Point p => p -> Int -> Int
axisFromDepth p depth = depth `mod` k
where k = dimension p

toList :: KdTree p -> [p]
toList t = F.foldr (:) [] t

subtrees :: KdTree p -> [KdTree p]
subtrees KdEmpty = [KdEmpty]
subtrees t@(KdNode l x r axis) = subtrees l ++ [t] ++ subtrees r

nearestNeighbor :: Point p => KdTree p -> p -> Maybe p
nearestNeighbor KdEmpty probe = Nothing
nearestNeighbor (KdNode KdEmpty p KdEmpty _) probe = Just p
nearestNeighbor (KdNode l p r axis) probe =
if xProbe <= xp then doStuff l r else doStuff r l
where xProbe = coord axis probe
xp = coord axis p
doStuff tree1 tree2 =
let candidates1 = case nearestNeighbor tree1 probe of
Nothing -> [p]
Just best1 -> [best1, p]
sphereIntersectsPlane = (xProbe - xp)^2 <= dist2 probe p
candidates2 = if sphereIntersectsPlane
then candidates1 ++ maybeToList (nearestNeighbor tree2 probe)
else candidates1 in
Just . L.minimumBy (compareDistance probe) \$ candidates2

-- |invariant tells whether the KD tree property holds for a given tree and
-- all its subtrees.
-- Specifically, it tests that all points in the left subtree lie to the left
-- of the plane, p is on the plane, and all points in the right subtree lie to
-- the right.
invariant :: Point p => KdTree p -> Bool
invariant KdEmpty = True
invariant (KdNode l p r axis) = leftIsGood && rightIsGood
where x = coord axis p
leftIsGood = all ((<= x) . coord axis) (toList l)
rightIsGood = all ((>= x) . coord axis) (toList r)

invariant' :: Point p => KdTree p -> Bool
invariant' = all invariant . subtrees

instance Arbitrary Point3d where
arbitrary = do
x <- arbitrary
y <- arbitrary
z <- arbitrary
return (Point3d x y z)

```