{- |
Module : Data.Set.BKTree
Copyright : (c) Josef Svenningsson 2007
(c) Henning Günter 2007
License : BSD-style
Maintainer : josef.svenningsson@gmail.com
Stability : Alpha quality. Interface may change without notice.
Portability : portable
Burkhard-Keller trees provide an implementation of sets which apart
from the ordinary operations also has an approximate member search,
allowing you to search for elements that are of a distance @n@ from
the element you are searching for. The distance is determined using
a metric on the type of elements. Therefore all elements must
implement the 'Metric' type class, rather than the more usual
'Ord'.
Useful metrics include the manhattan distance between two points,
the Levenshtein edit distance between two strings, the number of
edges in the shortest path between two nodes in an undirected graph
and the Hamming distance between two binary strings. Any euclidean
space also has a metric. However, in this module we use int-valued
metrics and that's not compatible with the metrics of euclidean
spaces which are real-values.
The worst case complexity of many of these operations is quite bad,
but the expected behavior varies greatly with the metric. For
example, the discrete metric (@distance x y | y == x = 0 |
otherwise = 1@) makes BK-trees behave abysmally. The metrics
mentioned above should give good performance characteristics.
-}
module Data.Set.BKTree
(-- The main type
BKTree
-- Metric
,Metric(..)
--
,null,size,empty
,fromList,singleton
,insert
,member,memberDistance
,delete
,union,unions
,elems,elemsDistance
,closest
#ifdef DEBUG
,runTests
#endif
)where
import qualified Data.IntMap as M
import qualified Data.List as L hiding (null)
import Prelude hiding (null)
import Data.Array.IArray (Array,array,listArray,(!),assocs)
import Data.Array.Unboxed (UArray)
#ifdef DEBUG
import qualified Prelude
import Test.QuickCheck
import Text.Printf
import System.Exit
#endif
-- | A type is 'Metric' if is has a function 'distance' which has the following
-- properties:
--
-- * @'distance' x y >= 0@
--
-- * @'distance' x y == 0@ if and only if @x == y@
--
-- * @'distance' x y == 'distance' y x@
--
-- * @'distance' x z <= 'distance' x y + 'distance' y z@
--
-- All types of elements to 'BKTree' must implement 'Metric'.
--
-- This definition of a metric deviates from the mathematical one in that it
-- returns an integer instead of a real number. The reason for choosing
-- integers is that I wanted to avoid the rather unpredictable rounding
-- of floating point numbers.
class Eq a => Metric a where
distance :: a -> a -> Int
instance Metric Int where
distance i j = abs (i - j)
-- Fishy instance. Maybe I shouldn't have it.
-- Or generalize Metric to use integer?
instance Metric Integer where
distance i j = fromInteger (abs (i - j))
instance Metric Char where
distance i j = abs (fromEnum i - fromEnum j)
hirschberg :: Eq a => [a] -> [a] -> Int
hirschberg xs [] = length xs
hirschberg xs ys = let
lxs = length xs
lys = length ys
start_arr :: UArray Int Int
start_arr = listArray (1,lys) [1..lys]
in (L.foldl' (\arr (i,xi) -> let
narr :: UArray Int Int
narr = array (1,lys) (snd $ L.mapAccumL
(\(s,c) ((j,el),yj) -> let
nc = minimum
[s + (if xi==yj then 0 else 1)
,el + 1
,c + 1
]
in ((el,nc),(j,nc)))
(i-1,i)
(zip (assocs arr) ys)
)
in narr
) start_arr (zip [1..] xs))!lys
instance Eq a => Metric [a] where
distance = hirschberg
-- --------
-- BKTrees
-- --------
-- | The type of Burkhard-Keller trees.
data BKTree a = Node a !Int (M.IntMap (BKTree a))
| Empty
#ifdef DEBUG
deriving Show
#endif
-- | Test if the tree is empty.
null :: BKTree a -> Bool
null (Empty) = True
null (Node _ _ _) = False
-- | Size of the tree.
size :: BKTree a -> Int
size (Empty) = 0
size (Node _ s _) = s
-- | The empty tree.
empty :: BKTree a
empty = Empty
-- | The tree with a single element
singleton :: a -> BKTree a
singleton a = Node a 1 M.empty
-- | Inserts an element into the tree. If an element is inserted several times
-- it will be stored several times.
insert :: Metric a => a -> BKTree a -> BKTree a
insert a Empty = Node a 1 M.empty
insert a (Node b size map) = Node b (size+1) map'
where map' = M.insertWith recurse d (Node a 1 M.empty) map
d = distance a b
recurse _ tree = insert a tree
-- | Checks whether an element is in the tree.
member :: Metric a => a -> BKTree a -> Bool
member a Empty = False
member a (Node b _ map)
| d == 0 = True
| otherwise = case M.lookup d map of
Nothing -> False
Just tree -> member a tree
where d = distance a b
-- | Approximate searching. @'memberDistance' n a tree@ will return true if
-- there is an element in @tree@ which has a 'distance' less than or equal to
-- @n@ from @a@.
memberDistance :: Metric a => Int -> a -> BKTree a -> Bool
memberDistance n a Empty = False
memberDistance n a (Node b _ map)
| d <= n = True
| otherwise = any (memberDistance n a) (M.elems subMap)
where d = distance a b
subMap = case M.split (d-n-1) map of
(_,mapRight) ->
case M.split (d+n+1) mapRight of
(mapCenter,_) -> mapCenter
-- | Removes an element from the tree. If an element occurs several times in
-- the tree then only one occurrence will be deleted.
delete :: Metric a => a -> BKTree a -> BKTree a
delete a Empty = Empty
delete a t@(Node b _ map)
| d == 0 = unions (M.elems map)
| otherwise = Node b sz subtrees
where d = distance a b
subtrees = M.update (Just . delete a) d map
sz = sum (L.map size (M.elems subtrees)) + 1
-- | Returns all the elements of the tree
elems :: BKTree a -> [a]
elems Empty = []
elems (Node a _ imap) = a : concatMap elems (M.elems imap)
-- | @'elemsDistance' n a tree@ returns all the elements in @tree@ which are
-- at a 'distance' less than or equal to @n@ from the element @a@.
elemsDistance :: Metric a => Int -> a -> BKTree a -> [a]
elemsDistance n a Empty = []
elemsDistance n a (Node b _ imap)
= (if d <= n then (b :) else id) $
concatMap (elemsDistance n a) (M.elems subMap)
where d = distance a b
subMap = case M.split (d-n-1) imap of
(_,mapRight) ->
case M.split (d+n+1) mapRight of
(mapCenter,_) -> mapCenter
-- | Constructs a tree from a list
fromList :: Metric a => [a] -> BKTree a
fromList xs = constructTree (\a -> Just (a,[])) xs
-- | Merges several trees
unions :: Metric a => [BKTree a] -> BKTree a
unions xs = constructTree split xs
where split Empty = Nothing
split (Node a _ imap) = Just (a,M.elems imap)
constructTree extract [] = Empty
constructTree extract (a:as)
= case extract a of
Nothing -> constructTree extract as
Just (piv,rest) ->
(\imap -> Node piv (1 + sum (map size (M.elems imap))) imap) $
M.fromAscList $
map recurse $
L.groupBy ((==) `on` fst) $
L.sortBy (compare `on` fst) $
concatMap (mkDist piv) $
as ++ rest
where mkDist piv m = case extract m of
Just (a,_) -> [(distance piv a,m)]
Nothing -> []
recurse bs@((k,_):_) = (k, constructTree extract (map snd bs))
-- | Merges two trees
union :: Metric a => BKTree a -> BKTree a -> BKTree a
union t1 t2 = unions [t1,t2]
-- | @'closest' a tree@ returns the element in @tree@ which is closest to
-- @a@ together with the distance. Returns @Nothing@ if the tree is empty.
closest :: Metric a => a -> BKTree a -> Maybe (a,Int)
closest a Empty = Nothing
closest a tree@(Node b _ _) = Just (closeLoop a (b,distance a b) tree)
closeLoop a candidate Empty = candidate
closeLoop a candidate@(b,d) (Node x _ imap)
= L.foldl' (closeLoop a) newCand (M.elems subMap)
where newCand = if j >= d
then candidate
else (x,j)
j = distance a x
subMap = case M.split (d-j-1) imap of
(_,mapRight) ->
case M.split (d+j+1) mapRight of
(mapCenter,_) -> mapCenter
-- Helper functions
on rel f x y = rel (f x) (f y)
#ifdef DEBUG
-- Testing
-- N.B. This code requires QuickCheck 2.0
-- We use a more standard implementation of the levenshtein edit distance
-- to check the hirschberg algorithm
levenshtein :: Eq a => [a] -> [a] -> Int
levenshtein xs ys = let
lxs = length xs
lys = length ys
d x y cx cy = minimum
[dist!(x-1,y-1) + (if cx == cy then 0 else 1)
,dist!(x-1,y) + 1
,dist!(x,y-1) + 1
]
dist :: Array (Int,Int) Int
dist = array ((0,0),(lxs,lys))
( [((0,0),0)]
++ [((x,0),x) | x <- [1..lxs]]
++ [((0,y),y) | y <- [1..lys]]
++ [ ((x,y),d x y cx cy)
| (x,cx) <- zip [1..] xs
, (y,cy) <- zip [1..] ys])
in dist!(lxs,lys)
-- These properties are all rather weaker than I would like.
-- Think of something better.
prop_levenshtein xs ys = distance xs ys == levenshtein xs (ys :: [Int])
prop_levenshteinRepeat (NonZero (NonNegative n)) (NonZero (NonNegative m)) =
distance (replicate n (0::Int)) (replicate m 0) == distance n m
prop_levenshteinLength xs =
forAll (vectorOf (length xs) arbitrary) $ \ys ->
distance xs ys == length xs && allDifferent xs ys
|| distance xs ys < length (xs :: [Int])
where allDifferent xs ys = all (==False) (zipWith (==) xs ys)
-- Semantics of BKTrees. Just a boring list of integers
sem tree = L.sort (elems tree) :: [Int]
-- For testing functions that transform trees
trans f xs = sem (f (fromList xs))
-- Tests for individual functions
prop_empty n = not (member (n::Int) empty)
prop_null xs = null (fromList xs) == Prelude.null (xs :: [Int])
prop_singleton n = elems (fromList [n]) == [n :: Int]
prop_fromList xs = sem (fromList xs) == L.sort xs
prop_insert n xs =
trans (insert n) xs == L.sort (n:xs)
prop_member n xs = member n (fromList xs) == L.elem (n::Int) xs
prop_memberDistance dist n xs =
let d = dist `mod` 5
ref = L.any (\e -> distance n e <= d) xs
in collect ref $
memberDistance d n (fromList xs) ==
L.any (\e -> distance n e <= d) (xs :: [Int])
prop_delete n xs =
trans (delete n) xs ==
L.sort (removeFirst (xs :: [Int]))
where removeFirst [] = []
removeFirst (a:as) | a == n = as
| otherwise = a : removeFirst as
prop_elems xs = L.sort (elems (fromList xs)) == L.sort (xs::[Int])
prop_elemsDistance dist n xs =
let d = dist `mod` 5 in
L.sort (elemsDistance d n (fromList xs)) ==
L.sort (filter (\e -> distance n e <= d) (xs::[Int]))
prop_unions xss =
sem (unions (map fromList xss)) ==
L.sort (concat (xss::[[Int]]))
prop_union xs ys =
sem (union (fromList xs) (fromList ys)) ==
L.sort (xs ++ (ys::[Int]))
prop_closest n xs =
case (closest n (fromList xs),xs) of
(Nothing,[]) -> True
(Just (_,d),ys) -> d == minimum (map (distance n) (ys::[Int]))
_ -> False
-- Testing the relations between operations
prop_insertDelete n xs =
trans (delete n . insert n) xs == L.sort (xs::[Int])
prop_sizeEmpty = size empty == 0
prop_sizeFromList xs = size (fromList xs) == length (xs :: [Int])
prop_sizeSucc n xs = size (insert (n::Int) tree) == size tree + 1
where tree = fromList xs
prop_sizeDelete n xs
= size (delete (n::Int) tree) ==
size tree - (if n `member` tree then 1 else 0)
where tree = fromList xs
prop_sizeUnion xs ys = size (union treeXs treeYs) == size treeXs + size treeYs
where (treeXs,treeYs) = (fromList xs, fromList (ys :: [Int]))
prop_sizeUnions xss = size (unions trees) == sum (map size trees)
where trees = map fromList (xss :: [[Int]])
-- All the tests
tests = [("empty", quickCheck' prop_empty)
,("null", quickCheck' prop_null)
,("singleton", quickCheck' prop_singleton)
,("fromList", quickCheck' prop_fromList)
,("insert", quickCheck' prop_insert)
,("member", quickCheck' prop_member)
,("memberDistance", quickCheck' prop_memberDistance)
,("delete", quickCheck' prop_delete)
,("elems", quickCheck' prop_elems)
,("elemsDistance", quickCheck' prop_elemsDistance)
,("unions", quickCheck' prop_unions)
,("union", quickCheck' prop_union)
,("closest", quickCheck' prop_closest)
,("size/empty", quickCheck' prop_sizeEmpty)
,("size/fromList", quickCheck' prop_sizeFromList)
,("size/succ", quickCheck' prop_sizeSucc)
,("size/delete", quickCheck' prop_sizeDelete)
,("size/union", quickCheck' prop_sizeUnion)
,("size/unions", quickCheck' prop_sizeUnions)
,("insert/delete", quickCheck' prop_insertDelete)
,("levenshtein", quickCheck' prop_levenshtein)
,("levenshtein repeat",quickCheck' prop_levenshteinRepeat)
,("levenshtein length",quickCheck' prop_levenshteinLength)
]
runTests = mapM_ runTest tests
where runTest (s,a) = do printf "%-25s :" s
b <- a
if b
then return ()
else exitFailure
#endif