The Knights Tour

The Knight's Tour is a mathematical problem involving a knight on a chessboard. The knight is placed on the empty board and, moving according to the rules of chess, must visit each square exactly once.

Here are some Haskell implementations.

One

--
-- Quick implementation by dmwit on #haskell
-- Faster, shorter, uses less memory than the Python version.
--

import Control.Arrow
import Control.Monad
import Data.List
import Data.Maybe
import Data.Ord
import System.Environment
import qualified Data.Map as M

sortOn f = map snd . sortBy (comparing fst) . map (f &&& id)

clip coord size = coord >= 0 && coord < size
valid size solution xy@(x, y) = and [clip x size, clip y size, isNothing (M.lookup xy solution)]
neighbors size solution xy = length . filter (valid size solution) $ sequence moves xy

moves = do
    f <- [(+), subtract]
    g <- [(+), subtract]
    (x, y) <- [(1, 2), (2, 1)]
    [f x *** g y]

solve size solution n xy = do
    guard (valid size solution xy)
    let solution'   = M.insert xy n solution
        sortedMoves = sortOn (neighbors size solution) (sequence moves xy)
    if n == size * size
        then [solution']
        else sortedMoves >>= solve size solution' (n+1)

printBoard size solution = board [0..size-1] where
    sqSize    = size * size
    elemSize  = length (show sqSize)
    separator = intercalate (replicate elemSize '-') (replicate (size + 1) "+")
    pad n s   = replicate (elemSize - length s) ' ' ++ s
    elem xy   = pad elemSize . show $ solution M.! xy
    line y    = concat  . intersperseWrap "|" $ [elem (x, y) | x <- [0..size-1]]
    board     = unlines . intersperseWrap separator . map line
    intersperseWrap s ss = s : intersperse s ss ++ [s]

go size = case solve size M.empty 1 (0, 0) of
    []    -> "No solution found"
    (s:_) -> printBoard size s

main = do
    args <- getArgs
    name <- getProgName
    putStrLn $ case map reads args of
        []             -> go 8
        [[(size, "")]] -> go size
        _              -> "Usage: " ++ name ++ " <size>"

Using Continuations

An efficient version (some 10x faster than the example Python solution) using continuations.

This is about as direct a translation of the Python algorithm as you'll get without sticking the whole thing in IO. The Python version prints the board and exits immediately upon finding it, so it can roll back changes if that doesn't happen. Instead, this version sets up an exit continuation using callCC and calls that to immediately return the first solution found. The Logic version below takes around 50% more time.

import Control.Monad.Cont
import Control.Monad.ST

import Data.Array.ST
import Data.List
import Data.Ord
import Data.Ix

import System.Environment

type Square  = (Int, Int)
type Board s = STUArray s (Int,Int) Int
type ChessM r s = ContT r (ST s)
type ChessK r s = String -> ChessM r s ()

successors :: Int -> Board s -> Square -> ChessM r s [Square]
successors n b = sortWith (fmap length . succs) <=< succs
 where
 sortWith f l = map fst `fmap` sortBy (comparing snd)
                        `fmap` mapM (\x -> (,) x `fmap` f x) l
 succs (i,j) = filterM (empty b)
                 [ (i', j') | (dx,dy) <- [(1,2),(2,1)]
                            , i' <- [i+dx,i-dx] , j' <- [j+dy, j-dy]
                            , inRange ((1,1),(n,n)) (i',j') ]

empty :: Board s -> Square -> ChessM r s Bool
empty b s = fmap (<1) . lift $ readArray b s

mark :: Square -> Int -> Board s -> ChessM r s ()
mark s k b = lift $ writeArray b s k

tour :: Int -> Int -> ChessK r s -> Square -> Board s -> ChessM r s ()
tour n k exit s b | k > n*n   = showBoard n b >>= exit
                  | otherwise = successors n b s >>=
                                mapM_ (\x -> do mark x k b
                                                tour n (k+1) exit x b
                                                -- failed
                                                mark x 0 b)

showBoard :: Int -> Board s -> ChessM r s String
showBoard n b = fmap unlines . forM [1..n] $ \i ->
                 fmap unwords . forM [1..n] $ \j ->
                   pad `fmap` lift (readArray b (i,j))
 where
 k = ceiling . logBase 10 . fromIntegral $ n*n + 1
 pad i = let s = show i in replicate (k-length s) ' ' ++ s

main = do (n:_) <- map read `fmap` getArgs
          s <- stToIO . flip runContT return $
               (do b <- lift $ newArray ((1,1),(n,n)) 0
                   mark (1,1) 1 b
                   callCC $ \k -> tour n 2 k (1,1) b >> fail "No solution!")
          putStrLn s

LogicT monad

A very short implementation using the LogicT monad

16 lines of code. 7 imports.

import Control.Monad.Logic
 
import Data.List
import Data.Maybe
import Data.Ord
import Data.Ix
import qualified Data.Map as Map
import System.Environment
 
successors n b = sortWith (length . succs) . succs
 where sortWith f = map fst . sortBy (comparing snd) . map (\x -> (x, f x))
       succs (i,j) = [ (i', j') | (dx,dy) <- [(1,2),(2,1)]
                                , i' <- [i+dx,i-dx] , j' <- [j+dy, j-dy]
                                , isNothing (Map.lookup (i',j') b)
                                , inRange ((1,1),(n,n)) (i',j') ]
 
tour n k s b | k > n*n   = return b
             | otherwise = do next <- msum . map return $ successors n b s
                              tour n (k+1) next $ Map.insert next k b
 
showBoard n b = unlines . map (\i -> unwords . map (\j ->
                  pad . fromJust $ Map.lookup (i,j) b) $ [1..n]) $ [1..n]
 where k = ceiling . logBase 10 . fromIntegral $ n*n + 1
       pad i = let s = show i in replicate (k-length s) ' ' ++ s
 
main = do (n:_) <- map read `fmap` getArgs
          let b = observe . tour n 2 (1,1) $ Map.singleton (1,1) 1
          putStrLn $ showBoard n b