The Knights Tour
From HaskellWiki
DonStewart (Talk  contribs) 
(Improved ContT r (ST s) code) 

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

<haskell> 
<haskell> 

−  import Control.Applicative ((<$>)) 

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

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

Line 85:  Line 86:  
import Data.Ord 
import Data.Ord 

import Data.Ix 
import Data.Ix 

−  import Data.Map (Map, lookup, singleton, insert) 

import System.Environment 
import System.Environment 

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type Square = (Int, Int) 
type Square = (Int, Int) 

type Board s = STUArray s (Int,Int) Int 
type Board s = STUArray s (Int,Int) Int 

−  
type ChessM r s = ContT r (ST s) 
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 :: Int > Board s > Square > ChessM r s [Square] 

−  successors n b s = sortWith (fmap length . succs) =<< succs s 
+  successors n b = sortWith (fmap length . succs) <=< succs 
where 
where 

−  sortWith f l = map fst <$> sortBy (comparing snd) <$> mapM (\x > (,) x <$> f x) l 
+  sortWith f l = map fst `fmap` sortBy (comparing snd) 
−  succs (i,j) = filterM (empty b) [ (i', j')  (dx,dy) < [(1,2),(2,1)] 
+  `fmap` mapM (\x > (,) x `fmap` f x) l 
−  , i' < [i+dx,idx] , j' < [j+dy, jdy] 
+  succs (i,j) = filterM (empty b) 
−  , inRange ((1,1),(n,n)) (i',j') ] 
+  [ (i', j')  (dx,dy) < [(1,2),(2,1)] 
−  +  , i' < [i+dx,idx] , j' < [j+dy, jdy] 

−  stop :: Square > Board s > ChessM r s Int 
+  , inRange ((1,1),(n,n)) (i',j') ] 
−  stop s b = lift $ readArray b s 

empty :: Board s > Square > ChessM r s Bool 
empty :: Board s > Square > ChessM r s Bool 

Line 110:  Line 110:  
mark s k b = lift $ writeArray b s k 
mark s k b = lift $ writeArray b s k 

−  tour :: Int > Int > (Board s > ChessM r s ()) > Square > Board s > ChessM r s () 
+  tour :: Int > Int > ChessK r s > Square > Board s > ChessM r s () 
−  tour n k exit s b  k > n*n = exit b 
+  tour n k exit s b  k > n*n = showBoard n b >>= exit 
−   otherwise = do ss < successors n b s 
+   otherwise = successors n b s >>= 
−  try ss 
+  mapM_ (\x > do mark x k b 
−  where 
+  tour n (k+1) exit x b 
−  try [] = return () 
+   failed 
−  try (x:xs) = do mark x k b 
+  mark x 0 b) 
−  tour n (k+1) exit x b 

−   failed 

−  mark x 0 b 

−  try xs 

showBoard :: Int > Board s > ChessM r s String 
showBoard :: Int > Board s > ChessM r s String 

−  showBoard n b = fmap (unlines . map unwords) . sequence . map sequence 
+  showBoard n b = fmap unlines . forM [1..n] $ \i > 
−  $ [ [ fmt `fmap` stop (i,j) b  i < [1..n] ]  j < [1..n] ] 
+  fmap unwords . forM [1..n] $ \j > 
+  pad `fmap` lift (readArray b (i,j)) 

where 
where 

−  fmt i  i < 10 = ' ': show i 
+  k = floor . log . fromIntegral $ n*n 
−   otherwise = show i 
+  pad i = let s = show i in replicate (klength s) ' ' ++ s 
main = do (n:_) < map read `fmap` getArgs 
main = do (n:_) < map read `fmap` getArgs 

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(do b < lift $ newArray ((1,1),(n,n)) 0 
(do b < lift $ newArray ((1,1),(n,n)) 0 

mark (1,1) 1 b 
mark (1,1) 1 b 

−  callCC $ \exit > tour n 2 exit (1,1) b >> fail "No solution!" 
+  callCC $ \k > tour n 2 k (1,1) b >> fail "No solution!") 
−  showBoard n b) 

putStrLn s 
putStrLn s 

+  
</haskell> 
</haskell> 

Revision as of 02:18, 1 December 2008
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.
Contents 
1 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..size1] 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..size1]] 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>"
2 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,idx] , j' < [j+dy, jdy] , 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 = floor . log . fromIntegral $ n*n pad i = let s = show i in replicate (klength 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
3 LogicT monad
A very short implementation using the LogicT monad
19 lines of code. 8 imports.
import Control.Monad.Logic import Prelude hiding (lookup) import Data.List hiding (lookup, insert) import Data.Maybe import Data.Ord import Data.Ix import Data.Map (Map, lookup, singleton, insert) 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,idx] , j' < [j+dy, jdy] , empty (i',j') b, inRange ((1,1),(n,n)) (i',j') ] empty s = isNothing . lookup s choose = msum . map return tour n k s b  k > n*n = return b  otherwise = do next < choose $ successors n b s tour n (k+1) next (insert next k b) showBoard n b = unlines . map unwords $ [ [ fmt . fromJust $ lookup (i,j) b  i < [1..n] ]  j < [1..n] ] where fmt i  i < 10 = ' ': show i  otherwise = show i main = do (n:_) < map read `fmap` getArgs let b = observe . tour n 2 (1,1) $ singleton (1,1) 1 putStrLn $ showBoard n b