Difference between revisions of "The Knights Tour"
(Improved ContT r (ST s) code) 
m (link to FBackTrack module) 

(4 intermediate revisions by 2 users not shown)  
Line 11:  Line 11:  
__TOC__ 
__TOC__ 

−  == 
+  == First Solution == 
<haskell> 
<haskell> 

Line 123:  Line 123:  
pad `fmap` lift (readArray b (i,j)) 
pad `fmap` lift (readArray b (i,j)) 

where 
where 

−  k = 
+  k = ceiling . logBase 10 . fromIntegral $ n*n + 1 
pad i = let s = show i in replicate (klength s) ' ' ++ s 
pad i = let s = show i in replicate (klength s) ' ' ++ s 

Line 139:  Line 139:  
A very short implementation using [http://hackage.haskell.org/cgibin/hackagescripts/package/logict the LogicT monad] 
A very short implementation using [http://hackage.haskell.org/cgibin/hackagescripts/package/logict the LogicT monad] 

−  +  16 lines of code. 7 imports. 

<haskell> 
<haskell> 

import Control.Monad.Logic 
import Control.Monad.Logic 

−  
+  
−  import Prelude hiding (lookup) 

+  import Data.List 

−  import Data.List hiding (lookup, insert) 

import Data.Maybe 
import Data.Maybe 

import Data.Ord 
import Data.Ord 

import Data.Ix 
import Data.Ix 

−  import Data.Map 
+  import qualified Data.Map as Map 
import System.Environment 
import System.Environment 

−  
+  
successors n b = sortWith (length . succs) . succs 
successors n b = sortWith (length . succs) . succs 

−  where 

+  where sortWith f = map fst . sortBy (comparing snd) . map (\x > (x, f x)) 

−  sortWith f = map fst . sortBy (comparing snd) . map (\x > (x, f x)) 

+  succs (i,j) = [ (i', j')  (dx,dy) < [(1,2),(2,1)] 

−  succs (i,j) = [ (i', j')  (dx,dy) < [(1,2),(2,1)] 

+  , i' < [i+dx,idx] , j' < [j+dy, jdy] 

−  +  , 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 (klength 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 

+  </haskell> 

−  empty s = isNothing . lookup s 

+  == Oleg Kiselyov's Solution == 

−  choose = msum . map return 

+  Oleg [http://www.haskell.org/pipermail/haskellcafe/2008December/051277.html provided a solution] on haskellcafe: 

+  
+  <blockquote> 

+  It seems the following pure functional (except for the final printout) 

+  version of the search has almost the same performance as the Dan 

+  Doel's latest version with the unboxed arrays and callCC. For the board of 

+  size 40, Dan Doel's version takes 0.047s on my computer; the version 

+  below takes 0.048s. For smaller boards, the difference is 

+  imperceptible. Interestingly, the file sizes of the compiled 

+  executables (ghc O2, ghc 6.8.2) are similar too: 606093 bytes for Dan 

+  Doel's version, and 605938 bytes for the version below. 

+  
+  The version below is essentially Dan Doel's earlier version. Since 

+  the problem involves only pure search (rather than committed choice), 

+  I took the liberty of substituting [http://okmij.org/ftp/Haskell/FBackTrack.hs FBackTrack] (efficient MonadPlus) 

+  for LogicT. FBackTrack can too be made the instance of LogicT; there 

+  has not been any demand for that though. 

+  </blockquote> 

+  
+  <haskell> 

+  import Data.List 

+  import Data.Ord 

+  import qualified Data.IntMap as Map 

+  import System.Environment 

+  import FBackTrack 

+  import Control.Monad 

+  
+   Emulate the 2dimensional map as a nested 1dimensional map 

+  initmap n = Map.fromList $ (1,Map.singleton 1 1):[ (k,Map.empty)  k < [2..n] ] 

+  notMember (i,j) m = Map.notMember j $ Map.findWithDefault undefined i m 

+  insrt (i,j) v m = Map.update (Just . Map.insert j v) i m 

+  lkup (i,j) m = Map.findWithDefault undefined j $ 

+  Map.findWithDefault undefined i m 

+  
+  
+  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] 

+  , i' >= 1, j' >= 1, i' <= n, j' <= n 

+  , notMember (i',j') b ] 

tour n k s b  k > n*n = return b 
tour n k s b  k > n*n = return b 

−   otherwise = do next < 
+   otherwise = do next < foldl1 mplus.map return $ successors n b s 
−  tour n (k+1) next 
+  tour n (k+1) next $ insrt next k b 
−  showBoard n b = unlines . map unwords 

+  
−  $ [ [ fmt . fromJust $ lookup (i,j) b  i < [1..n] ]  j < [1..n] ] 

+  showBoard n b = unlines . map (\i > unwords . map (\j > 

−  where 

+  pad $ lkup (i,j) b) $ [1..n]) $ [1..n] 

−  fmt i  i < 10 = ' ': show i 

+  where k = length . show $ n*n + 1 

−  +  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 

−  let b = 
+  let (b:_) = runM Nothing . tour n 2 (1,1) $ initmap n 
putStrLn $ showBoard n b 
putStrLn $ showBoard n b 

+  
</haskell> 
</haskell> 
Latest revision as of 10:10, 2 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.
First Solution

 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>"
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 = ceiling . logBase 10 . fromIntegral $ n*n + 1
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
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,idx] , j' < [j+dy, jdy]
, 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 (klength 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
Oleg Kiselyov's Solution
Oleg provided a solution on haskellcafe:
It seems the following pure functional (except for the final printout) version of the search has almost the same performance as the Dan Doel's latest version with the unboxed arrays and callCC. For the board of size 40, Dan Doel's version takes 0.047s on my computer; the version below takes 0.048s. For smaller boards, the difference is imperceptible. Interestingly, the file sizes of the compiled executables (ghc O2, ghc 6.8.2) are similar too: 606093 bytes for Dan Doel's version, and 605938 bytes for the version below.
The version below is essentially Dan Doel's earlier version. Since the problem involves only pure search (rather than committed choice), I took the liberty of substituting FBackTrack (efficient MonadPlus) for LogicT. FBackTrack can too be made the instance of LogicT; there has not been any demand for that though.
import Data.List
import Data.Ord
import qualified Data.IntMap as Map
import System.Environment
import FBackTrack
import Control.Monad
 Emulate the 2dimensional map as a nested 1dimensional map
initmap n = Map.fromList $ (1,Map.singleton 1 1):[ (k,Map.empty)  k < [2..n] ]
notMember (i,j) m = Map.notMember j $ Map.findWithDefault undefined i m
insrt (i,j) v m = Map.update (Just . Map.insert j v) i m
lkup (i,j) m = Map.findWithDefault undefined j $
Map.findWithDefault undefined i m
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]
, i' >= 1, j' >= 1, i' <= n, j' <= n
, notMember (i',j') b ]
tour n k s b  k > n*n = return b
 otherwise = do next < foldl1 mplus.map return $ successors n b s
tour n (k+1) next $ insrt next k b
showBoard n b = unlines . map (\i > unwords . map (\j >
pad $ lkup (i,j) b) $ [1..n]) $ [1..n]
where k = length . show $ n*n + 1
pad i = let s = show i in replicate (klength s) ' ' ++ s
main = do (n:_) < map read `fmap` getArgs
let (b:_) = runM Nothing . tour n 2 (1,1) $ initmap n
putStrLn $ showBoard n b