[Haskell-cafe] The Knight's Tour: solutions please

[ajb at spamcop.net]

G'day all.

Quoting Don Stewart <dons at galois.com>:

> So, team, anyone want to implement a Knight's Tour solver in a list
> monad/list comprehension one liner? These little puzzles are made for
> fast languages with backtracking monads....

I conjecture that any one-liner won't be efficient.

Anyway, here's my ~30 min attempt.  The showBoard and main are both very
quick and dirty, and I'm sure someone can do much better.

I particularly like the fact that changing "Maybe" to "[]" will make
knightsTour return all tours starting at the upper left-hand corner,
rather than just one.  Warm fuzzy things rule.

Cheers,
Andrew Bromage

module Main where

import qualified Data.Set as S
import Data.List
import Data.Function
import Control.Arrow
import Control.Monad
import System

knightsTour :: Int -> Maybe [(Int,Int)]
knightsTour size
     = tour [(0,0)] (S.fromAscList [ (x,y) | x <- [0..size-1], y <-  
[0..size-1],
                         x /= 0 || y /= 0 ])
     where
         jumps = [(2,1),(1,2),(2,-1),(-1,2),(-2,1),(1,-2),(-2,-1),(-1,-2)]
         tour moves@(pos:_) blank
             | S.null blank = return (reverse moves)
             | otherwise = msum [ tour (npos:moves) (npos `S.delete` blank) |
                                         npos <- nextPositions pos ]
             where
                 nextPositions = map snd . sortBy (compare `on` fst) .
                                     map (length . neighbours &&& id) .
                                     neighbours
                 neighbours (x,y) = [ npos | (x',y') <- jumps,
                         let { npos = (x+x',y+y') }, npos `S.member` blank ]

showBoard :: Int -> [(Int,Int)] -> ShowS
showBoard size
     = inter bdr .
       map (inter ('|':) . map (shows . fst)) .
       groupBy ((==) `on` fst.snd) .
       sortBy (compare `on` snd) .
       zip [1..]
     where
         bdr = ('\n':) . inter ('+':) (replicate size (replicate width '-' ++))
                 . ('\n':)
         width = length . show $ size*size
         pad s = \r -> replicate (width - length (s "")) ' ' ++ s r
         inter sep xs = sep . foldr (.) id [ pad x . sep | x <- xs ]

main :: IO ()
main = do
         a <- getArgs
         size <- case a of
             [] -> return 8
             (s:_) -> return (read s)
         putStrLn $ case knightsTour size of
             Nothing -> "No solution found."
             Just b -> showBoard size b ""