# [Haskell-cafe] New Benchmark Under Review: Magic Squares

Daniel Fischer daniel.is.fischer at web.de
Tue Jul 4 20:20:18 EDT 2006

```Am Dienstag, 4. Juli 2006 18:20 schrieben Sie:
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> Daniel,
>
> > I have now tuned Josh Goldfoot's code without changing the order in
> > which the
> > magic squares are produced, for a 5x5 magic square, my machine took
> > 1/2 hours and used 2Mb memory (considering that the original code
> > did not
> > finish within 4 1/2 hours here, that should push time on the
> > benchmarking
> > machine under 3000s and put us in the lead, I hope).
>
> Thanks for your efforts on this project.  I'm actually more
> interested in using your earlier solution, since it is so much
> faster.  Right now, the magic square code rises in runtime from 1.5
> seconds to 4 hours with an increase of 1 in the square's dimension.
> I would much rather use a technique that had a more linear (or even
> exponential) increase!
>
> I would propose modifying the other entries (since there are only a
> handful) to match the output of your original solution.
>
> What do you think?

Cool, though the problem of exploding runtime remains, it's only pushed a
little further. Now I get a 5x5 magig square in 1 s, a 6x6 in 5.4 s, but 7x7
segfaulted after about 2 1/2 hours - out of memory, I believe.
And, as mentioned in passing, using 'intersect' in the first version is
slowing things down, so here is my currently fastest (undoubtedly, the
experts could still make it faster by clever unboxing):

import Data.Array.Unboxed
import Data.List
import System.Environment (getArgs)

main :: IO ()

msquare :: Int -> IO ()
msquare n = let mn = (n*(n*n+1)) `quot` 2
grd = listArray ((1,1),(n,n)) (repeat 0)
unus = [1 .. n*n]
ff  = findFewestMoves n mn grd unus
ini = Square grd unus ff (2*n*n)
allSquares = bestFirst (successorNodes n mn) [ini]
in  putStrLn \$ showGrid n . grid \$ head allSquares

data Square = Square { grid :: UArray (Int,Int) Int
, unused :: [Int]
, ffm :: ([Int], Int, Int, Int)
, priority :: !Int
} deriving Eq

instance Ord Square where
compare (Square g1 _ _ p1) (Square g2 _ _ p2)
= case compare p1 p2 of
EQ -> compare g1 g2
ot -> ot

showMat :: [[Int]] -> ShowS
showMat lns = foldr1 ((.) . (. showChar '\n')) \$ showLns
where
showLns =
map (foldr1 ((.) . (. showChar ' ')) . map shows) lns

showGrid :: Int -> UArray (Int,Int) Int -> String
showGrid n g = showMat [[g ! (r,c) | c <- [1 .. n]] | r <- [1 .. n]] ""

bestFirst :: (Square -> [Square]) -> [Square] -> [Square]
bestFirst _ [] = []
bestFirst successors (front:queue)
| priority front == 0 = front : bestFirst successors queue
| otherwise = bestFirst successors \$ foldr insert queue (successors front)

successorNodes n mn sq
= map (place sq n mn (r,c)) possibilities
where
(possibilities,_,r,c) = ffm sq

place :: Square -> Int -> Int -> (Int,Int) -> Int -> Square
place (Square grd unus _ _) n mn (r,c) k
= Square grd' uns moveChoices p
where
grd' = grd//[((r,c),k)]
moveChoices@(_,len,_,_) = findFewestMoves n mn grd' uns
uns = delete k unus
p = length uns + len

findFewestMoves :: Int -> Int -> UArray (Int,Int) Int -> [Int] ->
([Int],Int,Int,Int)
findFewestMoves n mn grid unus
| null unus = ([],0,0,0)
| otherwise = (movelist, length movelist, mr, mc)
where
openSquares = [(r,c) | r <- [1 .. n], c <- [1 .. n], grid ! (r,c) ==
0]
pm = possibleMoves n mn grid unus
openMap = map (\(x,y) -> (pm x y,x,y)) openSquares
mycompare (a,_,_) (b,_,_) = compare (length a) (length b)
(movelist,mr,mc) = minimumBy mycompare openMap

possibleMoves :: Int -> Int -> UArray (Int,Int) Int -> [Int] -> Int -> Int ->
[Int]
possibleMoves n mn grid unus r c
| grid ! (r,c) /= 0 = []
| otherwise = takeWhile (<= ma) \$ dropWhile (< mi) unus
where
cellGroups
| r == c && r + c == n + 1 = [d1, d2, theRow, theCol]
| r == c = [d1, theRow, theCol]
| r + c == n + 1 = [d2, theRow, theCol]
| otherwise = [theRow, theCol]
d1 = diag1 grid n
d2 = diag2 grid n
theRow = gridRow grid n r
theCol = gridCol grid n c
lows = scanl (+) 0 unus
higs = scanl (+) 0 \$ reverse unus
rge :: [Int] -> (Int,Int)
rge cg = let k = count0s cg - 1
lft = mn - sum cg
in (lft - (higs!!k),lft - (lows!!k))
(mi,ma) = foldr1 mima \$ map rge cellGroups
mima (a,b) (c,d) = (max a c, min b d)

gridRow, gridCol :: UArray (Int,Int) Int -> Int -> Int -> [Int]
diag1, diag2 :: UArray (Int,Int) Int -> Int -> [Int]
gridRow grid n r = [grid ! (r,i) | i <- [1 .. n]]
gridCol grid n c = [grid ! (i,c) | i <- [1 .. n]]
diag1 grid n     = [grid ! (i,i) | i <- [1 .. n]]
diag2 grid n     = [grid ! (i,n+1-i) | i <- [1 .. n]]
count0s :: [Int] -> Int
count0s = length . filter (== 0)

>
> - -Brent
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Cheers,
Daniel

--

"In My Egotistical Opinion, most people's C programs should be
indented six feet downward and covered with dirt."
-- Blair P. Houghton

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