El problema de las N torres

El problema de las N torres consiste en colocar N torres en un tablero con N filas y N columnas de forma que no haya dos torres en la misma fila ni en la misma columna.

Cada solución del problema de puede representar mediante una matriz con ceros y unos donde los unos representan las posiciones ocupadas por las torres y los ceros las posiciones libres. Por ejemplo,

   ( 0 1 0 )
   ( 1 0 0 )
   ( 0 0 1 )

( 0 1 0 )

( 1 0 0 )

( 0 0 1 )

representa una solución del problema de las 3 torres.

Definir las funciones

   torres  :: Int -> [Matrix Int]
   nTorres :: Int -> Integer

1 2	torres :: Int -> [Matrix Int] nTorres :: Int -> Integer

tales que
+ (torres n) es la lista de las soluciones del problema de las n torres. Por ejemplo,

      λ> torres 3
      [( 1 0 0 )
       ( 0 1 0 )
       ( 0 0 1 )
      ,( 1 0 0 )
       ( 0 0 1 )
       ( 0 1 0 )
      ,( 0 1 0 )
       ( 1 0 0 )
       ( 0 0 1 )
      ,( 0 1 0 )
       ( 0 0 1 )
       ( 1 0 0 )
      ,( 0 0 1 )
       ( 1 0 0 )
       ( 0 1 0 )
      ,( 0 0 1 )
       ( 0 1 0 )
       ( 1 0 0 )
      ]

λ> torres 3

[( 1 0 0 )

( 0 1 0 )

( 0 0 1 )

,( 1 0 0 )

( 0 0 1 )

( 0 1 0 )

,( 0 1 0 )

( 1 0 0 )

( 0 0 1 )

,( 0 1 0 )

( 0 0 1 )

( 1 0 0 )

,( 0 0 1 )

( 1 0 0 )

( 0 1 0 )

,( 0 0 1 )

( 0 1 0 )

( 1 0 0 )

]

(nTorres n) es el número de soluciones del problema de las n torres. Por ejemplo,

      λ> nTorres 3
      6
      λ> length (show (nTorres (10^4)))
      35660

λ> nTorres 3

λ> length (show (nTorres (10^4)))

35660

Soluciones

import Data.List (genericLength, sort, permutations)
import Data.Matrix 

-- 1ª definición de torres
-- =======================

torres1 :: Int -> [Matrix Int]
torres1 n = 
    [permutacionAmatriz n p | p <- sort (permutations [1..n])]

permutacionAmatriz :: Int -> [Int] -> Matrix Int
permutacionAmatriz n p =
    matrix n n f
    where f (i,j) | (i,j) `elem` posiciones = 1
                  | otherwise               = 0
          posiciones = zip [1..n] p    

-- 2ª definición de torres
-- =======================

torres2 :: Int -> [Matrix Int]
torres2 = map fromLists . permutations . toLists . identity

-- El cálculo con la definición anterior es:
--    λ> identity 3
--    ( 1 0 0 )
--    ( 0 1 0 )
--    ( 0 0 1 )
--    
--    λ> toLists it
--    [[1,0,0],[0,1,0],[0,0,1]]
--    λ> permutations it
--    [[[1,0,0],[0,1,0],[0,0,1]],
--     [[0,1,0],[1,0,0],[0,0,1]],
--     [[0,0,1],[0,1,0],[1,0,0]],
--     [[0,1,0],[0,0,1],[1,0,0]],
--     [[0,0,1],[1,0,0],[0,1,0]],
--     [[1,0,0],[0,0,1],[0,1,0]]]
--    λ> map fromLists it
--    [( 1 0 0 )
--     ( 0 1 0 )
--     ( 0 0 1 )
--    ,( 0 1 0 )
--     ( 1 0 0 )
--     ( 0 0 1 )
--    ,( 0 0 1 )
--     ( 0 1 0 )
--     ( 1 0 0 )
--    ,( 0 1 0 )
--     ( 0 0 1 )
--     ( 1 0 0 )
--    ,( 0 0 1 )
--     ( 1 0 0 )
--     ( 0 1 0 )
--    ,( 1 0 0 )
--     ( 0 0 1 )
--     ( 0 1 0 )
--    ]

-- 1ª definición de nTorres
-- ========================

nTorres1 :: Int -> Integer
nTorres1 = genericLength . torres1

-- 2ª definición de nTorres
-- ========================

nTorres2 :: Int -> Integer
nTorres2 n = product [1..fromIntegral n]

-- Comparación de eficiencia
-- =========================

--    λ> nTorres1 9
--    362880
--    (4.22 secs, 693,596,128 bytes)
--    λ> nTorres2 9
--    362880
--    (0.00 secs, 0 bytes)

import Data.List (genericLength, sort, permutations)

import Data.Matrix

-- 1ª definición de torres

-- =======================

torres1 :: Int -> [Matrix Int]

torres1 n =

[permutacionAmatriz n p | p <- sort (permutations [1..n])]

permutacionAmatriz :: Int -> [Int] -> Matrix Int

permutacionAmatriz n p =

matrix n n f

where f (i,j) | (i,j) `elem` posiciones = 1

| otherwise = 0

posiciones = zip [1..n] p

-- 2ª definición de torres

-- =======================

torres2 :: Int -> [Matrix Int]

torres2 = map fromLists . permutations . toLists . identity

-- El cálculo con la definición anterior es:

-- λ> identity 3

-- ( 1 0 0 )

-- ( 0 1 0 )

-- ( 0 0 1 )

-- λ> toLists it

-- [[1,0,0],[0,1,0],[0,0,1]]

-- λ> permutations it

-- [[[1,0,0],[0,1,0],[0,0,1]],

-- [[0,1,0],[1,0,0],[0,0,1]],

-- [[0,0,1],[0,1,0],[1,0,0]],

-- [[0,1,0],[0,0,1],[1,0,0]],

-- [[0,0,1],[1,0,0],[0,1,0]],

-- [[1,0,0],[0,0,1],[0,1,0]]]

-- λ> map fromLists it

-- [( 1 0 0 )

-- ( 0 1 0 )

-- ( 0 0 1 )

-- ,( 0 1 0 )

-- ( 1 0 0 )

-- ( 0 0 1 )

-- ,( 0 0 1 )

-- ( 0 1 0 )

-- ( 1 0 0 )

-- ,( 0 1 0 )

-- ( 0 0 1 )

-- ( 1 0 0 )

-- ,( 0 0 1 )

-- ( 1 0 0 )

-- ( 0 1 0 )

-- ,( 1 0 0 )

-- ( 0 0 1 )

-- ( 0 1 0 )

-- ]

-- 1ª definición de nTorres

-- ========================

nTorres1 :: Int -> Integer

nTorres1 = genericLength . torres1

-- 2ª definición de nTorres

-- ========================

nTorres2 :: Int -> Integer

nTorres2 n = product [1..fromIntegral n]

-- Comparación de eficiencia

-- =========================

-- λ> nTorres1 9

-- 362880

-- (4.22 secs, 693,596,128 bytes)

-- λ> nTorres2 9

-- 362880

-- (0.00 secs, 0 bytes)

5 Comentarios

import Data.List (permutations)
import Data.Matrix 

torres  :: Int -> [Matrix Int]
torres n = map (matriz n) (lista n)
 
lista :: Int -> [[(Int,Int)]]
lista n = map (zip [1..n]) (permutations [1..n])
 
matriz :: Int -> [(Int,Int)] -> Matrix Int
matriz n xs = matrix n n (\(i,j)-> f i j)
  where f i j | (i,j) `elem` xs = 1
              | otherwise       = 0
 
nTorres :: Int -> Integer
nTorres n = product [1..(fromIntegral n)]

import Data.List (permutations)

import Data.Matrix

torres :: Int -> [Matrix Int]

torres n = map (matriz n) (lista n)

lista :: Int -> [[(Int,Int)]]

lista n = map (zip [1..n]) (permutations [1..n])

matriz :: Int -> [(Int,Int)] -> Matrix Int

matriz n xs = matrix n n (\(i,j)-> f i j)

where f i j | (i,j) `elem` xs = 1

| otherwise = 0

nTorres :: Int -> Integer

nTorres n = product [1..(fromIntegral n)]

Responder

import Data.Matrix 

torres :: Int -> [Matrix Int]
torres n =
  foldl (*) (identity n) . map (uncurry (permMatrix n)) <$> permutaciones [1..n]

permutaciones :: [Int] -> [[(Int,Int)]]
permutaciones [] = [[]]
permutaciones p@(x:xs) =
  pxs ++ concat [map ((x,y):) pxs | y <- xs]
  where pxs = permutaciones xs

nTorres :: Int -> Integer
nTorres n = product [1..toInteger n]

import Data.Matrix

torres :: Int -> [Matrix Int]

torres n =

foldl (*) (identity n) . map (uncurry (permMatrix n)) <$> permutaciones [1..n]

permutaciones :: [Int] -> [[(Int,Int)]]

permutaciones [] = [[]]

permutaciones p@(x:xs) =

pxs ++ concat [map ((x,y):) pxs | y <- xs]

where pxs = permutaciones xs

nTorres :: Int -> Integer

nTorres n = product [1..toInteger n]

Responder

Una pequeña modificación de la solución de marjimcom que mejora ligeramente la eficacia.

import Data.List (permutations)
import Data.Matrix 

torres :: Int -> [Matrix Int]
torres n = aux n <$> permutations [1..n]

aux :: Int -> [Int] -> Matrix Int
aux n xs =  foldl (+) z [setElem 1 x z | x <- zip [1..n] xs]
  where z = zero n n

--    *Main> length (torres2 11)
--    39916800
--    (4.32 secs, 18,843,026,656 bytes)
--    *Main> length (torres 11)
--    39916800
--    (3.68 secs, 15,330,346,728 bytes)

import Data.List (permutations)

import Data.Matrix

torres :: Int -> [Matrix Int]

torres n = aux n <$> permutations [1..n]

aux :: Int -> [Int] -> Matrix Int

aux n xs = foldl (+) z [setElem 1 x z | x <- zip [1..n] xs]

where z = zero n n

-- *Main> length (torres2 11)

-- 39916800

-- (4.32 secs, 18,843,026,656 bytes)

-- *Main> length (torres 11)

-- 39916800

-- (3.68 secs, 15,330,346,728 bytes)

Responder

torres  :: Int -> [Matrix Int]
torres n = map (creaMatriz n) (permutations [1..n])
     where creaMatriz a xs = matrix a a f        
              where f x | elem x t = 1
                        | otherwise = 0
                    t = zip xs [1..a]
nTorres :: Int -> Integer
nTorres x = product [1..fromIntegral x]

torres :: Int -> [Matrix Int]

torres n = map (creaMatriz n) (permutations [1..n])

where creaMatriz a xs = matrix a a f

where f x | elem x t = 1

| otherwise = 0

t = zip xs [1..a]

nTorres :: Int -> Integer

nTorres x = product [1..fromIntegral x]

Responder

import Data.Matrix
import Data.List

colocaTorres :: Matrix Int -> [Matrix Int]
colocaTorres p = [ setElem 1 (i,j) p | i <- [1..nrows p]
                                     , j <- [1..ncols p]
                                     , all (==0) (getCol j p)
                                     , all (==0) (getRow i p)]

torres :: Int -> [Matrix Int]
torres n = nub (aux n (zero n n))
       where aux :: Int -> Matrix Int -> [Matrix Int]
             aux 0 p = [p]
             aux x p = concatMap (aux (x-1)) (colocaTorres p)

nTorres :: Int -> Integer
nTorres = genericLength . torres

nTorres' :: Int -> Integer
nTorres' n = product [1..fromIntegral n]

import Data.Matrix

import Data.List

colocaTorres :: Matrix Int -> [Matrix Int]

colocaTorres p = [ setElem 1 (i,j) p | i <- [1..nrows p]

, j <- [1..ncols p]

, all (==0) (getCol j p)

, all (==0) (getRow i p)]

torres :: Int -> [Matrix Int]

torres n = nub (aux n (zero n n))

where aux :: Int -> Matrix Int -> [Matrix Int]

aux 0 p = [p]

aux x p = concatMap (aux (x-1)) (colocaTorres p)

nTorres :: Int -> Integer

nTorres = genericLength . torres

nTorres' :: Int -> Integer

nTorres' n = product [1..fromIntegral n]

Responder

El problema de las N torres

Soluciones

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