Decidir si existe un subconjunto con suma dada
Sea S un conjunto finito de números naturales y m un número natural. El problema consiste en determinar si existe un subconjunto de S cuya suma es m. Por ejemplo, si S = [3,34,4,12,5,2] y m = 9, existe un subconjunto de S, [4,5], cuya suma es 9. En cambio, no hay ningún subconjunto de S que sume 13.
Definir la función
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existeSubSuma :: [Int] -> Int -> Bool |
tal que (existeSubSuma xs m) se verifica si existe algún subconjunto de xs que sume m. Por ejemplo,
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existeSubSuma [3,34,4,12,5,2] 9 == True existeSubSuma [3,34,4,12,5,2] 13 == False existeSubSuma ([3,34,4,12,5,2]++[20..400]) 13 == False existeSubSuma ([3,34,4,12,5,2]++[20..400]) 654 == True existeSubSuma [1..200] (sum [1..200]) == True |
Soluciones
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import Data.List (subsequences, sort) import Data.Array (Array, array, listArray, (!)) -- 1ª definición (Calculando todos los subconjuntos) -- ================================================= existeSubSuma1 :: [Int] -> Int -> Bool existeSubSuma1 xs n = any (\ys -> sum ys == n) (subsequences xs) -- 2ª definición (por recursión) -- ============================= existeSubSuma2 :: [Int] -> Int -> Bool existeSubSuma2 _ 0 = True existeSubSuma2 [] _ = False existeSubSuma2 (x:xs) n | n < x = existeSubSuma2 xs n | otherwise = existeSubSuma2 xs (n-x) || existeSubSuma2 xs n -- 3ª definición (por programación dinámica) -- ========================================= existeSubSuma3 :: [Int] -> Int -> Bool existeSubSuma3 xs n = matrizExisteSubSuma3 xs n ! (length xs,n) -- (matrizExisteSubSuma3 xs m) es la matriz q tal que q(i,j) se verifica -- si existe algún subconjunto de (take i xs) que sume j. Por ejemplo, -- λ> elems (matrizExisteSubSuma3 [1,3,5] 9) -- [True,False,False,False,False,False,False,False,False,False, -- True,True, False,False,False,False,False,False,False,False, -- True,True, False,True, True, False,False,False,False,False, -- True,True, False,True, True, True, True, False,True, True] -- Con las cabeceras, -- 0 1 2 3 4 5 6 7 8 9 -- [] [True,False,False,False,False,False,False,False,False,False, -- [1] True,True, False,False,False,False,False,False,False,False, -- [1,3] True,True, False,True, True, False,False,False,False,False, -- [1,3,5] True,True, False,True, True, True, True, False,True, True] matrizExisteSubSuma3 :: [Int] -> Int -> Array (Int,Int) Bool matrizExisteSubSuma3 xs n = q where m = length xs v = listArray (1,m) xs q = array ((0,0),(m,n)) [((i,j), f i j) | i <- [0..m], j <- [0..n]] f _ 0 = True f 0 _ = False f i j | j < v ! i = q ! (i-1,j) | otherwise = q ! (i-1,j-v!i) || q ! (i-1,j) -- 4ª definición (ordenando y por recursión) -- ========================================= existeSubSuma4 :: [Int] -> Int -> Bool existeSubSuma4 xs = aux (sort xs) where aux _ 0 = True aux [] _ = False aux (y:ys) n | y <= n = aux ys (n-y) || aux ys n | otherwise = False -- 5ª definición (ordenando y dinámica) -- ==================================== existeSubSuma5 :: [Int] -> Int -> Bool existeSubSuma5 xs n = matrizExisteSubSuma5 (reverse (sort xs)) n ! (length xs,n) matrizExisteSubSuma5 :: [Int] -> Int -> Array (Int,Int) Bool matrizExisteSubSuma5 xs n = q where m = length xs v = listArray (1,m) xs q = array ((0,0),(m,n)) [((i,j), f i j) | i <- [0..m], j <- [0..n]] f _ 0 = True f 0 _ = False f i j | v ! i <= j = q ! (i-1,j-v!i) || q ! (i-1,j) | otherwise = False -- Equivalencia -- ============ prop_existeSubSuma :: [Int] -> Int -> Bool prop_existeSubSuma xs n = all (== existeSubSuma2 ys m) [ existeSubSuma3 ys m , existeSubSuma4 ys m , existeSubSuma5 ys m ] where ys = map abs xs m = abs n -- La comprobación es -- λ> quickCheck prop_existeSubSuma -- +++ OK, passed 100 tests. -- Comparación de eficiencia: -- ========================== -- λ> let xs = [1..22] in existeSubSuma1 xs (sum xs) -- True -- (7.76 secs, 3,892,403,928 bytes) -- λ> let xs = [1..22] in existeSubSuma2 xs (sum xs) -- True -- (0.02 secs, 95,968 bytes) -- λ> let xs = [1..22] in existeSubSuma3 xs (sum xs) -- True -- (0.03 secs, 6,055,200 bytes) -- λ> let xs = [1..22] in existeSubSuma4 xs (sum xs) -- True -- (0.01 secs, 98,880 bytes) -- λ> let xs = [1..22] in existeSubSuma5 xs (sum xs) -- True -- (0.02 secs, 2,827,560 bytes) -- λ> let xs = [1..200] in existeSubSuma2 xs (sum xs) -- True -- (0.01 secs, 182,280 bytes) -- λ> let xs = [1..200] in existeSubSuma3 xs (sum xs) -- True -- (8.89 secs, 1,875,071,968 bytes) -- λ> let xs = [1..200] in existeSubSuma4 xs (sum xs) -- True -- (0.02 secs, 217,128 bytes) -- λ> let xs = [1..200] in existeSubSuma5 xs (sum xs) -- True -- (8.66 secs, 1,875,087,976 bytes) -- -- λ> and [existeSubSuma2 [1..20] n | n <- [1..sum [1..20]]] -- True -- (2.82 secs, 323,372,512 bytes) -- λ> and [existeSubSuma3 [1..20] n | n <- [1..sum [1..20]]] -- True -- (0.65 secs, 221,806,376 bytes) -- λ> and [existeSubSuma4 [1..20] n | n <- [1..sum [1..20]]] -- True -- (4.12 secs, 535,153,152 bytes) -- λ> and [existeSubSuma5 [1..20] n | n <- [1..sum [1..20]]] -- True -- (0.73 secs, 238,579,696 bytes) |