{"id":7067,"date":"2022-06-03T06:00:13","date_gmt":"2022-06-03T04:00:13","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7067"},"modified":"2022-06-05T17:16:58","modified_gmt":"2022-06-05T15:16:58","slug":"descomposiciones-con-sumandos-1-o-2","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/descomposiciones-con-sumandos-1-o-2\/","title":{"rendered":"Descomposiciones con sumandos 1 \u00f3 2"},"content":{"rendered":"<p>Definir la funciones<\/p>\n<pre lang=\"text\">\n   sumas  :: Int -> [[Int]]\n   nSumas :: Int -> Integer\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li><code>(sumas n)<\/code> es la lista de las descomposiciones de <code>n<\/code> como sumas cuyos sumandos son 1 \u00f3 2. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n      sumas 1            ==  [[1]]\n      sumas 2            ==  [[1,1],[2]]\n      sumas 3            ==  [[1,1,1],[1,2],[2,1]]\n      sumas 4            ==  [[1,1,1,1],[1,1,2],[1,2,1],[2,1,1],[2,2]]\n      length (sumas 26)  ==  196418\n      length (sumas 33)  ==  5702887\n<\/pre>\n<ul>\n<li><code>(nSumas n)<\/code> es el n\u00famero de descomposiciones de <code>n<\/code> como sumas cuyos sumandos son 1 \u00f3 2. Por ejemplo, <\/li>\n<\/ul>\n<pre lang=\"text\">\n      nSumas 4                      ==  5\n      nSumas 123                    ==  36726740705505779255899443\n      length (show (nSumas 123456)) ==  25801\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List  (genericIndex, genericLength)\nimport Data.Array ((!), array)\nimport Test.QuickCheck (Positive(Positive), quickCheckWith)\n\n-- 1\u00aa soluci\u00f3n de sumas\n-- ====================\n\nsumas1 :: Int -> [[Int]]\nsumas1 0 = [[]]\nsumas1 1 = [[1]]\nsumas1 n = [1:xs | xs <- sumas1 (n-1)] ++ [2:xs | xs <- sumas1 (n-2)]\n\n-- 2\u00aa soluci\u00f3n de sumas\n-- ====================\n\nsumas2 :: Int -> [[Int]]\nsumas2 0 = [[]]\nsumas2 1 = [[1]]\nsumas2 n = map (1:) (sumas2 (n-1)) ++ map (2:) (sumas2 (n-2))\n\n-- 3\u00aa soluci\u00f3n de sumas\n-- ====================\n\nsumas3 :: Int -> [[Int]]\nsumas3 n = v ! n\n  where v = array (0,n) [(i, f i) | i <- [0..n]]\n        f 0 = [[]]\n        f 1 = [[1]]\n        f k = map (1:) (v!(k-1)) ++ map (2:) (v!(k-2))\n \n-- 4\u00aa soluci\u00f3n de sumas\n-- ====================\n\nsumas4 :: Int -> [[Int]]\nsumas4 n = sucSumas !! n\n\n-- sucSumas es la sucesi\u00f3n cuyo n-\u00e9simo elemento es la lista de las\n-- descomposiciones de n como sumas cuyos sumandos son 1 \u00f3 2. Por\n-- ejemplo,\n--    \u03bb> take 4 sucSumas\n--    [[[]],[[1]],[[1,1],[2]],[[1,1,1],[1,2],[2,1]]]\n--    \u03bb> mapM_ print (take 5 sucSumas)\n--    [[]]\n--    [[1]]\n--    [[1,1],[2]]\n--    [[1,1,1],[1,2],[2,1]]\n--    [[1,1,1,1],[1,1,2],[1,2,1],[2,1,1],[2,2]]\nsucSumas :: [[[Int]]]\nsucSumas = [[]] : [[1]] : zipWith f (tail sucSumas) sucSumas\n  where f xs ys = map (1:) xs ++ map (2:) ys\n\n-- Comprobaci\u00f3n de equivalencia de sumas\n-- =====================================\n\n-- La propiedad es\nprop_sumas :: Positive Int -> Bool\nprop_sumas (Positive n) =\n  all (== sumas1 n)\n      [sumas2 n,\n       sumas3 n,\n       sumas4 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheckWith (stdArgs {maxSize=20}) prop_sumas\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia de sumas\n-- ==================================\n\n-- La comparaci\u00f3n es\n--    \u03bb> length (sumas1 28)\n--    514229\n--    (2.79 secs, 1,739,784,512 bytes)\n--    \u03bb> length (sumas2 28)\n--    514229\n--    (1.33 secs, 1,512,291,248 bytes)\n--    \u03bb> length (sumas3 28)\n--    514229\n--    (0.20 secs, 165,215,800 bytes)\n--    \u03bb> length (sumas4 28)\n--    514229\n--    (0.17 secs, 165,201,592 bytes)\n--\n--    \u03bb> length (sumas3 33)\n--    5702887\n--    (2.16 secs, 1,830,761,864 bytes)\n--    \u03bb> length (sumas4 33)\n--    5702887\n--    (1.44 secs, 1,830,749,832 bytes)\n\n-- Definici\u00f3n de sumas\n-- ===================\n\n-- La cuarta soluci\u00f3n es m\u00e1s eficiente y es la que usaremos en lo\n-- sucesivo:\nsumas :: Int -> [[Int]]\nsumas = sumas4\n\n-- 1\u00aa soluci\u00f3n de nSumas\n-- =====================\n\nnSumas1 :: Int -> Integer\nnSumas1 = genericLength . sumas2\n\n-- 2\u00aa soluci\u00f3n de nSumas\n-- =====================\n\nnSumas2 :: Int -> Integer\nnSumas2 0 = 1\nnSumas2 1 = 1\nnSumas2 n = nSumas2 (n-1) + nSumas2 (n-2)\n\n-- 3\u00aa soluci\u00f3n de nSumas\n-- =====================\n\nnSumas3 :: Int -> Integer\nnSumas3 n = v ! n\n  where v = array (0,n) [(i,f i) | i <- [0..n]]\n        f 0 = 1\n        f 1 = 1\n        f k = v ! (k-1) + v ! (k-2)\n\n-- 4\u00aa soluci\u00f3n de nSumas\n-- =====================\n\nnSumas4 :: Int -> Integer\nnSumas4 n = aux `genericIndex` n\n  where aux = 1 : 1 : zipWith (+) aux (tail aux) \n\n-- Comprobaci\u00f3n de equivalencia de nSumas\n-- ======================================\n\n-- La propiedad es\nprop_nSumas :: Positive Int -> Bool\nprop_nSumas (Positive n) =\n  all (== nSumas1 n)\n      [nSumas2 n,\n       nSumas3 n,\n       nSumas4 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheckWith (stdArgs {maxSize=20}) prop_nSumas\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia de nSumas\n-- ===================================\n\n-- La comparaci\u00f3n es\n--    \u03bb> nSumas1 33\n--    5702887\n--    (17.32 secs, 23,140,562,600 bytes)\n--    \u03bb> nSumas2 33\n--    5702887\n--    (3.48 secs, 1,870,676,904 bytes)\n--    \u03bb> nSumas3 33\n--    5702887\n--    (0.00 secs, 152,960 bytes)\n--    \u03bb> nSumas4 33\n--    5702887\n--    (0.00 secs, 139,456 bytes)\n--    \n--    \u03bb> length (show (nSumas3 (2*10^5)))\n--    41798\n--    (1.41 secs, 1,895,295,528 bytes)\n--    \u03bb> length (show (nSumas4 (2*10^5)))\n--    41798\n--    (2.39 secs, 1,834,998,800 bytes)\n\n-- Nota. El valor de (nSumas n) es el n-\u00e9simo t\u00e9rmino de la sucesi\u00f3n de\n-- Fibonacci 1, 1, 2, 3, 5, 8, ...\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Descomposiciones_con_sumandos_1_o_2.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funciones sumas :: Int -> [[Int]] nSumas :: Int -> Integer tales que (sumas n) es la lista de las descomposiciones de n como sumas cuyos sumandos son 1 \u00f3 2. Por ejemplo, sumas 1 == [[1]] sumas 2 == [[1,1],[2]] sumas 3 == [[1,1,1],[1,2],[2,1]] sumas 4 == [[1,1,1,1],[1,1,2],[1,2,1],[2,1,1],[2,2]] length (sumas 26) ==&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[2],"tags":[572],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7067"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=7067"}],"version-history":[{"count":2,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7067\/revisions"}],"predecessor-version":[{"id":7078,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7067\/revisions\/7078"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7067"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7067"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7067"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}