{"id":4878,"date":"2019-03-28T08:58:23","date_gmt":"2019-03-28T06:58:23","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4878"},"modified":"2019-04-04T06:51:27","modified_gmt":"2019-04-04T04:51:27","slug":"subconjuntos-divisibles","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/subconjuntos-divisibles\/","title":{"rendered":"Subconjuntos divisibles"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\"> \n  subconjuntosDivisibles :: [Int] -> [[Int]]\n<\/pre>\n<p>tal que (subconjuntosDivisibles xs) es la lista de todos los subconjuntos de xs en los que todos los elementos tienen un factor  com\u00fan mayor que 1. Por ejemplo,<\/p>\n<pre lang=\"text\"> \n  subconjuntosDivisibles []         ==  [[]]\n  subconjuntosDivisibles [1]        ==  [[]]\n  subconjuntosDivisibles [3]        ==  [[3],[]]\n  subconjuntosDivisibles [1,3]      ==  [[3],[]]\n  subconjuntosDivisibles [3,6]      ==  [[3,6],[3],[6],[]]\n  subconjuntosDivisibles [1,3,6]    ==  [[3,6],[3],[6],[]]\n  subconjuntosDivisibles [2,3,6]    ==  [[2,6],[2],[3,6],[3],[6],[]]\n  subconjuntosDivisibles [2,3,6,8]  ==  [[2,6,8],[2,6],[2,8],[2],[3,6],[3],[6,8],[6],[8],[]]\n  length (subconjuntosDivisibles [1..10])  ==  41\n  length (subconjuntosDivisibles [1..20])  ==  1097\n  length (subconjuntosDivisibles [1..30])  ==  33833\n  length (subconjuntosDivisibles [1..40])  ==  1056986\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (foldl1', subsequences)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsubconjuntosDivisibles :: [Int] -> [[Int]]\nsubconjuntosDivisibles xs = filter esDivisible (subsequences xs)\n\n-- (esDivisible xs) se verifica si todos los elementos de xs tienen un\n-- factor com\u00fan mayor que 1. Por ejemplo,\n--    esDivisible [6,10,22]  ==  True\n--    esDivisible [6,10,23]  ==  False\nesDivisible :: [Int] -> Bool\nesDivisible [] = True\nesDivisible xs = mcd xs > 1\n\n-- (mcd xs) es el m\u00e1ximo com\u00fan divisor de xs. Por ejemplo,\n--    mcd [6,10,22]  ==  2\n--    mcd [6,10,23]  ==  1\nmcd :: [Int] -> Int\nmcd = foldl1' gcd\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nsubconjuntosDivisibles2 :: [Int] -> [[Int]]\nsubconjuntosDivisibles2 []     = [[]]\nsubconjuntosDivisibles2 (x:xs) = [x:ys | ys <- yss, esDivisible (x:ys)] ++ yss\n  where yss = subconjuntosDivisibles2 xs\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nsubconjuntosDivisibles3 :: [Int] -> [[Int]]\nsubconjuntosDivisibles3 []     = [[]]\nsubconjuntosDivisibles3 (x:xs) = filter esDivisible (map (x:) yss) ++ yss\n  where yss = subconjuntosDivisibles3 xs\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> length (subconjuntosDivisibles [1..21])\n--    1164\n--    (3.83 secs, 5,750,416,768 bytes)\n--    \u03bb> length (subconjuntosDivisibles2 [1..21])\n--    1164\n--    (0.01 secs, 5,400,232 bytes)\n--    \u03bb> length (subconjuntosDivisibles3 [1..21])\n--    1164\n--    (0.01 secs, 5,264,928 bytes)\n--    \n--    \u03bb> length (subconjuntosDivisibles2 [1..40])\n--    1056986\n--    (6.95 secs, 8,845,664,672 bytes)\n--    \u03bb> length (subconjuntosDivisibles3 [1..40])\n--    1056986\n--    (6.74 secs, 8,727,141,792 bytes)\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nAbejas, cantores,<br \/>\nno a la miel, sino a las flores.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n subconjuntosDivisibles :: [Int] -> [[Int]] tal que (subconjuntosDivisibles xs) es la lista de todos los subconjuntos de xs en los que todos los elementos tienen un factor com\u00fan mayor que 1. Por ejemplo, subconjuntosDivisibles [] == [[]] subconjuntosDivisibles [1] == [[]] subconjuntosDivisibles [3] == [[3],[]] subconjuntosDivisibles [1,3] == [[3],[]] subconjuntosDivisibles [3,6] ==&#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":[7],"tags":[8,38,359,10,11,6,88],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4878"}],"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=4878"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4878\/revisions"}],"predecessor-version":[{"id":4915,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4878\/revisions\/4915"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4878"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4878"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4878"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}