{"id":2666,"date":"2016-12-08T06:00:25","date_gmt":"2016-12-08T04:00:25","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=2666"},"modified":"2016-12-15T08:40:00","modified_gmt":"2016-12-15T06:40:00","slug":"suma-de-los-maximos-de-los-subconjuntos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/suma-de-los-maximos-de-los-subconjuntos\/","title":{"rendered":"Suma de los m\u00e1ximos de los subconjuntos"},"content":{"rendered":"<p>Los subconjuntos distinto del vac\u00edo del conjunto {3, 2, 5}, junto con sus  m\u00e1ximos elementos, son<\/p>\n<pre lang=\"text\">\n   {3}       su m\u00e1ximo es 3\n   {2}       su m\u00e1ximo es 2\n   {5}       su m\u00e1ximo es 5\n   {3, 2}    su m\u00e1ximo es 3\n   {3, 5}    su m\u00e1ximo es 5\n   {2, 5}    su m\u00e1ximo es 5\n   {3, 2, 5} su m\u00e1ximo es 5\n<\/pre>\n<p>Por tanto, la suma de los m\u00e1ximos elementos de los subconjuntos de {3, 2, 5} es 3 + 2 + 5 + 3 + 5 + 5 + 5 = 28.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   sumaMaximos :: [Integer] -> Integer\n<\/pre>\n<p>tal que (sumaMaximos xs) es la suma de los m\u00e1ximos elementos de los subconjuntos de xs. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   sumaMaximos [3,2,5]    ==  28\n   sumaMaximos [4,1,6,3]  ==  71\n   sumaMaximos [1..100]   ==  125497409422594710748173617332225\n   length (show (sumaMaximos [1..10^5]))  ==  30108\n   sumaMaximos [1..10^5] `mod` (10^7)     ==  4490625\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (sort, subsequences)\n\n-- 1\u00aa definici\u00f3n\nsumaMaximos :: [Integer] -> Integer\nsumaMaximos xs =\n  sum [maximum ys | ys <- tail (subsequences xs)]\n\n-- 2\u00aa definici\u00f3n\nsumaMaximos2 :: [Integer] -> Integer\nsumaMaximos2 =\n  sum . map maximum . tail . subsequences\n\n-- 3\u00aa definici\u00f3n\nsumaMaximos3 :: [Integer] -> Integer\nsumaMaximos3 xs =\n  sum (zipWith (*) (sort xs) potenciasDeDos)\n\npotenciasDeDos :: [Integer]\npotenciasDeDos = iterate (*2) 1\n\n-- Comparaci\u00f3n de eficiencia\n--    \u03bb> sumaMaximos [1..20]\n--    19922945\n--    (2.74 secs, 703,957,848 bytes)\n--    \u03bb> sumaMaximos2 [1..20]\n--    19922945\n--    (2.06 secs, 680,728,696 bytes)\n--    \u03bb> sumaMaximos3 [1..20]\n--    19922945\n--    (0.01 secs, 0 bytes)\n<\/pre>\n<p>Basado en el art\u00edculo<br \/>\n<a href=\"http:\/\/www.geeksforgeeks.org\/sum-maximum-elements-subsets\/\">Sum of maximum elements of all subsets<\/a><br \/>\nde Utkarsh Trivedi en <a href=\"http:\/\/www.geeksforgeeks.org\">GeeksforGeeks<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Los subconjuntos distinto del vac\u00edo del conjunto {3, 2, 5}, junto con sus m\u00e1ximos elementos, son {3} su m\u00e1ximo es 3 {2} su m\u00e1ximo es 2 {5} su m\u00e1ximo es 5 {3, 2} su m\u00e1ximo es 3 {3, 5} su m\u00e1ximo es 5 {2, 5} su m\u00e1ximo es 5 {3, 2, 5} su m\u00e1ximo es&#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":[4],"tags":[8,50,15,11,14,88,40,45,76],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2666"}],"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=2666"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2666\/revisions"}],"predecessor-version":[{"id":2702,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2666\/revisions\/2702"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2666"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2666"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2666"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}