{"id":4315,"date":"2018-11-21T06:00:56","date_gmt":"2018-11-21T04:00:56","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=4315"},"modified":"2019-01-19T12:22:33","modified_gmt":"2019-01-19T10:22:33","slug":"reconocimiento-de-particiones","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/reconocimiento-de-particiones\/","title":{"rendered":"Reconocimiento de particiones"},"content":{"rendered":"<p>Una <a href=\"http:\/\/bit.ly\/2Dw2GB4\">partici\u00f3n<\/a> de un conjunto es una divisi\u00f3n del mismo en subconjuntos disjuntos no vac\u00edos.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   esParticion :: Eq a => [[a]] -> Bool\n<\/pre>\n<p>tal que (esParticion xss) se verifica si xss es una partici\u00f3n; es decir sus elementos son listas no vac\u00edas disjuntas. Por ejemplo.<\/p>\n<pre lang=\"text\">\n   esParticion [[1,3],[2],[9,5,7]]  ==  True\n   esParticion [[1,3],[2],[9,5,1]]  ==  False\n   esParticion [[1,3],[],[9,5,7]]   ==  False\n   esParticion [[2,3,2],[4]]        ==  True\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List ((\\\\), intersect)\n\n-- 1\u00aa definici\u00f3n\n-- =============\n\nesParticion :: Eq a => [[a]] -> Bool\nesParticion xss =\n  [] `notElem` xss &&\n  and [disjuntos xs ys | xs <- xss, ys <- xss \\\\ [xs]] \n\ndisjuntos :: Eq a => [a] -> [a] -> Bool\ndisjuntos xs ys = null (xs `intersect` ys)\n\n-- 2\u00aa definici\u00f3n\n-- =============\n\nesParticion2 :: Eq a => [[a]] -> Bool\nesParticion2 []       = True\nesParticion2 (xs:xss) =\n  not (null xs) &&\n  and [disjuntos xs ys | ys <- xss] &#038;&#038;\n  esParticion2 xss\n\n-- 3\u00aa definici\u00f3n\n-- =============\n\nesParticion3 :: Eq a => [[a]] -> Bool\nesParticion3 []       = True\nesParticion3 (xs:xss) =\n  not (null xs) &&\n  all (disjuntos xs) xss &&\n  esParticion3 xss\n\n-- Equivalencia\nprop_equiv :: [[Int]] -> Bool\nprop_equiv xss =\n  and [esParticion xss == f xss | f <- [ esParticion2\n                                       , esParticion3]]\n\n-- Comprobaci\u00f3n\n--    \u03bb> quickCheck prop_equiv\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia:\n--    \u03bb> esParticion [[x] | x <- [1..3000]]\n--    True\n--    (4.37 secs, 3,527,956,400 bytes)\n--    \u03bb> esParticion2 [[x] | x <- [1..3000]]\n--    True\n--    (1.26 secs, 1,045,792,552 bytes)\n--    \u03bb> esParticion3 [[x] | x <- [1..3000]]\n--    True\n--    (1.30 secs, 1,045,795,272 bytes)\n--    \u03bb> esParticion3 [[x] | x <- [1..3000]]\n--    True\n--    (1.30 secs, 1,045,795,272 bytes)\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nSent\u00eda los cuatro vientos,<br \/>\nen la encrucijada<br \/>\nde su pensamiento.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Una partici\u00f3n de un conjunto es una divisi\u00f3n del mismo en subconjuntos disjuntos no vac\u00edos. Definir la funci\u00f3n esParticion :: Eq a => [[a]] -> Bool tal que (esParticion xss) se verifica si xss es una partici\u00f3n; es decir sus elementos son listas no vac\u00edas disjuntas. Por ejemplo. esParticion [[1,3],[2],[9,5,7]] == True esParticion [[1,3],[2],[9,5,1]] ==&#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":[5],"tags":[41,100,8,270,181,27,141,11,6],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4315"}],"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=4315"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4315\/revisions"}],"predecessor-version":[{"id":4596,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/4315\/revisions\/4596"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=4315"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=4315"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=4315"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}