{"id":2662,"date":"2016-12-07T06:00:00","date_gmt":"2016-12-07T04:00:00","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=2662"},"modified":"2016-12-14T07:12:27","modified_gmt":"2016-12-14T05:12:27","slug":"elementos-que-respetan-la-ordenacion","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/elementos-que-respetan-la-ordenacion\/","title":{"rendered":"Elementos que respetan la ordenaci\u00f3n"},"content":{"rendered":"<p>Se dice que un elemento x de una lista xs respeta la ordenaci\u00f3n si x es mayor o igual que todos lo que tiene delante en xs y es menor o igual que todos lo que tiene detr\u00e1s en xs. Por ejemplo, en la lista lista [3,2,1,4,6,5,7,9,8] el n\u00famero 4 respeta la ordenaci\u00f3n pero el n\u00famero 5 no la respeta (porque es mayor que el 6 que est\u00e1 delante).<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   respetuosos :: Ord a => [a] -> [a]\n<\/pre>\n<p>tal que (respetuosos xs) es la lista de los elementos de xs que respetan la ordenaci\u00f3n. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   respetuosos [3,2,1,4,6,4,7,9,8]  ==  [4,7]\n   respetuosos [2,1,3,4,6,4,7,8,9]  ==  [3,4,7,8,9]\n   respetuosos \"abaco\"              ==  \"aco\"\n   respetuosos \"amor\"               ==  \"amor\"\n   respetuosos \"romanos\"            ==  \"s\"\n   respetuosos [1..9]               ==  [1,2,3,4,5,6,7,8,9]\n   respetuosos [9,8..1]             ==  []\n<\/pre>\n<p>Comprobar con QuickCheck que para cualquier lista de enteros xs se verifican las siguientes propiedades:<\/p>\n<ul>\n<li>todos los elementos de (sort xs) respetan la ordenaci\u00f3n y<\/li>\n<li>en la lista (nub (reverse (sort xs))) hay como m\u00e1ximo un elemento que respeta la ordenaci\u00f3n.<\/li>\n<\/ul>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (inits, nub, sort, tails)\nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n (por comprensi\u00f3n):\nrespetuosos :: Ord a => [a] -> [a]\nrespetuosos xs =\n  [z | k <- [0..n-1]\n     , let (ys,z:zs) = splitAt k xs\n     , all (<=z) ys\n     , all (>=z) zs]\n  where n = length xs\n\n-- 2\u00aa definici\u00f3n (por recursi\u00f3n):\nrespetuosos2 :: Ord a => [a] -> [a]\nrespetuosos2 xs = aux [] [] xs\n  where aux zs _  []      = reverse zs\n        aux zs ys (x:xs)\n          | all (<=x) ys &#038;&#038; all (>=x) xs = aux (x:zs) (x:ys) xs\n          | otherwise                    = aux zs     (x:ys) xs\n\n-- 3\u00aa definici\u00f3n\nrespetuosos3 :: Ord a => [a] -> [a]\nrespetuosos3 xs = [ x | (ys,x,zs) <- zip3 (inits xs) xs (tails xs)\n                      , all (<=x) ys\n                      , all (x<=) zs ]\n\n-- 4\u00aa soluci\u00f3n\nrespetuosos4 :: Ord a =>[a] ->[a]\nrespetuosos4 xs =\n  [x | (a, x, b) <- zip3 (scanl1 max xs) xs (scanr1 min xs)\n     , a <= x &#038;&#038; x <= b]\n\n-- Comparaci\u00f3n de eficiencia\n--    \u03bb> length (respetuosos [1..3000])\n--    3000\n--    (3.31 secs, 1,140,407,224 bytes)\n--    \u03bb> length (respetuosos2 [1..3000])\n--    3000\n--    (2.85 secs, 587,082,160 bytes)\n--    \u03bb> length (respetuosos3 [1..3000])\n--    3000\n--    (2.12 secs, 785,446,880 bytes)\n--    \u03bb> length (respetuosos4 [1..3000])\n--    3000\n--    (0.02 secs, 0 bytes)\n\n-- 1\u00aa propiedad\nprop_respetuosos1 :: [Int] -> Bool\nprop_respetuosos1 xs =\n  respetuosos (sort xs) == sort xs\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_respetuosos1\n--    +++ OK, passed 100 tests.\n\n-- La 2\u00aa propiedad\nprop_respetuosos2 :: [Int] -> Bool\nprop_respetuosos2 xs =\n  length (respetuosos (nub (reverse (sort xs)))) <= 1\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_respetuosos2\n--    +++ OK, passed 100 tests.\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Se dice que un elemento x de una lista xs respeta la ordenaci\u00f3n si x es mayor o igual que todos lo que tiene delante en xs y es menor o igual que todos lo que tiene detr\u00e1s en xs. Por ejemplo, en la lista lista [3,2,1,4,6,5,7,9,8] el n\u00famero 4 respeta la ordenaci\u00f3n pero el&#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":[41,8,28,11,6,32,73,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2662"}],"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=2662"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2662\/revisions"}],"predecessor-version":[{"id":2696,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2662\/revisions\/2696"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2662"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2662"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2662"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}