{"id":6331,"date":"2021-05-03T06:00:19","date_gmt":"2021-05-03T04:00:19","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6331"},"modified":"2021-05-10T08:29:22","modified_gmt":"2021-05-10T06:29:22","slug":"ordenacion-de-los-pares-de-enteros","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/ordenacion-de-los-pares-de-enteros\/","title":{"rendered":"Ordenaci\u00f3n de los pares de enteros"},"content":{"rendered":"<p>Los pares de n\u00fameros enteros se pueden ordenar por la suma de los valores absolutos de sus componentes. Los primeros pares en dicha ordenaci\u00f3n son<\/p>\n<pre lang=\"text\">\n   (0,0),\n   (-1,0),(0,-1),(0,1),(1,0),\n   (-2,0),(-1,-1),(-1,1),(0,-2),(0,2),(1,-1),(1,1),(2,0), ...\n<\/pre>\n<p>Definir la lista<\/p>\n<pre lang=\"text\">\n   pares :: [(Integer, Integer)]\n<\/pre>\n<p>cuyos elementos son los pares de n\u00fameros enteros con la ordenaci\u00f3n anterior. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> take 38 pares\n   [(0,0),(-1,0),(0,-1),(0,1),(1,0),(-2,0),(-1,-1),(-1,1),(0,-2),\n    (0,2),(1,-1),(1,1),(2,0),(-3,0),(-2,-1),(-2,1),(-1,-2),(-1,2),\n    (0,-3),(0,3),(1,-2),(1,2),(2,-1),(2,1),(3,0),(-4,0),(-3,-1),\n    (-3,1),(-2,-2),(-2,2),(-1,-3),(-1,3),(0,-4),(0,4),(1,-3),(1,3),\n    (2,-2),(2,2)]\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (sort)\nimport Test.QuickCheck (Property, (==>), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\npares :: [(Integer, Integer)]\npares =\n  concat [sort (concat [aux (x,n-x) | x <- [0..n]])\n         | n <- [0..]]\n  where\n    aux (0,0) = [(0,0)]\n    aux (0,y) = [(0,y), (0,-y)]\n    aux (x,0) = [(x,0), (-x,0)]\n    aux (x,y) = [(x,y), (-x,y), (x,-y), (-x,-y)]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\npares2 :: [(Integer,Integer)]\npares2 = concatMap paresEnterosSuma [0..]\n\n-- (paresEnterosSuma n) es la lista de pares de enteros (x,y) tales que la suma\n-- de los valores absolutos de x e y es igual a n. Por ejemplo,\n--    \u03bb> paresEnterosSuma 3\n--    [(-3,0),(-2,-1),(-2,1),(-1,-2),(-1,2),(0,-3),(0,3),(1,-2),(1,2),\n--     (2,-1),(2,1),(3,0)]\nparesEnterosSuma :: Integer -> [(Integer,Integer)]\nparesEnterosSuma n = concatMap aux [-n..n]\n  where aux k | m == 0    = [(k,0)]\n              | otherwise = [(k,-m),(k,m)]\n          where m = n - abs k\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_pares :: Int -> Property\nprop_pares n =\n  n >= 0 ==>\n  pares !! n == pares2 !! n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_pares\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> pares !! (10^7)\n--    (304,-1932)\n--    (9.15 secs, 11,031,475,520 bytes)\n--    \u03bb> pares2 !! (10^7)\n--    (304,-1932)\n--    (2.84 secs, 3,401,450,320 bytes)\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Los pares de n\u00fameros enteros se pueden ordenar por la suma de los valores absolutos de sus componentes. Los primeros pares en dicha ordenaci\u00f3n son (0,0), (-1,0),(0,-1),(0,1),(1,0), (-2,0),(-1,-1),(-1,1),(0,-2),(0,2),(1,-1),(1,1),(2,0), &#8230; Definir la lista pares :: [(Integer, Integer)] cuyos elementos son los pares de n\u00fameros enteros con la ordenaci\u00f3n anterior. Por ejemplo, \u03bb> take 38 pares [(0,0),(-1,0),(0,-1),(0,1),(1,0),(-2,0),(-1,-1),(-1,1),(0,-2),&#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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6331"}],"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=6331"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6331\/revisions"}],"predecessor-version":[{"id":6440,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6331\/revisions\/6440"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6331"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6331"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}