{"id":265,"date":"2009-11-12T12:11:55","date_gmt":"2009-11-12T12:11:55","guid":{"rendered":"http:\/\/elnostreraco.com\/blog\/?p=265"},"modified":"2014-03-23T14:51:33","modified_gmt":"2014-03-23T14:51:33","slug":"complexitat-problemes-computacio","status":"publish","type":"post","link":"https:\/\/elnostreraco.com\/blog\/complexitat-problemes-computacio\/","title":{"rendered":"Problemes que una m\u00e0quina no pot resoldre (P \u2260 NP => NP-Complets)"},"content":{"rendered":"<p>El 10 de novembre vaig llan\u00e7ar la pregunta <a href=\"http:\/\/postly.elnostreraco.com\/6-invitacions-a-google-wave-al-respondre-2-pr\">&#8220;Que vol dir complexitat P i NP?&#8221;<\/a> per als que volguessin invitacions a <a href=\"http:\/\/elnostreraco.com\/blog\/manual-google-wave-colaboratiu\/\">google wave<\/a>.<\/p>\n<p>La teoria de <a href=\"http:\/\/ca.wikipedia.org\/wiki\/Complexitat_computacional\">complexitat computacional<\/a> forma part de la teoria de la computaci\u00f3, que estudia els recursos necessaris durant el c\u00e0lcul per resoldre un problema donat. Els recursos m\u00e9s habituals s\u00f3n <strong>temps<\/strong> (quantes instruccions calen per resoldre el problema) i <strong>espai<\/strong> (quanta capacitat es necessita per resoldre el problema).<\/p>\n<blockquote><p>Un problema de molta complexitat de computaci\u00f3 i que s&#8217;ha estudiat molt la seva optimitzaci\u00f3 \u00e9s el del <a href=\"http:\/\/ca.wikipedia.org\/wiki\/Problema_del_viatjant_de_comer%C3%A7\">viatjant de comer\u00e7<\/a><\/p>\n<p><em>Donat un conjunt de llocs o ciutats, es tracta de trobar l&#8217;ordre a seguir per tal de tal que el cam\u00ed fet pel viatjant de comer\u00e7 des del punt de partida fins al punt d&#8217;arribada sigu el m\u00e9s curt possible.<\/em><\/p><\/blockquote>\n<p><a href=\"http:\/\/elnostreraco.com\/blog\/wp-content\/uploads\/2009\/11\/180px-Weighted_K4.svg_.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-890\" title=\"180px-Weighted_K4.svg\" src=\"http:\/\/elnostreraco.com\/blog\/wp-content\/uploads\/2009\/11\/180px-Weighted_K4.svg_.png\" alt=\"\" width=\"180\" height=\"150\" \/><\/a><\/p>\n<p>Un <strong>problema de decisi\u00f3<\/strong> \u00e9s un problema que agafa com a entrada una cadena i d\u00f3na com a sortida una resposta de tipus SI o NO. Si existeix un algorisme que \u00e9s capa\u00e7 de donar la resposta correcta per qualsevol cadena d&#8217;entrada de longitud n en com a m\u00e0xim c*nk passes, on k i c s\u00f3n constants independents de la longitud de la cadena, llavors diem que el problema es pot resoldre en temps polin\u00f2mic i per tant pertany a la classe P.<\/p>\n<p><strong>Exemples de c\u00e0lcul de complexitat<\/strong><\/p>\n<ul>\n<li>Extreure qualsevol element d&#8217;un vector. La indexaci\u00f3 d&#8217;un vector o array porta el mateix temps sigui quin sigui l&#8217;\u00edndex que es vulgui buscar. Per tant \u00e9s una operaci\u00f3 de complexitat <em>constant O(1).<\/em><\/li>\n<li>Buscar en un diccionari t\u00e9 una complexitat logar\u00edtmica. Es pot iniciar la cerca d&#8217;una paraula per la meitat del diccionari. Immediatament, se sap si s&#8217;ha trobat la paraula o, en cas contrari, en quina de les dues mitats s&#8217;ha de repetir el proc\u00e9s (\u00e9s un proc\u00e9s recursiu) fins a trobar el resultat. A cada (sub)cerca, el problema (les p\u00e0gines on pot estar la paraula) s&#8217;ha redu\u00eft a la mitat, lo que es correspon amb la funci\u00f3 logar\u00edtmica. Aquest procediment de cerca (conegut com cerca binaria en una estructura ordenada t\u00e9 una complexitat<em> logar\u00edtmica O(ln n).<\/em><\/li>\n<li>El proc\u00e9s m\u00e9s habitual per ordenar conjunts d&#8217;elements t\u00e9 complexitat quadr\u00e0tica. El procediment consisteix a crear una col\u00b7lecci\u00f3 buida d&#8217;elements. Se li afegeix, en ordre, el menor element del conjunt original que encara no s&#8217;hagi triat, el que implica fer un recorregut complet del conjunt original (<em>O(n), essent n el nombre d&#8217;elements del conjunt)<\/em>. Aquest recorregut sobre el conjunt original es realitza fins que tots els seus elements estan a la seq\u00fc\u00e8ncia de resultat. Es pot veure que s&#8217;ha de fer n seleccions (s&#8217;ordena tot el conjunt) cada una amb un cost n d&#8217;execuci\u00f3: el procediment \u00e9s d&#8217;orde<em> quadr\u00e0tica O(n\u00b2).<\/em> De tota manera hi ha diversos algorismes d&#8217;ordenaci\u00f3 amb millors resultats.<\/li>\n<\/ul>\n<p><strong>Turing va construir un model formal de computador, la<\/strong><strong> m\u00e0quina de Turing, i va demostrar que existien problemes que una m\u00e0quina no podia resoldre.<\/strong><\/p>\n<p>La m\u00e0quina de Turing (MTD) consta d&#8217;un cap\u00e7al lector\/escriptor i una cinta infinita en la que el cap\u00e7al llegix el contingut, esborra el contingut anterior i escriu un nou valor. Les operacions que es poden realitzar en aquesta m\u00e0quina es limiten a:\u00a0 avan\u00e7ar el cap\u00e7al lector\/escriptor cap a la dreta o cap a l&#8217;esquerra;<br \/>\nEl c\u00f2mput \u00e9s determinat a partir d&#8217;una taula d&#8217;estats de la forma:<\/p>\n<p style=\"text-align: center;\">(estat, valor) =&gt; (nou estat, nou valor, direcci\u00f3)<\/p>\n<p style=\"text-align: left;\"><strong>La classe P <\/strong>consisteix en els problemes en que les seves solucions poden ser trobades en un temps polin\u00f2mic en una m\u00e0quina de Turing.<\/p>\n<p>On una m\u00e0quina de Turing t\u00e9 un sol &#8220;cam\u00ed de computaci\u00f3&#8221; a seguir, una <strong>m\u00e0quina no determinista t\u00e9 un &#8220;arbre de computaci\u00f3&#8221;<\/strong>. En les m\u00e0quines no deterministes la funci\u00f3 de transici\u00f3 \u00e9s una funci\u00f3 multivaluada sobre els estats i els s\u00edmbols. Aqu\u00ed caldria imaginar-se que la m\u00e0quina &#8220;es divideix&#8221; en diverses c\u00f2pies, i cada una segueix una de les possibles transicions.<\/p>\n<p><strong>La classe NP<\/strong> consisteix en els problemes en que les seves solucions poden ser trobades en un temps polin\u00f2mic en una m\u00e0quina no determinista, i per els quals la seva soluci\u00f3 positiva es pot verificar en relaci\u00f3 a la mida de l&#8217;entrada del problema en una m\u00e0quina de Turing.<\/p>\n<p>La dificultat dels problemes NP \u00e9s trobar la resposta de forma eficient, donada una forma eficient (en temps polin\u00f2mic) de verificar-la un cop trobada.<\/p>\n<p><strong>Es creu que P \u2260 NP<\/strong><br \/>\nEn inform\u00e0tica es parteix de la base que P \u2260 NP, ja que no s&#8217;ha trobat un algorisme de temps polinomial per un problema NP-complet. A dem\u00e9s si P = NP implicaria conseq\u00fc\u00e8ncies que es creuen falses. Intenta demostrar que P \u00e9s igual a NP. L&#8217;institut Clay et donaria 1 mil\u00b7li\u00f3 de d\u00f2lars.<\/p>\n<p><strong>Problemes &#8220;intractables&#8221; i NP-Complets<\/strong><br \/>\nTot i que el processament de les m\u00e0quines de Turing s\u00f3n m\u00e9s simples que les no Deterministes no vol dir que, per la mateixa entrada, un problema de la classe P tardi menys que un de la classe NP. Exemple: Un problema que requereix 10^1000n en temps i \u00e9s a la classe P \u00e9s polin\u00f2mic en el temps, per\u00f2 intractable a la pr\u00e0ctica. En canvi hi ha problemes que tot i tenir un temps en funci\u00f3 exponencial (NP) s\u00f3n tractables a la pr\u00e0ctica si l&#8217;entrada es prou petita.<\/p>\n<p><em>Els problemes que tenen solucions en temps polin\u00f2mics s\u00f3n solubles per m\u00e9s que uns quants valors. Els problemes poc escalables son els &#8220;EXPTIME-complet&#8221;. Assumint P \u2260 NP, els problemes NP-Complet s\u00f3n intractables.<\/em><\/p>\n<p><strong><br \/>\n<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>El 10 de novembre vaig llan\u00e7ar la pregunta &#8220;Que vol dir complexitat P i NP?&#8221; per als que volguessin invitacions a google wave. La teoria de complexitat computacional forma part de la teoria de la &hellip; <\/p>\n","protected":false},"author":1,"featured_media":890,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"class_list":["post-265","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-analytics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Problemes que una m\u00e0quina no pot resoldre (P \u2260 NP =&gt; NP-Complets) ~ Anal\u00edtica<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/elnostreraco.com\/blog\/complexitat-problemes-computacio\/\" \/>\n<meta property=\"og:locale\" content=\"ca_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Problemes que una m\u00e0quina no pot resoldre (P \u2260 NP =&gt; NP-Complets) ~ Anal\u00edtica\" \/>\n<meta property=\"og:description\" content=\"El 10 de novembre vaig llan\u00e7ar la pregunta &#8220;Que vol dir complexitat P i NP?&#8221; per als que volguessin invitacions a google wave. 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