{"id":429,"date":"2016-01-26T17:18:39","date_gmt":"2016-01-26T16:18:39","guid":{"rendered":"http:\/\/smart--grid.net\/?page_id=429"},"modified":"2022-12-03T22:57:05","modified_gmt":"2022-12-03T21:57:05","slug":"algorithme-de-dijkstra","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/es\/busqueda-de-ruta-de-teoria-de-grafos\/algoritmo-de-dijkstra\/","title":{"rendered":"Algoritmo de Dijkstra"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"429\" class=\"elementor elementor-429\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5f5a28e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5f5a28e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column 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class=\"ez-toc-section\" id=\"Algorithme-de-Dijkstra\"><\/span>Algorithme de Dijkstra<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"color: #000000;\">E. W. Dijkstra (1930-2002) a propos\u00e9 en 1959 un <a href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/\">algorithme<\/a> (nomm\u00e9 algorithme de Dijkstra) qui permet de d\u00e9terminer le <a href=\"https:\/\/complex-systems-ai.com\/es\/busqueda-de-ruta-de-teoria-de-grafos\/\">plus court chemin<\/a> entre deux sommets d&rsquo;un <a href=\"https:\/\/complex-systems-ai.com\/es\/teoria-de-grafos\/\">graphe<\/a> connexe pond\u00e9r\u00e9. L&rsquo;algorithme de Dijkstra est bas\u00e9 sur l&rsquo;observation suivante : une fois que nous d\u00e9terminons le chemin le plus court vers un sommet v, alors les chemins qui vont de v \u00e0 chacun de ses sommets adjacents pourraient \u00eatre le plus court chemin vers chacun de ces sommets voisins. L&rsquo;algorithme de Dijkstra est un algorithme de <a href=\"https:\/\/complex-systems-ai.com\/es\/algoritmico\/programacion-dinamica-2\/\">programmation dynamique<\/a> glouton, il visite toutes les solutions possibles.<\/span><\/p>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\">\n<p><strong>Conditions<\/strong><\/p>\n<ul>\n<li>Pas de longueur n\u00e9gative<\/li>\n<li>Arc ou ar\u00eate<\/li>\n<li>Nombre de sommets fini<\/li>\n<li>Une source (et accessoirement une cible) d\u00e9finie<\/li>\n<\/ul>\n<\/div>\n\n<p>L&rsquo;algorithme de Dijkstra prend en entr\u00e9e un graphe orient\u00e9 pond\u00e9r\u00e9 par des r\u00e9els positifs et un sommet source. Il s&rsquo;agit de construire progressivement un sous-graphe dans lequel sont class\u00e9s les diff\u00e9rents sommets par ordre croissant de leur distance minimale au sommet de d\u00e9part. La distance correspond \u00e0 la somme des poids des ar\u00eates emprunt\u00e9es.<\/p>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\"><b>Sommet visit\u00e9 : <\/b>Un sommet pour lequel nous avons d\u00e9termin\u00e9 le chemin le plus court. Une fois que nous avons d\u00e9fini un sommet comme VISITE, cela est d\u00e9finitif, et nous ne reviendrons plus sur ce sommet.<\/div>\n\n<div style=\"padding: 5px; background-color: #d5edff; border: 2px solid #3c95e8; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\"><strong>Sommet marqu\u00e9\u00a0<\/strong>: Un sommet pour lequel un chemin a \u00e9t\u00e9 trouv\u00e9. Nous marquons ce sommet comme CANDIDAT pour le chemin le plus court.<\/div>\n\n<p>Au d\u00e9part, on consid\u00e8re que les distances de chaque sommet au sommet de d\u00e9part sont infinies sauf pour le sommet de d\u00e9part pour lequel la distance est de 0. Le sous-graphe de d\u00e9part est l&rsquo;ensemble vide.<\/p>\n\n<p>Au cours de chaque it\u00e9ration, on choisit en dehors du sous-graphe un sommet de distance minimale et on l&rsquo;ajoute au sous-graphe (il devient un sommet visit\u00e9). Ensuite, on met \u00e0 jour les distances des sommets voisins de celui ajout\u00e9 (les sommets sont marqu\u00e9s). La mise \u00e0 jour s&rsquo;op\u00e8re comme suit\u00a0: la nouvelle distance du sommet voisin est le minimum entre <em>la distance existante<\/em> et celle obtenue en ajoutant <em>le poids de l&rsquo;arc entre sommet voisin et sommet ajout\u00e9 \u00e0 la distance du sommet ajout\u00e9<\/em>.<\/p>\n\n<p>On continue ainsi jusqu&rsquo;\u00e0 compl\u00e9ter enti\u00e8rement le sous-graphe (ou jusqu&rsquo;\u00e0 s\u00e9lection du sommet d&rsquo;arriv\u00e9e). Exemple :<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-6299 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/04\/dijkstra.png\" alt=\"plus court chemin algorithme de dijkstra source unique\" width=\"464\" height=\"503\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/04\/dijkstra.png 464w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/04\/dijkstra-277x300.png 277w\" sizes=\"(max-width: 464px) 100vw, 464px\" \/><\/figure>\n<\/div>\n\n<figure class=\"wp-block-table wikitable\">\n<table>\n<tbody>\n<tr>\n<th scope=\"col\">\u00a0distance<\/th>\n<th>\u00e0 A<\/th>\n<th scope=\"col\">\u00e0 B<\/th>\n<th scope=\"col\">\u00e0 C<\/th>\n<th scope=\"col\">\u00e0 D<\/th>\n<th scope=\"col\">\u00e0 E<\/th>\n<th scope=\"col\">\u00e0 F<\/th>\n<th scope=\"col\">\u00e0 G<\/th>\n<th scope=\"col\">\u00e0 H<\/th>\n<th scope=\"col\">\u00e0 I<\/th>\n<th scope=\"col\">\u00e0 J<\/th>\n<\/tr>\n<tr>\n<th>\u00e9tape initiale<\/th>\n<th>0<\/th>\n<th>\u221e<\/th>\n<th>\u221e<\/th>\n<th>\u221e<\/th>\n<th>\u221e<\/th>\n<th>\u221e<\/th>\n<th>\u221e<\/th>\n<th>\u221e<\/th>\n<th>\u221e<\/th>\n<th>\u221e<\/th>\n<\/tr>\n<tr>\n<th scope=\"row\">A(0)<\/th>\n<td>\u00a0<\/td>\n<td><u>85<\/u><\/td>\n<td>217<\/td>\n<td>\u221e<\/td>\n<td>173<\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">B(85<sub>A<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td>217<\/td>\n<td>\u221e<\/td>\n<td>173<\/td>\n<td><u>165<\/u><\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">F(165<sub>B<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td>217<\/td>\n<td>\u221e<\/td>\n<td><u>173<\/u><\/td>\n<td>&#8211;<\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<td>415<\/td>\n<td>\u221e<\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">E(173<sub>A<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td><u>217<\/u><\/td>\n<td>\u221e<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>\u221e<\/td>\n<td>\u221e<\/td>\n<td>415<\/td>\n<td>675<\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">C(217<sub>A<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>\u221e<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>403<\/td>\n<td><u>320<\/u><\/td>\n<td>415<\/td>\n<td>675<\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">H(320<sub>C<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>503<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td><u>403<\/u><\/td>\n<td>&#8211;<\/td>\n<td>415<\/td>\n<td><s>675<\/s>487<\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">G(403<sub>C<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>503<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td><u>415<\/u><\/td>\n<td>487<\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">I(415<sub>F<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>503<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td><u>487<\/u><\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">J(487<sub>H<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td><u>503<\/u><\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<tr>\n<th scope=\"row\">D(503<sub>H<\/sub>)<\/th>\n<td>\u00a0<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<td>&#8211;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/figure>\n\n<p>Pour des raisons pratiques, la r\u00e9solution de l&rsquo;algorithme de Dijkstra ne retourne qu&rsquo;un vecteur contenant le sommet visit\u00e9, la liste des pr\u00e9d\u00e9cesseurs (les sommets d\u00e9j\u00e0 valid\u00e9s) et les valeurs des plus courts chemins vers tous les autres sommets. Ce qui correspond \u00e0 une ligne courante du tableau pr\u00e9sent\u00e9. Gr\u00e2ce \u00e0 la colonne de gauche, nous pouvons cr\u00e9er un <a href=\"https:\/\/complex-systems-ai.com\/es\/teoria-de-grafos\/arboles-y-arboles\/\">arbre<\/a> des chemins les plus courts du sommet A \u00e0 tous les sommets.<\/p>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone wp-image-6299 size-full\" src=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/04\/dijkstra.png\" alt=\"plus court chemin algorithme de dijkstra source unique\" width=\"464\" height=\"503\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/04\/dijkstra.png 464w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2018\/04\/dijkstra-277x300.png 277w\" sizes=\"(max-width: 464px) 100vw, 464px\" \/><\/figure>\n<\/div>\n\n<div style=\"padding: 5px; background-color: #ffdcd3; border: 2px solid #ff7964; -moz-border-radius: 9px; -khtml-border-radius: 9px; -webkit-border-radius: 9px; border-radius: 9px;\"><strong> Optimalit\u00e9 <\/strong><br \/>Le sommet de d\u00e9part est le chemin le plus court pour arriver \u00e0 lui-m\u00eame. Ensuite, nous calculons tous les chemins de taille 1 partant de ce sommet, afin d&rsquo;en valider le plus court. Ce chemin est donc le plus court chemin \u00e0 partir du sommet de d\u00e9part car il n&rsquo;y a pas d&rsquo;ar\u00eate de poids n\u00e9gatif. Par r\u00e9currence, nous en d\u00e9duisons que l&rsquo;algorithme valide toujours un plus court chemin.<\/div>\n\n<div style=\"padding: 3px; border: 2px dotted #a5a5a5; background-color: #f6f9fa;\">\n<pre><strong>d[0]=0, d[i]=infini <\/strong>pour tout sommet autre qu'origine<strong>\ntant qu<\/strong>'il existe un sommet hors du sous-graphe <strong>P<\/strong>\n   choisir un sommet <strong>a<\/strong> hors de <strong>P<\/strong> de plus petite distance <strong>d[a]<\/strong>\n   mettre <strong>a<\/strong> dans <strong>P<\/strong>\n   <strong>pour<\/strong> chaque sommet <strong>b<\/strong> hors de <strong>P<\/strong> voisin de <strong>a<\/strong>\n      <strong>d[b]=min(d[b], d[a]+ poids(a,b))<\/strong>\n   <strong>fin<\/strong>\n<strong>fin<\/strong>\nretourner d\n<\/pre>\n<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>B\u00fasqueda de rutas P\u00e1gina de inicio Wiki Algoritmo de Dijkstra EW Dijkstra (1930-2002) propuso en 1959 un algoritmo (llamado algoritmo de Dijkstra) que permite \u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":362,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-429","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages\/429","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/comments?post=429"}],"version-history":[{"count":4,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages\/429\/revisions"}],"predecessor-version":[{"id":17921,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages\/429\/revisions\/17921"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/pages\/362"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/es\/wp-json\/wp\/v2\/media?parent=429"}],"curies":[{"name":"gracias","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}