From d483be52062dacb030cf65bde25c52d478bbe65a Mon Sep 17 00:00:00 2001 From: Orestis <orestis.malaspinas@pm.me> Date: Mon, 23 May 2022 22:57:19 +0200 Subject: [PATCH] added details --- slides/cours_25.md | 145 ++++++++++++++++++++++++---------- slides/figs/dijkstra_ex_0.png | Bin 0 -> 52682 bytes slides/figs/dijkstra_ex_1.png | Bin 0 -> 52768 bytes slides/figs/dijkstra_ex_2.png | Bin 0 -> 53087 bytes slides/figs/dijkstra_ex_3.png | Bin 0 -> 53095 bytes slides/figs/dijkstra_ex_4.png | Bin 0 -> 52634 bytes slides/figs/dijkstra_ex_5.png | Bin 0 -> 52597 bytes 7 files changed, 105 insertions(+), 40 deletions(-) create mode 100644 slides/figs/dijkstra_ex_0.png create mode 100644 slides/figs/dijkstra_ex_1.png create mode 100644 slides/figs/dijkstra_ex_2.png create mode 100644 slides/figs/dijkstra_ex_3.png create mode 100644 slides/figs/dijkstra_ex_4.png create mode 100644 slides/figs/dijkstra_ex_5.png diff --git a/slides/cours_25.md b/slides/cours_25.md index 1094882..45e9ade 100644 --- a/slides/cours_25.md +++ b/slides/cours_25.md @@ -380,12 +380,6 @@ pile parcours(précédent, s, t) ::: -# Algorithme de Dijkstra (exercice) - -## Trouver le chemin de `a` à tous lesautres sommets - - - # Algorithme de Dijkstra amélioré ## On peut améliorer l'algorithme @@ -412,8 +406,9 @@ pile parcours(précédent, s, t) ```C booléen est_vide(élément) // triviale -élément enfiler(élément, data) +élément enfiler(élément, data, priorité) data défiler(élément) +rien modifiier_priorité(élément, data, priotié) nombre priorité(data) // utilitaire ``` @@ -423,8 +418,9 @@ nombre priorité(data) // utilitaire ```C struct élément - data d - élément next + data + priorité + élément suivant ``` # Les files de priorité @@ -434,22 +430,22 @@ struct élément . . . ```C -élément enfiler(élément e, d) - élément = créer_élément(d) - si est_vide(e) - retourne élément +élément enfiler(élément, data, priorité) + n_élément = créer_élément(data, priorité) + si est_vide(élément) + retourne n_élément si priorité(d) > priorité(e.d) - élément.next = e - retourne e + n_élément.suivant = élément + retourne n_élément sinon - tmp = e - prev = e - tant que !est_vide(tmp) && priorité(d) < priorité(tmp.d) - prev = tmp - tmp = tmp.next - prev.next = élément - élément.next = tmp - retourne e + tmp = élément + préc = élément + tant que !est_vide(tmp) && priorité < tmp.priorité + préc = tmp + tmp = tmp.suivant + prev.suivant = n_élément + n_élément.suivant = tmp + retourne élément ``` # Les files de priorité @@ -459,30 +455,99 @@ struct élément . . . ```C -data défiler(élément e) - si est_vide(e) +data, élément défiler(élément) + si est_vide(élément) retourne AARGL! sinon - retourne e.d + tmp = élément.data + n_élément = élément.suivant + libérer(élément) + retourne tmp, n_élément ``` # Algorithme de Dijkstra avec file +```C +distance, précédent dijkstra(graphe, s, t): + distance[source] = 0 + fp = file_p_vide() + pour v dans sommets(graphe) + si v != s + distance[v] = infini + précédent[v] = indéfini + fp = enfiler(fp, v, distance[v]) + tant que !est_vide(fp) + u, fp = défiler(fp) + pour v dans voisinage de u + n_distance = distance[u] + w(i, v) + si n_distance < distance[v] + distance[v] = n_distance + précédent[v] = u + fp = changer_priorité(fp, v, n_distance) + retourne distance, précédent +``` + +# Algorithme de Dijkstra (exercice, 5min) + + + +* Donner la liste de priorité, puis... + +## A chaque étape donner: + +* Le tableau des distances à `a`; +* Le tableau des prédécessueurs; +* L'état de la file de priorité. + +# Algorithme de Dijkstra (corrigé) -function Dijkstra(graphe, s, t): + + +# Algorithme de Dijkstra (corrigé) + + + +# Algorithme de Dijkstra (corrigé) + + + +# Algorithme de Dijkstra (corrigé) + + + +# Algorithme de Dijkstra (corrigé) + + + +# Algorithme de Dijkstra (corrigé) + + + +# Complexité de l'algorithme de Dijkstra? + +\footnotesize + +. . . + +```C +distance, précédent dijkstra(graphe, s, t): distance[source] = 0 fp = file_p_vide() - pour chaque v dans sommets(graphe) +---------------------------------------------- + pour v dans sommets(graphe) si v != s - distance[v] = infinit - fp = enfiler(fq, v) - - while Q is not empty: // The main loop - u ← Q.extract_min() // Remove and return best vertex - for each neighbor v of u: // only v that are still in Q - alt ← dist[u] + Graph.Edges(u, v) - if alt < dist[v] - dist[v] ← alt - prev[v] ← u - Q.decrease_priority(v, alt) - return dist, prev +O(V) distance[v] = infini + fp = enfiler(fp, v, distance[v]) +---------------------------------------------- + tant que !est_vide(fp) +O(V) u, fp = défiler(fp) + ---------------------------------------------- + pour v dans voisinage de u + O(E) n_distance = distance[u] + w(i, v) + si n_distance < distance[v] + distance[v] = n_distance + fp = changer_priorité(fp, v, n_distance) + ---------------------------------------------- +---------------------------------------------- + retourne distance +``` diff --git a/slides/figs/dijkstra_ex_0.png 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