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Алгоритм Дейкстры — различия между версиями
Материал из CustisWiki
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| − | Алгоритм Дейкстры (Dijkstra) предназначен для решения задачи [[Поиск кратчайших путей в графе]].
| + | #REDIRECT [[discopal:{{PAGENAME}}]] |
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| − | Важным фактом, позволяющим исключить перебор, является то, что если у нас есть кратчайший путь от ''v'' до ''w'', проходящий через вершину ''y'', назовем его <m>(v \rightarrow w)^*</m>, то его первая часть от ''v'' до ''y'', <m>(v \rightarrow y)^*</m>, тоже будет кратчайшим путем. Действительно, если бы это было не так, т.е. существовал бы путь <m>(v \rightarrow y)^!</m> длины меньшей, чем <m>(v \rightarrow y)^*</m>, то можно было бы улучшить оптимальный путь <m>(v \rightarrow w)^*</m>, заменив в нем <m>(v \rightarrow y)^*</m> на <m>(v \rightarrow y)^!</m>
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| − | (См. [[#Рисунок 1]]).
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| − | В алгоритме Дейкстры, мы, итерационно поддерживаем два множества вершин:
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| − | ;Visited: множество вершин, до которых мы уже нашли кратчайший путь, ассоциированных со стоимостями кратчайших путей от стартовой вершины до них.
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| − | ;ToVisit: множество вершин, которые достижимы одним ребром из множества вершин ''Visited'', ассоциированных с верхними оценками стоимости пути до них.
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| − | На каждой итерации, мы выбираем из достижимых вершин вершину ''v'', самую ближайшую к стартовой вершине ''s'', и переносим ее из множества ''ToVisit'' в множество ''Visited'', увеличиваем множество «кандидатов» ''ToVisit'' ее соседями, и пересчитываем верхнюю оценку удаленности вершин из ''ToVisit'' до вершины ''s''.
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| − | В иллюстрации выполнения алгоритма, мы показываем изменение на каждой итерации хэш-таблиц ''Visited'' и ''ToVisit'', а в конце изображен входной граф, где сплошными линиями нарисованы имеющиеся ребра с ассоциированными длинами, а пунктиром — найденные кратчайшие пути.
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| − | Важно, что алгоритм работоспособен только в случае положительных расстояний, в противном случае, можно попробовать использовать [[алгоритм Флойда-Уоршолла]].
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| − | Код алгоритма, представлен в виде функции на языке [[Python]].
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| − | <code-python>
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| − | def dijkstra(G,start_node):
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| − | # хэш (узел -> стоимость) посещенных вершин
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| − | Visited={}
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| − | # хэш (узел -> стоимость) ближайших к посещенным
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| − | ToVisit={start_node:0}
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| − | # хэш (узел -> кратчайший путь)
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| − | Paths = {start_node:[start_node]}
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| − | # пока есть вершины, до которых не построен кратчайший путь
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| − | while ToVisit:
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| − | # выбираем ближайшую
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| − | v=argmin(ToVisit)
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| − | # до $v$ кратчайший путь найден
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| − | Visited[v]=ToVisit[v]; del ToVisit[v];
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| − | # для всех соседей вершины $v$
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| − | for w in G.neighbors(v):
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| − | # к которым еще не нашли кратчайший путь
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| − | if (w not in Visited):
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| − | # обновляем кратчайшие пути
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| − | vwLength = Visited[v] + G.get_edge(v,w)
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| − | if (w not in ToVisit) or (vwLength < ToVisit[w]):
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| − | ToVisit[w] = vwLength
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| − | Paths[w] = Paths[v]+[w]
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| − | print_frame(G, start_node, Visited, ToVisit, Paths)
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| − | return (Visited,Paths)
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| − | </code-python>
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| − | | + | |
| − | ==Трассировка алгоритма==
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| − | Последовательность итераций алгоритма показана ниже. В голубой цвет выкрашены вершины из «Visited» (причем указана стоимость их достижения), в желтый — из «ToVisit» (указана минимально известная стоимость их достижения для данной итерации). Также выделены ребра, принадлежащие кратчайшим путям.
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| − | ===Итерация 1===
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| − | <neato>
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| − | graph G{ rankdir=TB; overlap=scale; rankdir=TB; splines=true;
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| − | edge[len=3,fontcolor="blue",style="dashed"];
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| − | node[fontcolor="red"];
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| − |
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| − |
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| − | 0 [label="NY", ];
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| − | 1 [label="Moscow\n $15", style=filled, fillcolor="yellow" ];
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| − | 2 [label="Minsk\n Start\n $0", shape=box, periferies=2 style=filled, fillcolor="lightblue" ];
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| − | 3 [label="Berlin\n $20", style=filled, fillcolor="yellow" ];
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| − | 4 [label="Kiev\n $15", style=filled, fillcolor="yellow" ];
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| − |
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| − | 5 [label="Munich", ];
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| − |
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| − | 6 [label="Vladivostok", ];
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| − | 0 -- 1 [label="$60", ];
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| − | 0 -- 3 [label="$50", ];
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| − | 0 -- 6 [label="$30", ];
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| − | 1 -- 2 [label="$15", ];
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| − | 1 -- 3 [label="$50", ];
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| − | 1 -- 4 [label="$10", ];
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| − |
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| − | 1 -- 6 [label="$105", ];
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| − |
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| − | 2 -- 3 [label="$20", ];
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| − |
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| − | 2 -- 4 [label="$15", ];
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| − |
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| − | 3 -- 4 [label="$30", ];
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| − |
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| − | 3 -- 5 [label="$10", ];
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| − |
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| − | }
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| − | </neato>
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| − | | + | |
| − | ===Итерация 2===
| + | |
| − | <neato>
| + | |
| − | graph G{ rankdir=TB; overlap=scale; rankdir=TB; splines=true;
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| − | edge[len=3,fontcolor="blue",style="dashed"];
| + | |
| − | node[fontcolor="red"];
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| − |
| + | |
| − |
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| − | 0 [label="NY\n $75", style=filled, fillcolor="yellow" ];
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| − |
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| − | 1 [label="Moscow\n $15", style=filled, fillcolor="lightblue" ];
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| − |
| + | |
| − | 2 [label="Minsk\n Start\n $0", shape=box, periferies=2 style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 3 [label="Berlin\n $20", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − | 4 [label="Kiev\n $15", style=filled, fillcolor="yellow" ];
| + | |
| − |
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| − | 5 [label="Munich", ];
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| − |
| + | |
| − | 6 [label="Vladivostok\n $120", style=filled, fillcolor="yellow" ];
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| − |
| + | |
| − |
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| − | 0 -- 1 [label="$60", ];
| + | |
| − |
| + | |
| − | 0 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 0 -- 6 [label="$30", ];
| + | |
| − |
| + | |
| − | 1 -- 2 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 1 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 1 -- 4 [label="$10", ];
| + | |
| − |
| + | |
| − | 1 -- 6 [label="$105", ];
| + | |
| − |
| + | |
| − | 2 -- 3 [label="$20", ];
| + | |
| − |
| + | |
| − | 2 -- 4 [label="$15", ];
| + | |
| − |
| + | |
| − | 3 -- 4 [label="$30", ];
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| − |
| + | |
| − | 3 -- 5 [label="$10", ];
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| − |
| + | |
| − | }
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| − | </neato>
| + | |
| − | | + | |
| − | ===Итерация 3===
| + | |
| − | <neato>
| + | |
| − | graph G{ rankdir=TB; overlap=scale; rankdir=TB; splines=true;
| + | |
| − | edge[len=3,fontcolor="blue",style="dashed"];
| + | |
| − | node[fontcolor="red"];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 [label="NY\n $75", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − | 1 [label="Moscow\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 2 [label="Minsk\n Start\n $0", shape=box, periferies=2 style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 3 [label="Berlin\n $20", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − | 4 [label="Kiev\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 5 [label="Munich", ];
| + | |
| − |
| + | |
| − | 6 [label="Vladivostok\n $120", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 -- 1 [label="$60", ];
| + | |
| − |
| + | |
| − | 0 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 0 -- 6 [label="$30", ];
| + | |
| − |
| + | |
| − | 1 -- 2 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 1 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 1 -- 4 [label="$10", ];
| + | |
| − |
| + | |
| − | 1 -- 6 [label="$105", ];
| + | |
| − |
| + | |
| − | 2 -- 3 [label="$20", ];
| + | |
| − |
| + | |
| − | 2 -- 4 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 3 -- 4 [label="$30", ];
| + | |
| − |
| + | |
| − | 3 -- 5 [label="$10", ];
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| − |
| + | |
| − | }
| + | |
| − | </neato>
| + | |
| − | | + | |
| − | ===Итерация 4===
| + | |
| − | <neato>
| + | |
| − | graph G{ rankdir=TB; overlap=scale; rankdir=TB; splines=true;
| + | |
| − | edge[len=3,fontcolor="blue",style="dashed"];
| + | |
| − | node[fontcolor="red"];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 [label="NY\n $70", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − | 1 [label="Moscow\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 2 [label="Minsk\n Start\n $0", shape=box, periferies=2 style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 3 [label="Berlin\n $20", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 4 [label="Kiev\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 5 [label="Munich\n $30", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − | 6 [label="Vladivostok\n $120", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 -- 1 [label="$60", ];
| + | |
| − |
| + | |
| − | 0 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 0 -- 6 [label="$30", ];
| + | |
| − |
| + | |
| − | 1 -- 2 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 1 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 1 -- 4 [label="$10", ];
| + | |
| − |
| + | |
| − | 1 -- 6 [label="$105", ];
| + | |
| − |
| + | |
| − | 2 -- 3 [label="$20", style="bold"];
| + | |
| − |
| + | |
| − | 2 -- 4 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 3 -- 4 [label="$30", ];
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| − |
| + | |
| − | 3 -- 5 [label="$10", ];
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| − |
| + | |
| − | }
| + | |
| − | </neato>
| + | |
| − | | + | |
| − | ===Итерация 5===
| + | |
| − | <neato>
| + | |
| − | graph G{ rankdir=TB; overlap=scale; rankdir=TB; splines=true;
| + | |
| − | edge[len=3,fontcolor="blue",style="dashed"];
| + | |
| − | node[fontcolor="red"];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 [label="NY\n $70", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − | 1 [label="Moscow\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 2 [label="Minsk\n Start\n $0", shape=box, periferies=2 style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 3 [label="Berlin\n $20", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 4 [label="Kiev\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 5 [label="Munich\n $30", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 6 [label="Vladivostok\n $120", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 -- 1 [label="$60", ];
| + | |
| − |
| + | |
| − | 0 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 0 -- 6 [label="$30", ];
| + | |
| − |
| + | |
| − | 1 -- 2 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 1 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 1 -- 4 [label="$10", ];
| + | |
| − |
| + | |
| − | 1 -- 6 [label="$105", ];
| + | |
| − |
| + | |
| − | 2 -- 3 [label="$20", style="bold"];
| + | |
| − |
| + | |
| − | 2 -- 4 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 3 -- 4 [label="$30", ];
| + | |
| − |
| + | |
| − | 3 -- 5 [label="$10", style="bold"];
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| − |
| + | |
| − | }
| + | |
| − | </neato>
| + | |
| − | | + | |
| − | ===Итерация 6===
| + | |
| − | <neato>
| + | |
| − | graph G{ rankdir=TB; overlap=scale; rankdir=TB; splines=true;
| + | |
| − | edge[len=3,fontcolor="blue",style="dashed"];
| + | |
| − | node[fontcolor="red"];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 [label="NY\n $70", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 1 [label="Moscow\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 2 [label="Minsk\n Start\n $0", shape=box, periferies=2 style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 3 [label="Berlin\n $20", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 4 [label="Kiev\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 5 [label="Munich\n $30", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 6 [label="Vladivostok\n $100", style=filled, fillcolor="yellow" ];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 -- 1 [label="$60", ];
| + | |
| − |
| + | |
| − | 0 -- 3 [label="$50", style="bold"];
| + | |
| − |
| + | |
| − | 0 -- 6 [label="$30", ];
| + | |
| − |
| + | |
| − | 1 -- 2 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 1 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 1 -- 4 [label="$10", ];
| + | |
| − |
| + | |
| − | 1 -- 6 [label="$105", ];
| + | |
| − |
| + | |
| − | 2 -- 3 [label="$20", style="bold"];
| + | |
| − |
| + | |
| − | 2 -- 4 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 3 -- 4 [label="$30", ];
| + | |
| − |
| + | |
| − | 3 -- 5 [label="$10", style="bold"];
| + | |
| − |
| + | |
| − | }
| + | |
| − | </neato>
| + | |
| − | | + | |
| − | | + | |
| − | ===Итерация 7===
| + | |
| − | <neato>
| + | |
| − | graph G{ rankdir=TB; overlap=scale; rankdir=TB; splines=true;
| + | |
| − | edge[len=3,fontcolor="blue",style="dashed"];
| + | |
| − | node[fontcolor="red"];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 [label="NY\n $70", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 1 [label="Moscow\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 2 [label="Minsk\n Start\n $0", shape=box, periferies=2 style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 3 [label="Berlin\n $20", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 4 [label="Kiev\n $15", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 5 [label="Munich\n $30", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − | 6 [label="Vladivostok\n $100", style=filled, fillcolor="lightblue" ];
| + | |
| − |
| + | |
| − |
| + | |
| − | 0 -- 1 [label="$60", ];
| + | |
| − |
| + | |
| − | 0 -- 3 [label="$50", style="bold"];
| + | |
| − |
| + | |
| − | 0 -- 6 [label="$30", style="bold"];
| + | |
| − |
| + | |
| − | 1 -- 2 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 1 -- 3 [label="$50", ];
| + | |
| − |
| + | |
| − | 1 -- 4 [label="$10", ];
| + | |
| − |
| + | |
| − | 1 -- 6 [label="$105", ];
| + | |
| − |
| + | |
| − | 2 -- 3 [label="$20", style="bold"];
| + | |
| − |
| + | |
| − | 2 -- 4 [label="$15", style="bold"];
| + | |
| − |
| + | |
| − | 3 -- 4 [label="$30", ];
| + | |
| − |
| + | |
| − | 3 -- 5 [label="$10", style="bold"];
| + | |
| − |
| + | |
| − | }
| + | |
| − | </neato>
| + | |
| − | | + | |
| − | ==Рисунок 1==
| + | |
| − | | + | |
| − | <pic-svg>
| + | |
| − | <svg
| + | |
| − | xmlns:dc="http://purl.org/dc/elements/1.1/"
| + | |
| − | xmlns:cc="http://web.resource.org/cc/"
| + | |
| − | xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
| + | |
| − | xmlns:svg="http://www.w3.org/2000/svg"
| + | |
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Текущая версия на 20:43, 13 июня 2011
- REDIRECT discopal:Алгоритм Дейкстры
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