Алгоритм Борувки
Алгори́тм Бору́вки — это алгоритм нахождения минимального остовного дерева в графе.
Впервые был опубликован в 1926 году Отакаром Борувкой в качестве метода нахождения оптимальной электрической сети в Моравии. Несколько раз был переоткрыт, например Флореком, Перкалом и Соллином. Последний, кроме того, был единственным западным ученым из этого списка, и поэтому алгоритм часто называют алгоритмом Соллина, особенно в литературе по параллельным вычислениям.
Алгоритм
Работа алгоритма состоит из нескольких итераций, каждая из которых состоит в последовательном добавлении рёбер к остовному лесу графа, до тех пор, пока лес не превратится в дерево, то есть, лес, состоящий из одной компоненты связности.
В псевдокоде, алгоритм можно описать так:
- Изначально, пусть T — пустое множество ребер (представляющее собой остовный лес, в который каждая вершина входит в качестве отдельного дерева).
-
Пока T не является деревом (что эквивалентно условию: пока число рёбер в T меньше, чем V − 1, где V — число вершин в графе):
- Для каждой компоненты связности (то есть, дерева в остовном лесе) в подграфе с рёбрами T, найдём самое дешёвое ребро, связывающее эту компоненту с некоторой другой компонентой связности. (Предполагается, что веса рёбер различны, или как-то дополнительно упорядочены так, чтобы всегда можно было найти единственное ребро с минимальным весом).
- Добавим все найденные рёбра в множество T.
- Полученное множество рёбер T является минимальным остовным деревом входного графа.
Сложность алгоритма
На каждой итерации число деревьев в остовном лесу уменьшается по крайней мере в два раза, поэтому всего алгоритм совершает не более O(logV) итераций. Каждая итерация может быть реализована со сложностью O(E), поэтому общее время работы алгоритмы составляет O(ElogV) времени (здесь V и E — число вершин и рёбер в графе, соответственно).
Однако для некоторых видов графов, в частности, планарных, оно может быть уменьшено до O(E).[1] Существует также рандомизированный алгоритм построения минимального остовного дерева, основанный на алгоритме Борувки, работающий в среднем за линейное время.
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