Minimum weight spanning tree for Directed graph with parallel edges

Question

I would like to have the name of algorithms can be used to find Minimum weight spanning tree from a Directed cyclic graph with parallel edges. Information on any c++ libraries that can be used to obtain the same with their analysis on runtime and efficiency.

Answer 1

There's no such thing as minimum spanning tree for directed graphs. You probably have in mind minimum spanning aborescence ( https://en.wikipedia.org/wiki/Arborescence_(graph_theory) ).

For finding min cost aborescence there's an algorithm called Chu–Liu/Edmonds' algorithm. ( https://en.wikipedia.org/wiki/Edmonds%27_algorithm ) It can find the aborescne of minimum cost in O(VE) or O(E log V) or O(E + V log V) depending on implementation.