Why can't Prim's or Kruskal's algorithms be used on a directed graph?

Question

Prim's and Kruskal's algorithms are used to find the minimum spanning tree of a graph that is connected and undirected. Why can't they be used on a graph that is directed?

Answer 1

It's a minor miracle that these algorithms work in the first place -- most greedy algorithms just crash and burn on some instances. Assuming that you want to use them to find a minimum spanning arborescence (directed paths from one vertex to all others), then one problematic graph for Kruskal looks like this.

 5
  --> a
 /   / ^
s   1| |2
 \   v /
  --> b
 3

We'll take the a->b arc of cost 1, then get stuck because we really wanted s->b of cost 3 and b->a of cost 2.

For Prim, this graph is problematic.

 5
  --> a
 /   /
s   1|
 \   v
  --> b
 3

We'll take s->b of cost 3, but we really wanted s->a of cost 5 and a->b of cost 1.