I have been trying to learn Tarjan's algorithm from Wikipedia for 3 hours now, but I just can't make head or tail of it. :(
http://en.wikipedia.org/wiki/Tarjan's_strongly_connected_components_algorithm#cite_note-1
Why is it a subtree of the DFS tree? (actually DFS produces a forest? o_O)
And why does v.lowlink=v.index imply that v is a root?
Can someone please explain this to me / give the intuition or motivation behind this algorithm?
The idea is: When traversing the tree, every time you've searched through a branch and are backtracking, you check whether you've encountered an edge to an 'upper' node in the tree.
If you didn't (if (v.lowlink = v.index)), then you've just completed an SCC - it consists of the current node and all nodes on the stack. That's exactly a subtree of the DFS tree, except for the nodes in SCCs that were already completed.
If you did, you propagate this information to 'upper' nodes (v.lowlink := min(v.lowlink, w.lowlink)), because combined with the path in DFS tree the edge creates an 'upward' path.
DFS produces a forest, but you always consider one tree a time. An SCC is always included in one DFS tree, otherwise (being an SCC) there would be a path in both directions between both (all) trees in question - that's a contradiction.