Edge classification in a DFS

According to the book (Intro to Algorithm), in dfs, edges are classified as 4 kinds:

1. Tree Edge, if in edge (u,v), v is first discovered, then (u, v) is a tree edge.
2. Back Edge, if ......, v is discovere开发者_如何学God already and v is an ancestor, then it's a back edge.
3. Forward Edge, if ......, v is discovered already and v is a descendant of u, forward edge it is.
4. Cross Edge, all edges except for the above three.

My question is how can I identify whether v is u's ancestor or descendant when I'm trying to figure out if (u, v) is a back or forward edge?

If you really need it, you can check it by maintaining so called entry and exit times for each node. During the run of the algorithm, you increment a time variable (starting from 0, of course) each time you encounter a new vertex. The times entry_t(v), exit_t(v) are initially unset for all vertices.

When you first encounter a vertex, you set entry(v):=time. When you exit a vertex by an up edge (ie. poping the vertex from the stack), you set its exit(v):=time. With that, you have

• if entry(u) is set and exit(u) is not set, then u is ancestor of the current vertex (ie. vu is a back edge)
• if entry(u)>entry(current), then u is descendant from the current vertex (current->u is a forward edge)
• otherwise, it is a cross edge

Note that these relations are made for checking during the run of the algorithm. After the algorithm has completed, a check of ancestry is basically

u is_descendant_of v = entry(u)>entry(v) and exit(u)<=exit(v)