maksyuki 发表于 oj 分类,标签:

The Bottom of a Graph

We will use the following (standard) definitions from graph theory. Let V be a nonempty and finite set, its elements being called vertices (or nodes). Let E be a subset of the Cartesian product V×V, its elements being called edges. Then G=(V,E) is called a directed graph.
Let n be a positive integer, and let p=(e1,...,en) be a sequence of length n of edges eiE such that ei=(vi,vi+1) for a sequence of vertices (v1,...,vn+1). Then p is called a path from vertex v1 to vertex vn+1 in G and we say that vn+1 is reachable from v1, writing (v1→vn+1).
Here are some new definitions. A node v in a graph G=(V,E) is called a sink, if for every node w in G that is reachable from vv is also reachable from w. The bottom of a graph is the subset of all nodes that are sinks, i.e., bottom(G)={vV|wV:(v→w)(w→v)}. You have to calculate the bottom of certain graphs.


The input contains several test cases, each of which corresponds to a directed graph G. Each test case starts with an integer number v, denoting the number of vertices of G=(V,E), where the vertices will be identified by the integer numbers in the set V={1,...,v}. You may assume that 1<=v<=5000. That is followed by a non-negative integer e and, thereafter, e pairs of vertex identifiers v1,w1,...,ve,we with the meaning that (vi,wi)E. There are no edges other than specified by these pairs. The last test case is followed by a zero.


For each test case output the bottom of the specified graph on a single line. To this end, print the numbers of all nodes that are sinks in sorted order separated by a single space character. If the bottom is empty, print an empty line.

Sample Input

3 3
1 3 2 3 3 1
2 1
1 2

Sample Output

1 3



Ulm Local 2003



算法分析:本题要找所有满足”能够到达其他顶点w(w != u)的点u是否能够从w点到达”的所有的u点,即不考察那些u不可达的点的情况。由于同一个强连通分量中的顶点都是双向可达的,所以其中任意一个点对于同一个强连通分量中的其它点都是满足条件的,可以缩点。易知缩点后得到的图中出度为0的缩点都满足条件