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ZOJ Problem Set - 1979
The Bottom of a Graph

Time Limit: 5 Seconds      Memory Limit: 32768 KB

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 ei��E 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 v, v is also reachable from w. The bottom of a graph is the subset of all nodes that are sinks, i.e., bottom(G)={v��V|?w��V:(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

Source: University of Ulm Local Contest 2003
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