URI Online Judge | 1677

The Bottom of a Graph

Local Contest, University of Ulm DE Germany

Timelimit: 3

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 = {1,...,v}. You may assume that v(1 ≤ v ≤ 5000). That is followed by a non-negative integer e and, thereafter, e pairs of vertex identifiers v1W1,...,veWe 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 Sample Output

3 3

1 3 2 3 3 1

2 1

1 2


1 3