BOTTOM - The Bottom of a Graph

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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 $V \times V$, its elements being called edges. Then $G = (V, E)$ is called a directed graph.

Let $n$ be a positive integer, and let $p = (e_1, \ldots, e_n)$ be a sequence of length $n$ of edges $e_i \in E$ such that $e_i = (v_i, v_{i+1})$ for a sequence of vertices ($v_1, \ldots, v_{n+1}$). Then $p$ is called a path from vertex $v_1$ to vertex $v_{n+1}$ in $G$ and we say that $v_{n+1}$ is reachable from $v_1$, writing $(v_1 \to v_{n+1})$.

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., $\mathrm{bottom}(G) = \{v \in V \mid \forall w \in V : (v \to w) \Rightarrow (w \to v) \}$. You have to calculate the bottom of certain graphs.

Input Specification

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 $V = \{1, \ldots, v\}$. You may assume that $1 \le v \le 5000$. That is followed by a non-negative integer $e$ and, thereafter, $e$ pairs of vertex identifiers $v_1, w_1, \ldots, v_e, w_e$ with the meaning that $(v_i, w_i) \in E$. There are no edges other than specified by these pairs. The last test case is followed by a zero.

Output Specification

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

hide comments
evang12: 2022-01-10 02:26:24

For those who used Kosoraju's algorithm: what are the reasons that you didn't use Tarjan's algorithm? I'm new to SCCs, so any response I get would be appreciated.

lakshya1st: 2021-05-17 15:45:06

Same as CAPCITY only difference is print all SCC's with outdegree 0..!! :)

s_tank00_: 2020-10-21 13:50:09

@flyingduchman_ thank you for the help

s_tank00_: 2020-10-21 12:15:15 for clear question

utkarshgupta29: 2020-05-25 10:17:20

If you are getting TLE using JAVA then try using FAST IO and StringBuffer

luciferhell58: 2020-05-09 12:28:53

learnt a big lesson hashSet isn't automatically sorted :)

luciferhell58: 2020-05-09 12:04:26

is there something different for ouptut thing i am getting wrong answer for every case.

wittystranger: 2020-05-07 01:53:44

don't forget to sort the output.. costed me WA

vinty: 2020-03-27 14:28:48

So we have to find all the components(SCC) from which there are no outgoing edges. Correct me if I am wrong.

great_coder1: 2019-03-29 12:30:30

Getting tle i am only implementing kosaraju' algo. Please help me

Added by:Wanderley Guimarăes
Time limit:1s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ERL JS-RHINO
Resource:University of Ulm Local Contest 2003