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
0

Sample Output

1 3
2

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

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

dewa251202: 2018-09-27 05:13:55

I love this

abhimanyu_1998: 2018-09-20 03:48:17

time limit exceeds in java

aman_sachin200: 2018-06-17 22:00:51

Nice one!!!Try CAPCITY and TOUR after this!

sherlock11: 2018-06-08 10:29:56

if u want a clear understanding of SCC then this problem and CAPCITY are the problems that u are looking for.............if u are new with SCC then first read the concepts (kosaraju's algo) and then ......well u know what to do after that.............AC:)

karthik1997: 2017-12-18 09:18:35

Applied Kosaraju's algorithm . Really good problem . :)

Hey @asib_133012
The scc's are [3] , [4] and [1,2] . Since 3-4 edge exists , you cannot take 3 as sink . All other scc's have nodes that are actually sinks .
So the bottoms are 1,2,4 -> output :)

Last edit: 2017-12-18 09:19:28
vib_s02: 2017-10-29 09:46:51

@justforpractic

I think that answer should be 1 2

minaamir26: 2017-08-17 06:52:31

too strict time for java users

asib_133012: 2017-05-25 12:43:02

4 4
1 2 2 1 3 2 3 4
0
In this case what should be the ans? and why? please explain someone.thanks in advance

ayush_1997: 2017-03-10 21:17:24

learned the concept of sink vertex :)


Added by:Wanderley Guimarăes
Date:2007-09-21
Time limit:0.254s
Source limit:50000B
Memory limit:1536MB
Cluster: Cube (Intel G860)
Languages:All except: ERL JS-RHINO
Resource:University of Ulm Local Contest 2003