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
ayush: 2016-07-13 19:12:59

@code_master5 i somehow figured it out later that day, anyways thanks for coming up. :) a simple SCC indeed.

avisheksanvas: 2016-07-05 10:06:08

Simple SCC problem. The entire problem in one statement : (v→w)⇒(w→v)!
And @codedecode0111 , the space at the end of each line does not matter!
And for the order, if you do it the right way, it'll automatically be in Ascending Order! :)

Last edit: 2016-07-05 10:09:31
Rohit Agarwal: 2016-07-01 17:42:03

Should we print is descending order or ascending order? The output says sorted order but doesn't specify which one. Are both valid?
Can I get a WA if there's an extra space in the end of each line?

EDIT:
That was my mistake. Got AC. Make sure you sort them in ascending order and no spaces in the end of each line.

Last edit: 2016-07-03 15:58:17
code_master5: 2016-06-29 17:28:04

@ayush because if a vertex u is directed towards another vertex v (i.e. u->v), where u and v belong to different SCCs, then
1. u definitely cannot belong to a sink ( or bottom)
2. every vertex belonging to the same SCC as u cannot belong to a sink because we know that u is connected to every other vertex in that SCC, and hence we can reach v from each of those vertices.
for example, in your case,
1. 2->3 => 2 cannot be a sink
2. since 1->2 and 2->1 => (2,1) forms an SCC, and if we can reach 3 from 2, we can definitely reach 3 from 1 (via 2)
Hope it helped!!

ayush: 2016-06-29 16:09:49

if a vertex belongs to one component and has a neighbour of other component (by component i mean a SCC group), why the whole SCC group of that vertex is discarded, as given by test case:
3 3
1 2 2 1 2 3
0
output:
3
Please explain.Thanks in advance.

code_master5: 2016-06-26 21:56:03

Finally! I got ac.. But I was getting WA when I was using low[] values as id for an SCC (yeah! u guessed right, I was using Tarjan's Algo for finding SCC). I was assuming that low value will be different for each SCC. Can anyone explain where I went wrong!!!

Abhishek Naik: 2016-06-26 19:04:21

@kshubham02: Your comment was really very helpful. Got my code accepted! Thank you! :)

Last edit: 2016-06-26 19:04:57
code_master5: 2016-06-25 20:10:46

@deerishi If u're suggesting this case:
Input:
1 1
1 1
my code gives
Output:
1
and I guess this is right!!!

deerishi: 2016-06-25 01:19:56

Nice Problem. For those getting WA consider the case of a single vertex with a self loop.

code_master5: 2016-06-23 22:37:38

Please help me! I'm constantly getting WA. I've tried all the given testcases (including those in comments!!!). Help me find bugs in my last submission...


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
Resource:University of Ulm Local Contest 2003