193 - Graph Coloring

[ricky_bd]

what is the output of the following input
1
3 2
1 2
1 3

[cyfra]

Hi!

For this test case the answer is :

2
2 3

(these nodes can be colored black because they are not directly connected)

Good Luck

[rascle]

My algorithem is ...

Form the minimum degree node to the maxmum degree node,
if its all neighbors are not black,it is going to be colored.

Is it wrong ...??...

I can got correct results in many cases,but I got WA,WA,and WA...

[visser]

Nice heuristic, but nothing more than that...
Can you PROOF your algorithm is correct?

[rascle]

Sorry,I can't proof.....

This method broke into my heart occasionally.

I just don't know why it's wrong...^^....

Ex.

1
6 8
1 2
1 3
2 4
2 5
3 4
3 6
4 6
5 6

Step 1. Construct all neighbors for each node

node neighbors degree can_be_color
1 2 3 2 true
2 1 4 5 3 true
3 1 4 6 3 true
4 2 3 6 3 true
5 2 6 2 true
6 3 4 5 3 true

Step 2. Start from the node which has lowest degree.

The lowest degree in this case is "2"

So,we search from node 1 to node 6 to find the node which can be
colored and has degree "2"....we get node "1" , so all neighbors of
node 1 can not be colored...


node neighbors degree can_be_color
1 2 3 2 "colored"
2 1 4 5 3 "false forever"
3 1 4 6 3 "false forever"
4 2 3 6 3 true
5 2 6 2 true
6 3 4 5 3 true

Then we get node 5 which can be colored and has degree "2",so all
neighbors of node 5 can not be colored....

node neighbors degree can_be_color
1 2 3 2 "colored 1 "
2 1 4 5 3 "false forever"
3 1 4 6 3 "false forever"
4 2 3 6 3 true
5 2 6 2 "colored 2 "
6 3 4 5 3 "false forever"

Step 3. Now we have finished all node having degree 2,so we go to
degree 3..
We find node 4 can be colored and has degree "3",so we select it and
find that all node have been considered..............

node neighbors degree can_be_color
1 2 3 2 "colored 1 "
2 1 4 5 3 "false forever"
3 1 4 6 3 "false forever"
4 2 3 6 3 "colored 3 "
5 2 6 2 "colored 2 "
6 3 4 5 3 "false forever"


So the answer is

3
1 4 5

[rascle]

Sorry,I can't proof.....

This method broke into my heart occasionally.

I just don't know why it's wrong...^^....

Ex.

1
6 8
1 2
1 3
2 4
2 5
3 4
3 6
4 6
5 6

Step 1. Construct all neighbors for each node

node neighbors degree can_be_color
1 2 3 2 true
2 1 4 5 3 true
3 1 4 6 3 true
4 2 3 6 3 true
5 2 6 2 true
6 3 4 5 3 true

Step 2. Start from the node which has lowest degree.

The lowest degree in this case is "2"

So,we search from node 1 to node 6 to find the node which can be
colored and has degree "2"....we get node "1" , so all neighbors of
node 1 can not be colored...


node neighbors degree can_be_color
1 2 3 2 "colored"
2 1 4 5 3 "false forever"
3 1 4 6 3 "false forever"
4 2 3 6 3 true
5 2 6 2 true
6 3 4 5 3 true

Then we get node 5 which can be colored and has degree "2",so all
neighbors of node 5 can not be colored....

node neighbors degree can_be_color
1 2 3 2 "colored 1 "
2 1 4 5 3 "false forever"
3 1 4 6 3 "false forever"
4 2 3 6 3 true
5 2 6 2 "colored 2 "
6 3 4 5 3 "false forever"

Step 3. Now we have finished all node having degree 2,so we go to
degree 3..
We find node 4 can be colored and has degree "3",so we select it and
find that all node have been considered..............

node neighbors degree can_be_color
1 2 3 2 "colored 1 "
2 1 4 5 3 "false forever"
3 1 4 6 3 "false forever"
4 2 3 6 3 "colored 3 "
5 2 6 2 "colored 2 "
6 3 4 5 3 "false forever"


So the answer is

3
1 4 5

[sayeed]

/*@BEGIN_OF_SOURCE_CODE*/
#include<stdio.h>
#include<stdlib.h>

#define BLACK 2
#define WHITE 1

int vertex[102][102];
int flag[102],x[102],max,vertx;

void Coloring(int i,int count)
{
int j;
if(i==vertx){
if(max<count){
max =count;
for(j=0;j<vertx;j++)
if(flag[j]==BLACK) x[j] =1;
else x[j] = 0;
}
return;
}

for(j=0;j<vertx;j++)
if(vertex[j] == 1 && flag[j] == BLACK) break;

if(j==vertx){
flag =BLACK;
Coloring(i+1,count+1);
flag = 0;
}
else{
flag = WHITE;
Coloring(i+1,count);
flag = 0;


}

}
int main()

{
int edge,n,i,j,a,b,count;

scanf("%d",&n);

while(n-->0){
scanf("%d%d",&vertx,&edge);
max = 0;
for(i=0;i<vertx;i++) {
flag = 0,x = 0;
for(j = 0;j<vertx;j++)
vertex[j] = 0;
}
for(i=0;i<edge;i++){
scanf("%d%d",&a,&b);
vertex[a-1][b-1]=vertex[b-1][a-1]=1;

}

for(i =0 ;i<vertx;i++)
Coloring(i,0);



printf("%d\n",max);
for(i=0;i<vertx;i++)
if(x == 1) printf("%d ",i+1);
printf("\n");
}



return 0;
}

/*@END_OF_SOURCE_CODE*/

*******
Is the algorithm right?
Can anyone help me?

[Christian Schuster]

Hello,

I tried to solve 193 (and got AC in 0.01s), but I'm using some kind of probabilistic approach: I randomly select a node, remove it and its neighbours until there are no nodes left. I repeat this several times (100 is enough) to get the maximum node count.

I know that this problem (Maximum Independent Set) is NP-complete, but is there another (maybe a bit more deterministic) solution?

[mizocom2002]

i used the following heuristic to solve the problem:
find the node with minimum degree put it in the solution vector and delete it and all the nodes connected to it and so on till u finish all the nodes.
it generates the right answer for the test cases so can any body points out what is wrong with this algorithm and give me some difficult test cases, thanks.

Code: Select all

[cpp]

#include<iostream>
#include<fstream>
using namespace std;

int findmin();
void adjust(int x);

int a[101]={-1};
int b[101][101]={0};
int n;
int main()
{
	//ifstream ins;
	//ins.open("in.txt");
	//cin=ins;
	int m,k;
	int temp1,temp2;
	int f,t;
	int count;
	cin>>m;
	for(t=1;t<=m;t++)
	{
		count=0;
		//initialize a,b
		cin>>n>>k;
		for(f=1; f<=n;f++)
		{
			//a[f]=-1;
			for(int c=1;c<=n;c++)
				b[f][c]=0;
		}
		for(f=1;f<=k;f++)
		{
			cin>>temp1>>temp2;
			if(temp1 !=temp2)
			{
			b[temp1][temp2]=1;
			b[temp2][temp1]=1;
			}
		}
		for(f=1;f<=n;f++)
		{
			int y=findmin();
			if(y>n)
				break;
			a[f]=y;
			count++;
			for(int t=1;t<=n;t++)
			{
				if(b[y][t]==1)
					adjust(t);
			}
			adjust(y);

		}
		cout<<count<<endl;
		for(f=1;f<=count;f++)
		{if(f==count)
			cout<<a[f];
		else	
		cout<<a[f]<<" ";
		}
		cout<<endl;
		
	}
	return 0;


}


int findmin()
{
	int index=1000;
	int count;
	int mincount=1000;
	int i,j;
	for(i=1;i<=n;i++)
	{
		count=0;
		for(j=1;j<=n;j++)
		{
			if(b[i][j]==-1)
				break;
			if(b[i][j])
				count++;
		}
		if(j>n && count<mincount)
		{
			mincount=count;
			index=i;
		}
	}
	return index;
}
void adjust(int x)
{
	for(int i=1;i<=n;i++)
	{
		if(b[i][x]==1)
			b[i][x]=0;
		b[x][i]=-1;
	}
}

[/cpp]

[kmhasan]

so far as i know this algorithm will generate a maximal independent set, not necessarily the one that has maximum size.

maximal set: no other maximal set contains it
maximum set: the set whose cardinality is maximum

[nebula12]

Hi,
Can anyone provide their solution to the graph coloring problem and the source code(c,c++ or java). I'm working on a problem where I need this solution.

Thanks.

[bugzpodder]

I didnt think a greedy algorithm would work... so is this problem NP-Complete. any kool algorithms? I was looking at Lawler's algorithm...

[Sajid]

My program got WA in 0.039 sec.

Can anyone tell me some inputs and outputs?

[Maarten]

not if you don't provide us with some details about your program, algorithm, or implementation

[Sajid]

After taking the Graph, in a loop, I took the graph in another adjcency matrix and call the dfs, which call with the root (the root is 1 to n, end of the loop).

Code: Select all

for(i=1;i<=node;i++)
{
     temp=graph
     dfs(i)
}

The dfs visits every unvisited node, when it visits a node it colors white to its neighbours.

Finally, took the maximum number of Black nodes from the dfs(1->node)

is it, ok?