Question

We have a group of people and we know which pairs of people are friends. We want to give some of the people money in such a way that for every pair of friends exactly one gets money. Explain how this is a graph problem, and give pseudocode for an algorithm to determine if it possible. (language=c++)

Answer 1

Let's say that we have n people numbered from 1 to n. And there are m friends pair in the form of (a, b) are the persons.

Then, if we select a person x then we can not select any of his friend.

Now suppose we have graph of n vertices each representing one of the person and m edges connecting each pair of friends. Then, we have to divide the vertice into two sets, one of the set will contain those who will get money and the other will contain those who won't. So basically, we want our graph to not contain any edge between the people lying in the same set.

To do this, we can do a depth first search, and calculate an attribute st[x] for each x, which will store the set person x belong to.

And we want to have two sets. So, we will set st[x] to be either 1 or 2. We will start dfs from person number 1 and assign st[1]=1.

Then for each of the friends of person 1, we will assign st[x]=2. And then for friends of friends we will assign st[y]=1. And similarly continue doing dfs.

And once we visited a node which has already been assigned st[x] and is consistent with the calculation done till now, then such distribution is possible and else it is not.

//CPP CODE

#include <bits/stdc++.h>

using namespace std;

vector <int> v[100001];//adjacency list for the graph

int st[100001];//attribute of each of the person

bool dfs(int s, int assign)

{

    if(st[s]!=0)

    {

        /*If an already visited node is visited again, and is the value which is going to be assigned to

        it now is not same then the desired distribution is not possible */

        if(st[s]!=assign)return false;

        else return true;

    }

    st[s]=assign;

    bool res=true;

    for(auto it:v[s])

    {

        res&=dfs(it, (assign==1)?2:1);//if the parent is assigned 1, child will be assigned 2 and vice versa

    }

    return res;

}

int main()

{

    int n, m;

    

    cin>>n;//Enter number of persons

    cin>>m;//Enter number of friends pair

    int a, b;

    memset(st, 0, sizeof(st));

    for(int i=0;i<m;i++)

    {

        cin>>a>>b;// Enter a and b the friends pairs

        v[b].push_back(a);

        v[a].push_back(b);

    }

    bool ans=dfs(1, 1);//stores the boolean answer

    if(ans)cout<<"POSSIBLE"<<endl;

    else cout<<"NOT POSSIBLE"<<endl;

    return 0;

}

//CPP Code Ends here

/*

Happy Coding :)

*/