Problem:
Score : 500 points

Problem Statement

Rng has a connected undirected graph with N vertices.
Currently, there are M edges in the graph, and the i-th edge connects Vertices A_i and B_i.

Rng will add new edges to the graph by repeating the following operation:

Operation: Choose u and v (u \neq v) such that Vertex v can be reached by traversing exactly three edges from Vertex u, and add an edge connecting Vertices u and v. It is not allowed to add an edge if there is already an edge connecting Vertices u and v.

Find the maximum possible number of edges that can be added.

Constraints

2 \leq N \leq 10^5

1 \leq M \leq 10^5

1 \leq A_i,B_i \leq N

The graph has no self-loops or multiple edges.

The graph is connected.

Input

Input is given from Standard Input in the following format:

N M
A_1 B_1
A_2 B_2
:
A_M B_M

Output

Find the maximum possible number of edges that can be added.

Sample Input 1

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

Sample Output 1

4

If we add edges as shown below, four edges can be added, and no more.

Sample Input 2

5 5
1 2
2 3
3 1
5 4
5 1

Sample Output 2

5

Five edges can be added, for example, as follows:

Add an edge connecting Vertex 5 and Vertex 3.

Add an edge connecting Vertex 5 and Vertex 2.

Add an edge connecting Vertex 4 and Vertex 1.

Add an edge connecting Vertex 4 and Vertex 2.

Add an edge connecting Vertex 4 and Vertex 3.

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include<iostream>
#include<algorithm>
#include<vector>
#include<unordered_set>
#include<queue>
using namespace std;
struct E{
    int s{},t{};
    E()=default;
    E(int s,int t):s(s),t(t){}
};
bool operator==(E a,E b){
    if(a.s!=b.s)swap(b.s,b.t);
    return a.s==b.s&&a.t==b.t;
}
unordered_set<int> v[100001],u[100001][2];
vector<E> e;
int N,M,ans;
int main(){
    cin.tie(0);ios_base::sync_with_stdio(0);
    cin>>N>>M;
    for(int i{};i<M;++i){
        E x(0,0);cin>>x.s>>x.t;
        E y(x.t,x.s);
        e.push_back(x);
        v[x.s].insert(i);
        u[x.s][0].insert(x.t);
        v[x.t].insert(i);
        u[x.t][0].insert(x.s);
    }
    queue<E> q;
    for(int i{};i<e.size();++i){
        int A{e[i].s};
        for(auto& x:v[e[i].t]){
            if(x==i)continue;
            if(e[i].t!=e[x].s)swap(e[x].s,e[x].t);
            if(A==e[x].t)continue;
            int B{e[x].t};
            u[A][1].insert(B);
            u[B][1].insert(A);
            for(auto& y:v[B]){
                if(B!=e[y].s)swap(e[y].s,e[y].t);
                if(A==e[y].t)continue;
                q.emplace(A,e[y].t);
                ++ans;
            }
        }
    }
    while(q.size()){
        E x{q.front()};q.pop();
        int A{x.s},B{x.t};
        for(auto& y:u[B][1]){
            if(A!=y&&(u[A][0].find(y)==u[A][0].end())){
                u[A][0].insert(y);
                u[y][0].insert(A);
                q.emplace(A,y);
                ++ans;
            }
        }
        for(auto& y:u[A][1]){
            if(B!=y&&u[B][0].find(y)==u[B][0].end()){
                u[B][0].insert(y);
                u[y][0].insert(B);
                q.emplace(B,y);
                ++ans;
            }
        }
    }
    cout<<ans<<endl;
}