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# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace C
{
    class Program
    {
        static void Main(string[] args)
        {
            int[] nm = Console.ReadLine().Trim().Split(' ').Select(x => int.Parse(x)).ToArray();
            List<int>[] graph = new List<int>[nm[0]];
            bool[] update = new bool[nm[0]];
            List<Tuple<int, int>>[] spreader = new List<Tuple<int, int>>[nm[0]];

            for (int i = 0; i < nm[0]; i++)
            {
                graph[i] = new List<int>();
                spreader[i] = new List<Tuple<int, int>>();
                spreader[i].Add(Tuple.Create(i, 0));
            }

            for (int i = 0; i < nm[1]; i++)
            {
                int[] input = Console.ReadLine().Trim().Split(' ').Select(x => int.Parse(x)).ToArray();
                graph[input[0] - 1].Add(input[1] - 1);
                graph[input[1] - 1].Add(input[0] - 1);
            }

            int result = 0;

            for (int k = 0; k < 3; k++)
            {
                for (int i = 0; i < nm[0]; i++)
                {
                    for (int sp = 0; sp < spreader[i].Count(); sp++)
                    {
                        var v = spreader[i][sp];
                        if (v.Item2 < 3)
                        {
                            for (int gr = 0; gr < graph[i].Count; gr++)
                            {
                                var w = graph[i][gr];
                                if (!spreader[w].Contains(Tuple.Create(v.Item1, v.Item2 + 1)))
                                {
                                    spreader[w].Add(Tuple.Create(v.Item1, v.Item2 + 1));
                                }
                            }
                        }
                    }
                }
            }

            for (int i = 0; i < nm[0]; i++)
            {
                for (int sp = 0; sp < spreader[i].Count(); sp++)
                {
                    var v = spreader[i][sp];
                    if (v.Item2 == 3)
                    {
                        if (v.Item1 != i && !graph[v.Item1].Contains(i))
                        {
                            graph[v.Item1].Add(i);
                            graph[i].Add(v.Item1);
                            update[v.Item1] = true;
                            update[i] = true;
                            result++;
                        }
                    }
                }

            }

            while (update.Any(x => x))
            {
                for (int i = 0; i < nm[0]; i++)
                {
                    if (update[i])
                    {
                        update[i] = false;
                        for (int sp = 0; sp < spreader[i].Count(); sp++)
                        {
                            var v = spreader[i][sp];
                            if (v.Item2 < 3)
                            {
                                for (int gr = 0; gr < graph[i].Count; gr++)
                                {
                                    var w = graph[i][gr];
                                    if (!spreader[w].Contains(Tuple.Create(v.Item1, v.Item2 + 1)))
                                    {
                                        spreader[w].Add(Tuple.Create(v.Item1, v.Item2 + 1));
                                        if (v.Item2 + 1 == 3 && v.Item1 != w && !graph[v.Item1].Contains(w))
                                        {
                                            graph[v.Item1].Add(w);
                                            graph[w].Add(v.Item1);
                                            update[v.Item1] = true;
                                            update[w] = true;
                                            result++;
                                        }
                                    }
                                }
                            }
                        }
                    }
                }
            }

            Console.WriteLine(result);
#if DEBUG
            Console.ReadKey();
#endif
        }
    }
}