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 Ruby (2.3.3) (judge verdict: Time Limit Exceeded)

N, M = gets.split.map(&:to_i)
edge = []
M.times do |i|
  edge[i] = gets.split.map(&:to_i)
  edge[i][0] -= 1
  edge[i][1] -= 1
end

network = Array.new(N)
edge.each do |x|
  if network[x[0]].nil?
    network[x[0]] = []
  end
  if network[x[1]].nil?
    network[x[1]] = []
  end
  if network[x[0]][0].nil?
    network[x[0]][0] = []
  end
  if network[x[1]][0].nil?
    network[x[1]][0] = []
  end
  network[x[0]][0] << x[1]
  network[x[1]][0] << x[0]  
end

N.times do |i|
  network[i][1] = []
  network[i][0].each do |e|
    network[e][0].each do |ind|
      network[i][1] << ind if ind != i
    end
  end
  network[i][1].uniq
end

# p network[0]
# p network[network[0][1][0]]
N.times do |i|
  network[i][2] = []
  network[i][1].each do |e|
    network[e][0].each do |ind|
      network[i][2] << ind if ind != i
    end
  end
  network[i][2].uniq
end

ret = 0
flag = true
# p network[0]
while flag
  flag = false
  N.times do |i|
    network[i][2].each do |e|
      # network[i][2].delete(e)
      next if network[i][0].include?(e)
      # puts("e: " + [i, e].join(" "))
      ret += 1
      network[i][0] << e
      network[e][0] << i
      flag = true
      network[e][1].each do |ind|
        network[i][2] << ind if ind != i
      end
      network[i][2].uniq
      network[i][1].each do |ind|
        network[e][2] << ind if ind != e
      end
      network[2][2].uniq
    end
  end
end

puts ret