Problem:
Score : 600 points

Problem Statement

Kenkoooo found a simple connected graph.
The vertices are numbered 1 through n.
The i-th edge connects Vertex u_i and v_i, and has a fixed integer s_i.

Kenkoooo is trying to write a positive integer in each vertex so that the following condition is satisfied:

For every edge i, the sum of the positive integers written in Vertex u_i and v_i is equal to s_i.

Find the number of such ways to write positive integers in the vertices.

Constraints

2 \leq n \leq 10^5

1 \leq m \leq 10^5

1 \leq u_i < v_i \leq n

2 \leq s_i \leq 10^9

If i\neq j, then u_i \neq u_j or v_i \neq v_j.

The graph is connected.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

n m
u_1 v_1 s_1
:
u_m v_m s_m

Output

Print the number of ways to write positive integers in the vertices so that the condition is satisfied.

Sample Input 1

3 3
1 2 3
2 3 5
1 3 4

Sample Output 1

1

The condition will be satisfied if we write 1,2 and 3 in vertices 1,2 and 3, respectively.
There is no other way to satisfy the condition, so the answer is 1.

Sample Input 2

4 3
1 2 6
2 3 7
3 4 5

Sample Output 2

3

Let a,b,c and d be the numbers to write in vertices 1,2,3 and 4, respectively.
There are three quadruples (a,b,c,d) that satisfy the condition:

(a,b,c,d)=(1,5,2,3)

(a,b,c,d)=(2,4,3,2)

(a,b,c,d)=(3,3,4,1)

Sample Input 3

8 7
1 2 1000000000
2 3 2
3 4 1000000000
4 5 2
5 6 1000000000
6 7 2
7 8 1000000000

Sample Output 3

0

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Linq;
using System.Diagnostics;
using System.Collections.Generic;
using static System.Math;

class P
{
    static void Main()
    {
        int[] nm = Console.ReadLine().Split().Select(int.Parse).ToArray();
        //辺を追加していきながらValid性を検証してく
        //重み付きUF?
        //それぞれについて可能性が存在している全ての方向を保持しておく。
        //2つ繋がっている
        //生なので
        //グローバルに存在しているRange変数みたいなのが欲しいです。
        Vertex[] vs = Enumerable.Repeat(0, nm[0]).Select(_ => new Vertex()).ToArray();
        for (int i = 0; i < nm[1]; i++)
        {
            int[] uvs = Console.ReadLine().Split().Select(int.Parse).ToArray();
            vs[uvs[0] - 1].vs.Add(new int[] { uvs[1] - 1, uvs[2] });
            vs[uvs[1] - 1].vs.Add(new int[] { uvs[0] - 1, uvs[2] });
        }

        bool isRistricted = false;
        int ristrictedNum = 0;
        List<int> visitOrder = new List<int>();
        vs[0].sign = 1;;
        Queue<int> queue = new Queue<int>();
        queue.Enqueue(0);
        while (queue.Count != 0)
        {
            int index = queue.Dequeue();
            visitOrder.Add(index);
            int sign = vs[index].sign;
            int var = vs[index].var;
            foreach (var v in vs[index].vs)
            {
                if (vs[v[0]].sign == 0)
                {
                    vs[v[0]].sign = -sign;
                    vs[v[0]].var = v[1] - var;
                    queue.Enqueue(v[0]);
                }
                else if (vs[v[0]].sign == -sign)
                {
                    if (v[1] - (vs[v[0]].var + var) != 0) goto impossible;
                }
                else if (vs[v[0]].sign == sign)
                {
                    if ((v[1] - (vs[v[0]].var + var)) % 2 != 0) goto impossible;
                    int ristrictednum = (v[1] - (vs[v[0]].var + var)) / 2;
                    if (isRistricted) if (ristrictednum != ristrictedNum) goto impossible;
                    ristrictedNum = ristrictednum;
                    isRistricted = true;
                }
            }
        }

        if (isRistricted)
        {
            int[] value = new int[nm[0]];
            value[0] = ristrictedNum;
            foreach (var index in visitOrder)
            {
                foreach (var v in vs[index].vs)
                {
                    if(value[v[0]] == 0)
                    {
                        value[v[0]] =  v[1] - value[index];
                        if (value[v[0]] <= 0) goto impossible;
                    }
                    else
                    {
                        if (value[v[0]] == v[1] - value[index]) goto impossible;
                    }
                }
            }
            Console.WriteLine(1);
        }
        else
        {
            int upper = int.MaxValue;
            int lower = 0;
            foreach (var vertex in vs)
            {
                if (vertex.sign < 0) upper = Min(upper, vertex.var);
                if (0 < vertex.sign) lower = Max(lower, vertex.var);
            }
            Console.WriteLine(Max(0, upper - lower));
        }
        return;
        impossible:;
        Console.WriteLine(0);
    }
}
class Vertex
{
    public int sign = 0;
    public int var = 0;
    public List<int[]> vs = new List<int[]>();
}