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.Collections.Generic;
using System.Linq;
using System.Text;
using static System.Console;
using static System.Math;

namespace AtTest._600problems
{
    class SoundHound2018_Summer_Qual_E
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            int[] nm = ReadInts();
            int n = nm[0];
            int m = nm[1];
            List<int[]>[] graph = new List<int[]>[n];
            for (int i = 0; i < n; i++)
            {
                graph[i] = new List<int[]>();
            }
            for (int i = 0; i < m; i++)
            {
                int[] uvs = ReadInts();
                int u = uvs[0] - 1;
                int v = uvs[1] - 1;
                int s = uvs[2];
                graph[u].Add(new int[2] { v, s });
                graph[v].Add(new int[2] { u, s });
            }

            int?[][] conds = new int?[n][];
            for (int i = 0; i < n; i++)
            {
                conds[i] = new int?[2];
            }
            Queue<int[]> que = new Queue<int[]>();
            que.Enqueue(new int[3] { 0, 0, 1 });
            int min = 1;
            int max = int.MaxValue / 2;
            while (que.Count > 0)
            {
                int[] val = que.Dequeue();
                int index = val[0];
                int num = val[1];
                int next = val[2] == 1 ? 0 : 1;

                for (int i = 0; i < graph[index].Count; i++)
                {
                    int to = graph[index][i][0];
                    int s = graph[index][i][1];
                    if (conds[to][next] == null)
                    {
                        conds[to][next] = s - num;
                    }
                    else
                    {
                        if (conds[to][next] != s - num)
                        {
                            WriteLine(0);
                            return;
                        }
                    }
                    if (conds[to][0] != null && conds[to][1] != null)
                    {
                        int target = (int)(conds[to][0] - conds[to][1]);
                        if (target > 0 && target % 2 == 0)
                        {
                            target /= 2;
                            min = Max(min, target);
                            max = Min(max, target);
                        }
                        else
                        {
                            WriteLine(0);
                            return;
                        }
                    }
                    if (conds[to][0] != null)
                    {
                        max = Min(max, (int)conds[to][0] - 1);
                    }
                    if (max < min)
                    {
                        WriteLine(0);
                        return;
                    }
                    que.Enqueue(new int[3] { to, (int)conds[to][next], next });
                }
                graph[index].Clear();
            }

            WriteLine(max - min + 1);
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}