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: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
//using System.Text;
//using System.Text.RegularExpressions;
//using System.Globalization;
//using System.Diagnostics;
using static System.Console;
//using System.Numerics;
//using static System.Math;
//using pair = Pair<int, int>;

class Program
{
    static void Main()
    {
        //SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
        new Program().solve();
        Out.Flush();
    }
    Scanner cin = new Scanner();
    readonly int[] dd = { 0, 1, 0, -1, 0 }; //→↓←↑
    readonly int mod = 1000000007;
    void chmax<T>(ref T a, T b) where T : IComparable<T> { a = a.CompareTo(b) < 0 ? b : a; }
    void chmin<T>(ref T a, T b) where T : IComparable<T> { a = a.CompareTo(b) < 0 ? a : b; }


    struct Edge : IComparable<Edge>
    {
        public int from, to, id; public long cost;
        public Edge(int from, int to, long cost)
        {
            this.from = from;
            this.to = to;
            this.cost = cost;
            id = -1;
        }
        public void set_id(int id)
        {
            this.id = id;
        }
        public int CompareTo(Edge e)
        {
            return cost.CompareTo(e.cost);
        }
    }
    void dfs(int u, long d, int p)
    {
        if (flag > 0) return;
        A[u] = d;
        B[u] = p;
        //WriteLine(u + " " + A[u] + " " + B[u]);
        //頂点uはd + p * t
        foreach (var v in G[u])
        {
            long next_d = v.cost - d;
            int next_p = p * -1;
            //行ったことがない
            if (B[v.to] == 0)
            {
                dfs(v.to, next_d, next_p);
            }
            else if (B[v.to] == next_p)
            {
                if (A[v.to] != next_d)
                {
                    flag = 1;
                    return;
                }
            }
            else
            {
                var tt = next_d * -1 * next_p + A[v.to] * -1 * B[v.to];
                //WriteLine(" " + u + " " + tt);
                if (tt % 2 == 1)
                {
                    flag = 1;
                    return;
                }
                else
                {
                    id = v.to;
                    X = tt / 2;
                    flag = 2;
                    return;
                }

            }
        }
    }

    long[] J;
    List<Edge>[] G;
    long[] A;
    long[] B;
    int flag = 0;
    int id = -1;
    long X = -1;
    void solve()
    {
        int N = cin.nextint;
        int M = cin.nextint;
        G = Enumerable.Range(0, N).Select(i => new List<Edge>()).ToArray();

        for (int i = 0; i < M; i++)
        {
            int u = cin.nextint - 1;
            int v = cin.nextint - 1;
            var c = cin.nextlong;
            G[u].Add(new Edge(u, v, c));
            G[v].Add(new Edge(v, u, c));
        }
        A = new long[N];
        B = new long[N];

        dfs(0, 0, 1);
        //WriteLine(flag + " " + id + " " + X);

        if (flag == 1)
        {
            WriteLine(0);
        }
        else if (flag == 2)
        {
            J = new long[N];
            for (int i = 0; i < N; i++)
            {
                J[i] = long.MinValue;
            }
            dfs2(0, X);
            if (J.Min() > 0 && flag != 4)
            {
                WriteLine(1);
            }
            else
            {
                WriteLine(0);
            }
        }
        else
        {
            long tmin = long.MinValue;
            long tmax = long.MaxValue;
            for (int i = 0; i < N; i++)
            {
                if (B[i] == 1)
                {
                    chmax(ref tmin, 1 - A[i]);
                }
                else
                {
                    chmin(ref tmax, A[i] - 1);
                }
            }

            WriteLine(Math.Max(0, tmax - tmin + 1));
        }
    }
    void dfs2(int u, long val)
    {
        J[u] = val;
        foreach (var v in G[u])
        {
            if (J[v.to] == long.MinValue)
            {
                dfs2(v.to, v.cost - val);
            }
            else
            {
                //整合性とれてるか確認
                if (J[v.to] != v.cost - val)
                {
                    flag = 4;
                }
            }
        }
    }
}

class Scanner
{
    string[] s; int i;
    char[] cs = new char[] { ' ' };
    public Scanner() { s = new string[0]; i = 0; }
    public string[] scan { get { return ReadLine().Split(); } }
    public int[] scanint { get { return Array.ConvertAll(scan, int.Parse); } }
    public long[] scanlong { get { return Array.ConvertAll(scan, long.Parse); } }
    public double[] scandouble { get { return Array.ConvertAll(scan, double.Parse); } }
    public string next
    {
        get
        {
            if (i < s.Length) return s[i++];
            string st = ReadLine();
            while (st == "") st = ReadLine();
            s = st.Split(cs, StringSplitOptions.RemoveEmptyEntries);
            i = 0;
            return next;
        }
    }
    public int nextint { get { return int.Parse(next); } }
    public long nextlong { get { return long.Parse(next); } }
    public double nextdouble { get { return double.Parse(next); } }
}