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.Text;
using System.Collections.Generic;
using System.Linq;
class Solve{
    List<int>[] G;
    List<long>[] S;
    int[] A;
    long[] B;
    bool[] us;
    long x;
    public Solve(){}
    StringBuilder sb;
    ReadData re;
    public static int Main(){
        new Solve().Run();
        return 0;
    }
    void Run(){
        sb = new StringBuilder();
        re = new ReadData();
        Calc();
        Console.Write(sb.ToString());
    }
    void Calc(){
        int N = re.i();
        int M = re.i();
        G = new List<int>[N];
        S = new List<long>[N];
        for(int i=0;i<N;i++){
            G[i] = new List<int>();
            S[i] = new List<long>();
        }
        for(int i=0;i<M;i++){
            int f = re.i()-1;
            int t = re.i()-1;
            int s = re.i();
            G[f].Add(t);
            G[t].Add(f);
            S[f].Add(s);
            S[t].Add(s);
        }
        A = new int[N];
        B = new long[N];
        us = new bool[N];
        x = 0;
        dfs(0,1,0);
        if(x == -1){
            sb.Append("0\n");
        }
        else if(x != 0){
            for(int i=0;i<N;i++){
                if(A[i]*x+B[i] <= 0){
                    sb.Append("0\n");        
                    return;
                }
            }
            sb.Append("1\n");
        }
        else{
            long min = 1;
            long max = 100000000000;
            for(int i=0;i<N;i++){
                if(A[i] == 1){
                    long bf = -100000000000;
                    long bl = 100000000000;
                    while(bf != bl){
                        long bc = (bf+bl)/2 - ((bf+bl)%2 < 0 ? 1 : 0);
                        if(bc+B[i] > 0){
                            bl = bc;
                        }
                        else{
                            bf = bc+1;
                        }
                    }
                    min = Math.Max(min,bf);
                }
                else{
                    long bf = -100000000000;
                    long bl = 100000000000;
                    while(bf != bl){
                        long bc = (bf+bl+1)/2 - ((bf+bl+1)%2 < 0 ? 1 : 0);
                        if(-bc+B[i] > 0){
                            bf = bc;
                        }
                        else{
                            bl = bc-1;
                        }
                    }
                    max = Math.Min(max,bf);
                }
            }
            if(max < min){
                sb.Append("0\n");
            }
            else{
                sb.Append((max-min+1)+"\n");
            }
        }
    }
    void dfs(int v,int a,long b){
        us[v] = true;
        A[v] = a;
        B[v] = b;
        for(int i=0;i<G[v].Count;i++){
            int t = G[v][i];
            long s = S[v][i];
            if(us[t]){
                if(a != A[t]){
                    if(b+B[t] != s){
                        x = -1;
                    }
                }
                else if(a == 1){
                    if((b+B[t]+s)%2 != 0){
                        x = -1;
                    }
                    long xt = (s-(b+B[t]))/2;
                    if(xt <= 0){
                        x = -1;
                    }
                    else if(x == 0){
                        x = xt;
                    }
                    else if(x != xt){
                        x = -1;
                    }
                }
                else{
                    if((b+B[t]+s)%2 != 0){
                        x = -1;
                    }
                    long xt = ((b+B[t])-s)/2;
                    if(xt <= 0){
                        x = -1;
                    }
                    else if(x == 0){
                        x = xt;
                    }
                    else if(x != xt){
                        x = -1;
                    }
                }
            }
            else{
                dfs(t,-a,s-b);
            }
        }
    }
}
class ReadData{
    string[] str;
    int counter;
    public ReadData(){
        counter = 0;
    }
    public string s(){
        if(counter == 0){
            str = Console.ReadLine().Split(' ');
            counter = str.Length;
        }
        counter--;
        return str[str.Length-counter-1];
    }
    public int i(){
        return int.Parse(s());
    }
    public long l(){
        return long.Parse(s());
    }
    public double d(){
        return double.Parse(s());
    }
    public int[] ia(int N){
        int[] ans = new int[N];
        for(int j=0;j<N;j++){
            ans[j] = i();
        }
        return ans;
    }
    public int[] ia(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        int[] ans = new int[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = int.Parse(str[j]);
        }
        return ans;
    }
    public long[] la(int N){
        long[] ans = new long[N];
        for(int j=0;j<N;j++){
            ans[j] = l();
        }
        return ans;
    }
    public long[] la(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        long[] ans = new long[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = long.Parse(str[j]);
        }
        return ans;
    }
    public double[] da(int N){
        double[] ans = new double[N];
        for(int j=0;j<N;j++){
            ans[j] = d();
        }
        return ans;
    }
    public double[] da(){
        str = Console.ReadLine().Split(' ');
        counter = 0;
        double[] ans = new double[str.Length];
        for(int j=0;j<str.Length;j++){
            ans[j] = double.Parse(str[j]);
        }
        return ans;
    }
    public List<int>[] g(int N,int[] f,int[] t){
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<f.Length;j++){
            ans[f[j]].Add(t[j]);
            ans[t[j]].Add(f[j]);
        }
        return ans;
    }
    public List<int>[] g(int N,int M){
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<M;j++){
            int f = i()-1;
            int t = i()-1;
            ans[f].Add(t);
            ans[t].Add(f);
        }
        return ans;
    }
    public List<int>[] g(){
        int N = i();
        int M = i();
        List<int>[] ans = new List<int>[N];
        for(int j=0;j<N;j++){
            ans[j] = new List<int>();
        }
        for(int j=0;j<M;j++){
            int f = i()-1;
            int t = i()-1;
            ans[f].Add(t);
            ans[t].Add(f);
        }
        return ans;
    }
}
public static class Define{
    public const long mod = 1000000007;
}
public static class Debug{
    public static void Print(double[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(double[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(double[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
    public static void Print(long[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(long[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(long[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
    public static void Print(int[,,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                for(int l=0;l<k.GetLength(2);l++){
                    Console.Write(k[i,j,l]+" ");
                }
                Console.WriteLine();
            }
            Console.WriteLine();
        }
    }
    public static void Print(int[,] k){
        for(int i=0;i<k.GetLength(0);i++){
            for(int j=0;j<k.GetLength(1);j++){
                Console.Write(k[i,j]+" ");
            }
            Console.WriteLine();
        }
    }
    public static void Print(int[] k){
        for(int i=0;i<k.Length;i++){
            Console.WriteLine(k[i]);
        }
    }
}