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.Collections;
using System.Collections.Specialized;
using System.Linq;
using System.Text;
using System.IO;
using System.Reflection;
using static System.Math;
using System.Numerics;
static class Program{
	const int mod=(int)1e9+7;
	static List<Tuple<int,int>>[] li;
	static int[] b;
	static int z=0;
	static long u=0;
	static long[] h,d;
	static bool bo=false;
	static void Main(){
		Sc sc=new Sc();
		var s=sc.Ia;
		int n=s[0],m=s[1];
		li=new List<Tuple<int,int>>[n+1];
		b=new int[n+1];
		h=new long[n+1];
		d=new long[]{long.MaxValue,1};
		var c=new long[n+1];
		for(int i=1;i<=n;i++){
			li[i]=new List<Tuple<int,int>>();
			c[i]=long.MaxValue;
		}
		for(int i=0;i<m;i++){
			var e=sc.Ia;
			li[e[0]].Add(Tuple.Create(e[1],e[2]));
			li[e[1]].Add(Tuple.Create(e[0],e[2]));
		}
		h[1]=1;
		Fu(1,1);
		long ans=0;
		if(z!=0){
			h[z]=u>>1;
			b=new int[n+1];
			Fu2(z);
			h[0]=1;
			ans=(h.Min()>0&&z!=-1)?1:0;
		}
		else{
			b=new int[n+1];
			Fu2(1);
			if(d[0]+d[1]>0&&z!=-1){ans=1+d[0]-1+d[1]-1;}
		}
		Console.WriteLine("{0}",ans);
	}
	static void Fu2(int a){
		b[a]=1;
		for(int i=0;i<li[a].Count&&z!=-1;i++){
			if(b[li[a][i].Item1]==0){
				h[li[a][i].Item1]=li[a][i].Item2-h[a];
				Fu2(li[a][i].Item1);
			}
			else if(li[a][i].Item2-h[li[a][i].Item1]!=h[a]){z=-1;}
		}
	}
	static void Fu(int a,int g){
		b[a]=g;
		for(int i=0;i<li[a].Count;i++){
			if(b[li[a][i].Item1]==0){
				h[li[a][i].Item1]=li[a][i].Item2-h[a];
				d[(g^1)&1]=Min(d[(g^1)&1],h[li[a][i].Item1]);
				Fu(li[a][i].Item1,g+1);
				if(z!=0){
					if(!bo){
						u+=li[a][i].Item2*(b[z]%2==b[a]%2?1:-1);
						if(z==a){bo=true;}
					}
					break;
				}
			}
			else if(b[li[a][i].Item1]!=g-1){
				if((g-b[li[a][i].Item1])%2==0){
					u+=li[a][i].Item2;
					z=li[a][i].Item1;
					break;
				}
			}
		}
	}
}
public class Sc{
	public int I{get{return int.Parse(Console.ReadLine());}}
	public long L{get{return long.Parse(Console.ReadLine());}}
	public double D{get{return double.Parse(Console.ReadLine());}}
	public string S{get{return Console.ReadLine();}}
	public int[] Ia{get{return Array.ConvertAll(Console.ReadLine().Split(),int.Parse);}}
	public long[] La{get{return Array.ConvertAll(Console.ReadLine().Split(),long.Parse);}}
	public double[] Da{get{return Array.ConvertAll(Console.ReadLine().Split(),double.Parse);}}
	public string[] Sa{get{return Console.ReadLine().Split();}}
	public object[] Oa{get{return Console.ReadLine().Split();}}
	public int[] Ia2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),int.Parse);}}
	public int[] Ia3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),int.Parse);}
	public int[] Ia3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),int.Parse);}
	public long[] La2{get{return Array.ConvertAll(("0 "+Console.ReadLine()+" 0").Split(),long.Parse);}}
	public long[] La3(int a){return Array.ConvertAll((a.ToString()+" "+Console.ReadLine()).Split(),long.Parse);}
	public long[] La3(bool a,int b,bool c,int d){return Array.ConvertAll(((a?b.ToString()+" ":"")+Console.ReadLine()+(c?" "+d.ToString():"")).Split(),long.Parse);}
	public T[] Arr<T>(int n,Func<T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f();}return a;}
	public T[] Arr<T>(int n,Func<int,T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i);}return a;}
	public T[] Arr<T>(int n,Func<string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(Console.ReadLine().Split());}return a;}
	public T[] Arr<T>(int n,Func<int,string[],T> f){var a=new T[n];for(int i=0;i<n;i++){a[i]=f(i,Console.ReadLine().Split());}return a;}
}