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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <vector>
#include <string>
#include <cmath>
#include <algorithm>
#include <utility>
#include <queue>
#include <set>
#include <map>
#include <deque>
#include <iomanip>
#include <cstdio>

using namespace std;
typedef  long long ll;
typedef pair<int,int> PII;
typedef vector<int> VI;
typedef vector<VI> VVI;
#define  MP make_pair
#define  PB push_back
#define inf  1000000007
#define rep(i,n) for(int i=0;i<(int)(n);++i)

template<typename A, size_t N, typename T>
void Fill(A (&array)[N], const T &val){
    std::fill( (T*)array, (T*)(array+N), val );
}
vector<ll> p,q;
vector<bool>fp,fq;
vector<vector<pair<int,int> > > g;

bool flag = 0;
void dfs(int x,ll s,bool f){
    if(f){
        if(fp[x]==1){
            if(p[x]!=s)flag = 1;
            return ;
        }
        p[x] = s;
        fp[x] = 1;
        for(auto y:g[x]){
            dfs(y.first,y.second-s,0);
        }
    }else{
        if(fq[x]==1){
            if(q[x]!=s)flag = 1;
            return ;
        }
        q[x] = s;
        fq[x] = 1;
        for(auto y:g[x]){
            dfs(y.first,y.second-s,1);
        }
    }
}

int main(){
    int n,m;
    cin >> n >> m;
    p.resize(n);
    q.resize(n);
    g.resize(n);
    fp.resize(n);
    fq.resize(n);
    for(int i=0;i<m;i++){
        int u,v,s;
        cin >> u >> v >> s;
        u--;
        v--;
        g[u].PB(MP(v,s));
        g[v].PB(MP(u,s));
    }
    dfs(0,0,1);
    if(flag){
        cout << 0 << endl;
        return 0;
    }
    ll lo=0;
    ll up = inf;
    set<ll> st;
    for(int i=0;i<n;i++){
        if(fp[i]&&fq[i]){
            ll x = q[i]-p[i];
            if(x<=0){
                flag  =1;
            }else if(x%2!=0){
                flag = 1;
            }
            st.insert(x);
        }
    }
    if(flag||st.size()>1){
        cout << 0 << endl;
        return 0;
    }
    if(st.size()==1){
        cout << 1 << endl;
        return 0;
    }
    for(int i=0;i<n;i++){
        if(fp[i])lo = max(lo,-p[i]+1);
        if(fq[i])up = min(up,q[i]-1);
    }
    if(lo<0||up<0||lo>up){
        cout << 0 << endl;
        return 0;
    }
    cout << up-lo+1 << endl;
    return 0;
}