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<iomanip>
#include<math.h>
#include<algorithm>
#include<utility>
#include<queue>
#include<string.h>
#include<string>
#include<set>
#include<map>
#include<unordered_map>
#include<vector>
#include<functional>
using namespace std;
typedef long long ll;
typedef pair<ll,ll> P;
const ll INF=4e18;
const ll MOD=1e9+7;
const double EPS=1e-7;

ll N,M;
struct Edge{
  ll to,cost;
};
vector<Edge> edges[100010];
P state[100010];
ll cnt=0;
map<P,bool> used;
void Serch(ll n){
  for(auto e:edges[n]){
    if(used[P(min(n,e.to),max(n,e.to))])continue;
    if(state[e.to].first==-1){
      if(state[n].second==0){
        state[e.to] = P(-(e.cost-state[n].first),1);
      }else{
        state[e.to] = P(e.cost+state[n].first,0);
      }
      used[P(min(n,e.to),max(n,e.to))]=true;
      Serch(e.to);
    }else{
      used[P(min(n,e.to),max(n,e.to))]=true;
      cnt++;
      if(cnt>=2){
        return;
      }
    }
  }
}
int main(){
    cin>>N>>M;
    for(ll i=0;i<M;i++){
      ll u,v,s;
      cin>>u>>v>>s;
      u--;v--;
      edges[u].push_back(Edge{v,s});
      edges[v].push_back(Edge{u,s});
    }
    for(ll i=0;i<N;i++){
      state[i]=P(-1,0);
    }
    state[0]=P(0,1);
    Serch(0);
    if(cnt==0){
      ll d=1,u=INF;
      for(ll i=1;i<N;i++){
          //cout<<state[i].first<<","<<state[i].second<<endl;
        if(state[i].second==0){
          u=min(u,state[i].first-1);
        }else{
          d=max(d,state[i].first);
        }
      }
      cout<<max(0ll,u-d+1)<<endl;
    }else if(cnt==1){
      cout<<1<<endl;
    }else{
      cout<<0<<endl;
    }
    return 0;
}