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: Accepted)

#include <bits/stdc++.h>
using namespace std;
const int INF=1e9;
const int MOD=1e9+7;
//const int MOD=998244353;
const long long LINF=1e18;
#define int long long
//template
/*
考察
一点値決めればあとは自動で決まる気がする
0にiを入れたときの各値をiで表してあげて、可能の範囲を見て決定
0にiを入れてトポロジカルチックにやる？
とりあえずBFSとかで適当にやって、最後にその範囲とかからガンバルンバ
*/
template <typename T>
void fin(T a){
  cout<<a<<endl;
  exit(0);
}
//main
signed main(){
  int N,M;cin>>N>>M;
  typedef pair<int,int> P;
  std::vector<P> edge[N];
  while(M--){
    int u,v,s;cin>>u>>v>>s;u--;v--;
    edge[u].push_back(P(v,s));
    edge[v].push_back(P(u,s));
  }
  std::vector<P> val[N];
  val[0].push_back(P(0,1));
  queue<P> que;//親、子
  std::vector<int> used(N,0);
  que.push(P(-1,0));
  while(que.size()){
    P p=que.front();que.pop();
    int pa=p.first,now=p.second;
    if(used[now])continue;used[now]++;
    for(auto q:edge[now]){
      int po=q.first,we=q.second;
      if(po==pa)continue;
      val[po].push_back(P(we-val[now][0].first,-val[now][0].second));
      if(!used[po])que.push(P(now,po));
    }
  }
  int ans_i=-LINF,tmp;
  for(int i=0;i<N;i++){
    for(int j=1;j<val[i].size();j++){
      if(val[i][0]==val[i][j])continue;
      int n=val[i][0].second-val[i][j].second;
      int num=val[i][j].first-val[i][0].first;
      if(!n)fin(0);
      if(abs(num)%abs(n))fin(0);
      tmp=num/n;
      //cout<<i<<" "<<j<<" "<<val[i][0].first<<" "<<val[i][0].second<<" "<<val[i][j].first<<" "<<val[i][j].second<<" "<<ans_i<<" "<<tmp<<endl;
      if(ans_i>-LINF&&ans_i!=tmp)fin(0);
      ans_i=tmp;
    }
  }
  if(ans_i>-LINF){
    for(int i=0;i<N;i++)
      if(val[i][0].first+val[i][0].second*ans_i<=0)fin(0);
    fin(1);
  }
  int mn=-LINF,mx=LINF;
  for(int i=0;i<N;i++){
    int num=val[i][0].first,n=val[i][0].second;
    if(!n){
      if(val[i][0].first>0)continue;
      fin(0);
    }
    if(n>0){//num<n*aになるようにaを設定（mnを）
      mn=max(mn,(-num)/n+1);
    }
    else{//num<n*aになるようにaを設定(mxを)
      tmp=num/(-n);if(!(abs(num)%abs(n)))tmp--;
      mx=min(mx,tmp);
    }
  }
  //for(int i=0;i<N;i++)cout<<"VAL"<<val[i][0].first<<" "<<val[i][0].second<<endl;
  //cout<<mn<<" "<<mx<<endl;
  fin(max(0LL,mx-mn+1));
}