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 (Clang 3.8.0) (judge verdict: Wrong Answer)

#include<iostream>
#include<cstdio>
#include<cstring>
#include<limits>
#include<map>
#include<vector>
using namespace std;
typedef long long ll;

ll N,M;
struct edge {
  ll to,c;
  edge(){}
  edge(ll to,ll c):to(to),c(c){}
};
vector<edge> G[100001];
bool F[100001][2];
ll S[100001][2];

bool dfs(ll v, ll f){
  F[v][f] = true;
  for(edge e : G[v]) {
    if( F[e.to][1-f] ) {
      if (S[e.to][1-f] != e.c - S[v][f]) return false;
    } else {
      S[e.to][1-f] = e.c - S[v][f];
      if( !dfs(e.to, 1-f) ) return false;
    }
  }
  return true;
}

int main() {
  cin >> N >> M;
  for(ll i=0;i<M;i++){
    ll a,b,c;
     cin >> a >> b >> c; --a; --b;
    G[a].emplace_back( b,c);
    G[b].emplace_back( a,c);
  }

  if( !dfs(0,0) ) {
    cout << 0 << endl;
  }

  ll mini1 = INT_MAX, mini2 = INT_MAX;
  ll cnt = 0;
  for(ll i=0;i<N;i++){
    if( F[i][0] ) {
      mini1 = min( mini1, S[i][0] );
      cnt++;
    }
    if( F[i][1] ) {
      mini2 = min( mini2, S[i][1] );
      cnt++;
    }
  }
  

  if( cnt == 2 * N ) {
    cout << 1 << endl;
  } else {
    if( mini1 > mini2 )
      swap(mini1,mini2);
    mini1--;
    mini2 += mini1;
    cout << max(0LL,mini2) << endl;
  }
}