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 <bits/stdc++.h>

using namespace std;
typedef long long ll;
typedef pair<ll, ll> p_ll;

template<class T>
void debug(T itr1, T itr2) { auto now = itr1; while(now<itr2) { cout << *now << " "; now++; } cout << endl; }
#define repr(i,from,to) for (int i=(int)from; i<(int)to; i++)
#define all(vec) vec.begin(), vec.end()
#define rep(i,N) repr(i,0,N)
#define per(i,N) for (int i=(int)N-1; i>=0; i--)

const ll MOD = pow(10,9)+7;
const ll LLINF = pow(2,61)-1;
const int INF = pow(2,30)-1;

vector<ll> fac;
void c_fac(int x=pow(10,6)+10) { fac.resize(x,true); rep(i,x) fac[i] = i ? (fac[i-1]*i)%MOD : 1; }
ll inv(ll a, ll m=MOD) { ll b = m, x = 1, y = 0; while (b!=0) { int d = a/b; a -= b*d; swap(a,b); x -= y*d; swap(x,y); } return (x+m)%m; }
ll nck(ll n, ll k) { return fac[n]*inv(fac[k]*fac[n-k]%MOD)%MOD; }
ll gcd(ll a, ll b) { if (a<b) swap(a,b); return b==0 ? a : gcd(b, a%b); }
ll lcm(ll a, ll b) { return a/gcd(a,b)*b; }

// ----------------------------------------------------------------------
// ----------------------------------------------------------------------

struct edge { 
  int from; int to; 
  bool operator<(const edge& x) const { return from<x.from||(from==x.from&&to<x.to); }
};
vector<vector<int>> adj;
vector<ll> num, oe;
map<edge, ll> e;
bool modify = false;

bool dfs(int n=0, int p=-1) {
  for (auto x: adj[n]) {
    if (x==p) continue;
    if (num[x]!=LLINF) {
      if (num[x]+num[n]==e[{x,n}]) continue;
      else {
        if (modify) return false;
        else {
          if ((e[{x,n}]-(num[x]+num[n]))%2) return false;
          else { 
            ll v = (e[{x,n}]-(num[x]+num[n]))/2; 
            fill(all(num),LLINF); num[x] = v;
            fill(all(oe),0); oe[x] = 1;
            modify = true;
            return dfs(x,-1);
          }
        }
      }
    }
    else {
      oe[x] = oe[n]*-1;
      num[x] = e[{x,n}] - num[n];
      if (!dfs(x,n)) return false;
    }
  }
  return true;
}

// ----------------------------------------------------------------------
// ----------------------------------------------------------------------

int main() {
  int n, m; cin >> n >> m;
  adj.resize(n);
  oe.resize(n); oe[0] = 1;
  num.resize(n,LLINF); num[0] = 0;
  rep(i,m) {
    int u, v; cin >> u >> v; u--; v--;
    ll s; cin >> s;
    adj[u].push_back(v); adj[v].push_back(u);
    e[(edge){u,v}] = s; e[(edge){v,u}] = s;
  }

  if (!dfs()) cout << 0 << endl;
  else {
    if (modify) cout << 1 << endl;
    else {
      ll omax = -1*LLINF, omin = LLINF;
      ll emax = -1*LLINF, emin = LLINF;
      rep(i,n) {
        if (oe[i]==1) { omax = max(omax,num[i]); omin = min(omin,num[i]); }
        else { emax = max(emax,num[i]); emin = min(emin,num[i]); }
      }
      // cout << omin << "~" << omax << " " << emin << "~" << emax << endl;
      // debug(all(num));
      ll result = max((ll)0,omin+emin-1);
      cout << result << endl;
    }
  }
  return 0;
}