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>
#define f first
#define s second
#define mp make_pair
#define pb push_back
#define lp(i,a,n) for(int i=a;i<=n;++i)
#define lpd(i,a,n) for(int i=a;i>=n;--i)
#define mem(a,b) memset(a,b,sizeof a)
#define all(v) v.begin(),v.end()
#define println(a) cout <<(a) <<endl
#define sz(x) ((int)(x).size())
#define readi(x) scanf("%d",&x)
#define read2i(x,y) scanf("%d%d",&x,&y)
#define read3i(x,y,z) scanf("%d%d%d",&x,&y,&z)
#define mod 1000000007
#define eps 1e-8
#define infi 1000000000
#define infll 1000000000000000000ll
using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef vector<ll> vll;
typedef set<int> si;
typedef map<int,int> mii;

const int N = 100002;
int n,m,x[N],y[N],s[N],l[N],ms[N];
vector<pii> g[N];
bool v[N];
ll val[N];

void bfs(int x){
    queue<int> q;
    v[1] = true;
    q.push(1);
    val[1] = x;
    while(sz(q)){
        int i = q.front(); q.pop();
        for(pii j : g[i]) if(!v[j.f]){
            v[j.f] = true;
            l[j.f] = l[i] + 1;
            val[j.f] = j.s - val[i];
            q.push(j.f);
        }
    }
    lp(i,1,n) l[i] &= 1;
}

int main(){
    read2i(n,m);
    lp(i,1,n) ms[i] = infi;
    lp(i,1,m){
        read3i(x[i],y[i],s[i]);
        g[x[i]].pb({y[i], s[i]});
        g[y[i]].pb({x[i], s[i]});
        ms[x[i]] = min(ms[x[i]], s[i]);
        ms[y[i]] = min(ms[y[i]], s[i]);
    }

    bfs(1);

    si need;
    lp(i,1,m){
        if(l[x[i]] == l[y[i]]){
            ll d = s[i] - (val[x[i]] + val[y[i]]);
            if(d & 1) return cout <<0, 0;
            else if(d > 0 and l[x[i]]) return cout <<0, 0;
            else if(d < 0 and !l[x[i]]) return cout <<0, 0;
            else need.insert(d / 2);
        }
        else if(val[x[i]] + val[y[i]] != s[i]) return cout <<0, 0;
    }
    if(sz(need) > 1) return cout <<0, 0;
    if(sz(need) == 1){
        bfs(1 + *need.begin());
        lp(i,1,m) if(val[x[i]] <= 0 or val[y[i]] <= 0 or val[x[i]] + val[y[i]] != s[i])
            return cout <<0, 0;
        return cout <<1, 0;
    }

    ll mxSt = -infll, mnEnd = infll;
    lp(i,1,n){
        ll f,s;
        if(l[i]) f = val[i] - 1, s = val[i] - ms[i] + 1;
        else f = 1 - val[i], s = ms[i] - 1 - val[i];
        if(f > s) swap(f, s);
        mxSt = max(mxSt, f);
        mnEnd = min(mnEnd, s);
    }
    cout <<max(0ll, mnEnd - mxSt + 1);
}

/*
freopen("input.txt","r",stdin);
freopen("output.txt","w",stdout);
*/