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 <cstdio>
#include <vector>
#include <utility>
#include <tuple>
#include <cstring>
#include <iostream>
using namespace std;

int N, M;
using Graph = vector< vector< pair<int, int> > >;

const int INF = 1 << 28;
bool is_bipartite = true, exist_solution = true;
int color[100010], value[100010], start_val = -INF;

void dfs(Graph &G, int cur, int par=-1, int verify=false) {
    for(auto e : G[cur]) {
        int to, val; tie(to, val) = e;
        if(to == par) continue;

        if(color[to] >= 0) {
            if(color[to] == color[cur]) {
                is_bipartite = false;

                int diff = (value[cur] + value[to]) - val;
                if(abs(diff) % 2) {
                    exist_solution = false;
                }
                else {
                    int chg_val;
                    if(color[cur] == 1) {
                        chg_val = 1 + diff / 2 * (-1);
                    }
                    else {
                        chg_val = 1 + diff / 2;
                    }

                    if(start_val == -INF) {
                        start_val = chg_val;
                    }
                    else {
                        if(start_val != chg_val || chg_val <= 0) {
                            exist_solution = false;
                        }
                    }
                }
            }
        }
        else {
            color[to] = 1 - color[cur];
            value[to] = val - value[cur];
            if(verify && value[to] <= 0) exist_solution = false;
            dfs(G, to, cur);
        }
    }
}

int main() {
    scanf("%d%d", &N, &M);

    Graph G(N);
    for(int i=0; i<M; i++) {
        int u, v, s; scanf("%d%d%d", &u, &v, &s);
        u--; v--;
        G[u].push_back(make_pair(v, s));
        G[v].push_back(make_pair(u, s));
    }

    memset(color, -1, sizeof(color));
    value[0] = color[0] = 1;
    dfs(G, 0, -1, false);

    if(is_bipartite) {
        // color[i] == 1 側について、どれだけ上げないと成立しないか
        // color[i] == 0 側について、どれだけ余裕があるか
        int m1 = 0, m2 = INF;
        for(int i=0; i<N; i++) {
            if(color[i] == 1) {
                m1 = max(m1, -value[i] + 1);
            }
            else {
                m2 = min(m2,  value[i] - 1);
            }
        }

        if(m2 < 0 || m1 > m2) {
            cout << 0 << endl;
        }
        else {
            cout << m2 - m1 + 1 << endl;
        }
    }
    else {
        memset(color, -1, sizeof(color));
        value[0] = start_val;
        color[0] = 1;
        dfs(G, 0, -1, true);

        cout << exist_solution << endl;
    }
    return 0;
}