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 Rust (1.15.1) (judge verdict: Accepted)

use std::cmp;
use std::collections::VecDeque;

fn main() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());
    let n: usize = sc.read();
    let m: usize = sc.read();
    let mut graph = vec![vec![]; n];
    for _ in 0..m {
        let a = sc.read::<usize>() - 1;
        let b = sc.read::<usize>() - 1;
        let sum: i64 = sc.read();
        graph[a].push((b, sum));
        graph[b].push((a, sum));
    }

    let mut hint = vec![None; n];
    hint[0] = Some((0, true));
    let mut q = VecDeque::new();
    q.push_back(0);
    while let Some(v) = q.pop_front() {
        let (v_sum, is_add) = hint[v].unwrap();
        for &(next, sum) in graph[v].iter() {
            if let Some((next_sum, next_is_add)) = hint[next] {
                if next_is_add == !is_add && next_sum != sum - v_sum {
                    println!("0");
                    return;
                }
                if next_is_add == is_add {
                    let x2 = if is_add { 1 } else { -1 } * (sum - v_sum - next_sum);
                    if x2 <= 0 || x2 % 2 == 1 {
                        println!("0");
                        return;
                    }
                    let x = x2 / 2;
                    let mut num = vec![None; n];
                    num[0] = Some(x);
                    if check(0, &graph, &mut num) {
                        println!("1");
                    } else {
                        println!("0");
                    }
                    return;
                }
            } else {
                hint[next] = Some((sum - v_sum, !is_add));
                q.push_back(next);
            }
        }
    }

    let mut max_x = 1 << 50;
    let mut min_x = 1;
    for i in 0..n {
        let (sum, is_add) = hint[i].unwrap();
        if is_add {
            min_x = cmp::max(min_x, 1 - sum);
        } else {
            max_x = cmp::min(max_x, sum - 1);
        }
    }

    let ans = cmp::max(0, max_x - min_x + 1);
    println!("{}", ans);
}

fn check(v: usize, graph: &Vec<Vec<(usize, i64)>>, num: &mut Vec<Option<i64>>) -> bool {
    let v_num = num[v].unwrap();
    for &(next, sum) in graph[v].iter() {
        if let Some(next_num) = num[next] {
            if next_num + v_num != sum {
                return false;
            }
        } else {
            num[next] = Some(sum - v_num);
            if sum - v_num <= 0 {
                return false;
            }
            if !check(next, graph, num) {
                return false;
            }
        }
    }
    true
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> Self {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: ToString>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.to_string().as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}