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 x: i64 = sc.read();
        graph[a].push((b, x));
        graph[b].push((a, x));
    }

    let ans = solve(graph);
    println!("{}", ans);
}

fn solve(graph: Vec<Vec<(usize, i64)>>) -> i64 {
    let n = graph.len();
    let mut hints: Vec<Option<(i64, bool)>> = vec![None; n];
    hints[0] = Some((0, true));
    let mut q = VecDeque::new();
    q.push_back(0);
    while let Some(v) = q.pop_front() {
        let (sum1, is_add) = hints[v].unwrap();
        for &(next, edge) in graph[v].iter() {
            match hints[next] {
                Some((sum2, next_is_add)) => {
                    if is_add == next_is_add {
                        let x2 = edge - (sum1 + sum2);
                        let x2 = if is_add { x2 } else { -x2 };
                        if x2 <= 0 || x2 % 2 == 1 {
                            return 0;
                        }
                        let x = x2 / 2;
                        let mut num = vec![0; n];
                        num[0] = x;
                        return if check_dfs(0, &mut num, &graph) { 1 } else { 0 };
                    } else {
                        if sum1 + sum2 != edge {
                            return 0;
                        }
                    }
                }
                None => {
                    hints[next] = Some((edge - sum1, !is_add));
                    q.push_back(next);
                }
            }
        }
    }
    //    eprintln!("{:?}", hints);

    let mut upper_bound = std::i64::MAX;
    let mut lower_bound = 1;
    for hint in hints.into_iter() {
        let (s, is_add) = hint.unwrap();
        if is_add {
            lower_bound = cmp::max(lower_bound, -s + 1);
        } else {
            upper_bound = cmp::min(upper_bound, s);
        }
    }

    let ans = upper_bound - lower_bound;
    cmp::max(0, ans)
}

fn check_dfs(v: usize, num: &mut Vec<i64>, graph: &Vec<Vec<(usize, i64)>>) -> bool {
    for &(next, edge) in graph[v].iter() {
        if num[next] == 0 {
            num[next] = edge - num[v];
            if num[next] <= 0 {
                return false;
            }
            if !check_dfs(next, num, graph) {
                return false;
            }
        } else {
            if num[next] + num[v] != edge {
                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) -> IO<R, W> {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: std::ops::Deref<Target = str>>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.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()
    }
}