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 io::*;
use std::cmp::{max, min};
use std::collections::VecDeque;

fn main() {
    let con = read_string();
    let mut sc = Scanner::new(&con);

    let n: usize = sc.next();
    let m: usize = sc.next();

    let mut g: Vec<Vec<(usize, i64)>> = vec![vec![]; n];

    for _ in 0..m {
        let u = sc.next::<usize>() - 1;
        let v = sc.next::<usize>() - 1;
        let s: i64 = sc.next();

        g[u].push((v, s));
        g[v].push((u, s));
    }

    let mut q = VecDeque::new();
    q.push_back((0, 1, 0));

    let mut assign = vec![(0, 0); n];
    assign[0] = (1, 0);

    let mut only = None;

    while let Some((cur, sign, fact)) = q.pop_front() {
        for &(to, s) in g[cur].iter() {
            let nsign = -sign;
            let nfact = s - fact;

            if assign[to].0 == 0 {
                assign[to] = (nsign, nfact);
                q.push_back((to, nsign, nfact));
            } else {
                if assign[to] != (nsign, nfact) {
                    // println!("{:?}, {:?}", assign[to], (nsign, nfact));

                    if assign[to].0 == nsign {
                        println!("0");
                        return;
                    }

                    let (a, b) = if assign[to].0 > 0 {
                        (assign[to].1, nfact)
                    } else {
                        (nfact, assign[to].1)
                    };

                    // 2x = b-a;

                    if (b - a) % 2 != 0 {
                        println!("0");
                        return;
                    }

                    let x = (b - a) / 2;

                    if let Some(ox) = only {
                        if x != ox {
                            println!("0");
                            return;
                        }
                    } else {
                        only = Some(x);
                    }
                }
            }
        }
    }

    if let Some(x) = only {
        for i in 0..n {
            let mut up = std::i64::MAX;
            for &(_, s) in g[i].iter() {
                up = min(up, s - 1);
            }
            let (sign, fact) = assign[i];
            let label = sign * x + fact;

            if !(label >= 1 && label <= up) {
                println!("0");
                return;
            }
        }

        println!("1");
        return;
    }

    let mut lo = 1;
    let mut hi = 1000_000_001;

    for i in 0..n {
        let mut up = std::i64::MAX;
        for &(_, s) in g[i].iter() {
            up = min(up, s - 1);
        }

        let (sign, fact) = assign[i];

        // 1 <= sign * x + fact <= up

        // if sign == 1  => 1-fact <= x <= up-fact
        // if sign == -1 => -up+fact <= x <= fact-1

        let (nlo, nhi) = if sign > 0 {
            (1 - fact, up - fact)
        } else {
            (-up + fact, -1 + fact)
        };

        lo = max(lo, nlo);
        hi = min(hi, nhi);
    }

    // println!("[{}, {}]", lo, hi);
    println!("{}", max(0, hi - lo + 1));
}

//-----

pub mod io {
    use std;
    use std::str::FromStr;

    pub struct Scanner<'a> {
        iter: std::str::SplitWhitespace<'a>,
    }

    impl<'a> Scanner<'a> {
        pub fn new(s: &'a str) -> Scanner<'a> {
            Scanner {
                iter: s.split_whitespace(),
            }
        }

        pub fn next<T: FromStr>(&mut self) -> T {
            let s = self.iter.next().unwrap();
            if let Ok(v) = s.parse::<T>() {
                v
            } else {
                panic!("Parse error")
            }
        }

        pub fn next_vec_len<T: FromStr>(&mut self) -> Vec<T> {
            let n: usize = self.next();
            self.next_vec(n)
        }

        pub fn next_vec<T: FromStr>(&mut self, n: usize) -> Vec<T> {
            (0..n).map(|_| self.next()).collect()
        }
    }

    pub fn read_string() -> String {
        use std::io::Read;

        let mut s = String::new();
        std::io::stdin().read_to_string(&mut s).unwrap();
        s
    }

    pub fn read_line() -> String {
        let mut s = String::new();
        std::io::stdin().read_line(&mut s).unwrap();
        s.trim_right().to_owned()
    }
}