Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices. The vertices are numbered 1 to N, and the i-th edge connects Vertex a_i and Vertex b_i.

Takahashi loves the number 3. He is seeking a permutation p_1, p_2, \ldots , p_N of integers from 1 to N satisfying the following condition:

For every pair of vertices (i, j), if the distance between Vertex i and Vertex j is 3, the sum or product of p_i and p_j is a multiple of 3.

Here the distance between Vertex i and Vertex j is the number of edges contained in the shortest path from Vertex i to Vertex j.

Help Takahashi by finding a permutation that satisfies the condition.

Constraints

2\leq N\leq 2\times 10^5

1\leq a_i, b_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
a_2 b_2
\vdots
a_{N-1} b_{N-1}

Output

If no permutation satisfies the condition, print -1.

Otherwise, print a permutation satisfying the condition, with space in between.
If there are multiple solutions, you can print any of them.

Sample Input 1

5
1 2
1 3
3 4
3 5

Sample Output 1

1 2 5 4 3

The distance between two vertices is 3 for the two pairs (2, 4) and (2, 5).

p_2 + p_4 = 6

p_2\times p_5 = 6

Thus, this permutation satisfies the condition.

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::io::{stdin, Read};

#[allow(unused_macros)]
macro_rules! parse {
    ($it: ident ) => {};
    ($it: ident, ) => {};
    ($it: ident, $var:ident : $t:tt $($r:tt)*) => {
        let $var = parse_val!($it, $t);
        parse!($it $($r)*);
    };
    ($it: ident, mut $var:ident : $t:tt $($r:tt)*) => {
        let mut $var = parse_val!($it, $t);
        parse!($it $($r)*);
    };
}

#[allow(unused_macros)]
macro_rules! parse_val {
    ($it: ident, [$t:tt; $len:expr]) => {
        (0..$len).map(|_| parse_val!($it, $t)).collect::<Vec<_>>();
    };
    ($it: ident, ($($t: ty),*)) => {
        ($(parse_val!($it, $t)),*)
    };
    ($it: ident, $t: ty) => {
        $it.next().unwrap().parse::<$t>().unwrap()
    };
}

fn solve(s: &str) {
    let mut it = s.split_whitespace();
    parse!(it, n: usize);
    let mut g = vec![vec![]; n];
    for _ in 0..(n - 1) {
        parse!(it, a: usize, b: usize);
        g[a - 1].push(b - 1);
        g[b - 1].push(a - 1);
    }
    let mut types = vec![false; n];
    let mut q = std::collections::VecDeque::new();
    q.push_back((0, false));
    let mut seen = vec![false; n];
    seen[0] = true;
    while let Some((u, t)) = q.pop_front() {
        types[u] = t;
        for &v in g[u].iter() {
            if !seen[v] {
                q.push_back((v, !t));
                seen[v] = true;
            }
        }
    }
    let neven = types.iter().filter(|x| !**x).count();
    let nodd = n - neven;
    let majority = neven < nodd;
    let (na, nb) = if majority {
        (nodd, neven)
    } else {
        (neven, nodd)
    };
    let nac = n / 3 + if n % 3 != 0 { 1 } else { 0 };
    let nbc = n / 3 + if n % 3 == 2 { 1 } else { 0 };
    let mut ret = vec![0; n];
    let mut cura = 1;
    let mut curb = 2;
    let mut curc = 3;
    if nbc <= nb {
        for i in 0..n {
            let cur = if types[i] == majority {
                if cura <= n {
                    &mut cura
                } else {
                    &mut curc
                }
            } else {
                if curb <= n {
                    &mut curb
                } else {
                    &mut curc
                }
            };
            ret[i] = *cur;
            *cur += 3;
        }
    } else {
        for i in 0..n {
            let cur = if types[i] == majority {
                if cura <= n {
                    &mut cura
                } else if curb <= n {
                    &mut curb
                } else {
                    &mut curc
                }
            } else {
                &mut curc
            };
            ret[i] = *cur;
            *cur += 3;
        }
    }
    println!(
        "{}",
        ret.iter()
            .map(|x| x.to_string())
            .collect::<Vec<_>>()
            .join(" ")
    );
}

fn main() {
    let mut s = String::new();
    stdin().read_to_string(&mut s).unwrap();
    solve(&s);
}

#[cfg(test)]
mod tests {
    use super::*;
    #[test]
    fn test_input() {
        let s = "
        ";
        solve(s);
    }
}