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)

#[allow(unused_imports)]
use std::cmp::{Ordering};

#[allow(unused_imports)]
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};

#[allow(unused_imports)]
use std::io::{stdin, stdout, BufWriter, Write};

#[allow(unused_imports)]
use std::iter::FromIterator;

#[allow(unused_macros)]
macro_rules! min {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::min($x, min!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! max {
    ($x: expr) => ($x);
    ($x: expr, $($z: expr),+) => {{
        std::cmp::max($x, max!($($z),*))
    }}
}

#[allow(unused_macros)]
macro_rules! get { 
    ([$t: ty]) => { 
        { 
            let mut line = String::new(); 
            stdin().read_line(&mut line).unwrap(); 
            line.split_whitespace().map(|t|t.parse::<$t>().unwrap()).collect::<Vec<_>>()
        }
    };
    ([$t: ty];$n: expr) => {
        (0..$n).map(|_|get!([$t])).collect::<Vec<_>>()
    };
    ($t: ty) => {
        {
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            line.trim().parse::<$t>().unwrap()
        }
    };
    ($($t: ty),*) => {
        { 
            let mut line = String::new();
            stdin().read_line(&mut line).unwrap();
            let mut iter = line.split_whitespace();
            ($(iter.next().unwrap().parse::<$t>().unwrap(),)*)
        }
    };
    ($t: ty; $n: expr) => {
        (0..$n).map(|_|get!($t)).collect::<Vec<_>>()
    };
    ($($t: ty),*; $n: expr) => {
        (0..$n).map(|_|get!($($t),*)).collect::<Vec<_>>()
    };
}

#[allow(unused_macros)]
macro_rules! debug {
    ($($a:expr),*) => {
        #[cfg(debug_assertions)]
        writeln!(&mut std::io::stderr(), concat!("[DEBUG] ", $(stringify!($a), "={:?} "),*), $($a),*).unwrap();
    }
}

const BIG_STACK_SIZE: bool = true;

#[allow(dead_code)]
fn main() {
    use std::thread;
    if BIG_STACK_SIZE {
        thread::Builder::new()
            .stack_size(32 * 1024 * 1024)
            .name("solve".into())
            .spawn(solve)
            .unwrap()
            .join()
            .unwrap();
    } else {
        solve();
    }
}

fn solve() {
    let n = get!(usize);
    let ab = get!(usize, usize; n-1);

    let mut nodes = vec![Vec::new(); n+1];
    for (a,b) in ab {
        nodes[a].push(b);
        nodes[b].push(a);
    }

    let mut cand = vec![Vec::new(); 3];
    for i in 1..n+1 {
        cand[i%3].push(i);
    }

    let mut vec = vec![Vec::new();2];
    fn dfs(node: usize, nodes: &Vec<Vec<usize>>, parent: usize, result: &mut Vec<Vec<usize>>, color: usize) {
        result[color].push(node);
        for &next in &nodes[node] {
            if next == parent {
                continue;
            }
            dfs(next, nodes, node, result, (color+1)%2);
        }
    }
    dfs(1,&nodes,0,&mut vec,0);
    let mut res = vec![0;n+1];
    if vec[0].len() > n/3 && vec[1].len() > n/3 {
        for i in 0..2 {
            for &e in &vec[i] {
                res[e] = if let Some(val) = cand[i+1].pop() {
                    val
                } else {
                    cand[0].pop().unwrap()
                };
            }
        }
    } else  if vec[0].len() > n/3 {
        for &e in &vec[1] {
            res[e] = cand[0].pop().unwrap();
        }
        for &e in &vec[0] {
            res[e] = if let Some(val) = cand[1].pop() {
                val
            } else if let Some(val) = cand[2].pop() {
                val
            } else {
                cand[0].pop().unwrap()
            };
        }
    } else {
        for &e in &vec[0] {
            res[e] = cand[0].pop().unwrap();
        }
        for &e in &vec[1] {
            res[e] = if let Some(val) = cand[1].pop() {
                val
            } else if let Some(val) = cand[2].pop() {
                val
            } else {
                cand[0].pop().unwrap()
            };
        }
    }
    for i in 1..n+1 {
        print!("{} ", res[i]);
    }
    println!("");
}