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 (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports, non_snake_case, dead_code)]
use proconio::{
    input, fastout,
    marker::{Chars, Bytes, Usize1}
};

struct Graph {
    size: usize,
    G: Vec<Vec<usize>>,
    bic: Vec<usize>,
}

impl Graph {
    fn new(size:usize, G:Vec<Vec<usize>>) -> Self {
        let bic = vec![0; size];
        Graph {
            size, G, bic
        }
    }

    fn dfs(&mut self, v:usize, p:usize) {
        let mut chi = Vec::new();
        for &x in &self.G[v] {
            if x == p { continue };
            self.bic[x] = 1 ^ self.bic[v];
            chi.push(x);
        }
        for &x in &chi {
            self.dfs(x, v);
        }
    }
}

#[fastout]
fn main() {
    input! {
        N: usize,
        edges: [(Usize1, Usize1); N-1]
    }
    let mut G = vec![vec![]; N];
    for &(u, v) in &edges {
        G[u].push(v);
        G[v].push(u)
    }
    let mut graph = Graph::new(N, G);
    graph.dfs(0, 0);
    if graph.bic.iter().sum::<usize>() * 2 > N {
        for i in 0..N {
            graph.bic[i] ^= 1;
        }
    }
    let mut one = 1;
    let mut two = 2;
    let mut three = 3;
    let mut ans = vec![0; N];
    if graph.bic.iter().sum::<usize>() * 3 > N {
        for i in 0..N {
            if graph.bic[i] > 0 && one <= N {
                ans[i] = one;
                one += 3;
            } else if graph.bic[i] == 0 && two <= N {
                ans[i] = two;
                two += 3;
            } else {
                ans[i] = three;
                three += 3;
            }
        }
    } else {
        for i in 0..N {
            if graph.bic[i] == 0 && one <= N {
                ans[i] = one;
                one += 3;
            } else if graph.bic[i] == 0 && two <= N {
                ans[i] = two;
                two += 3;
            } else {
                ans[i] = three;
                three += 3;
            }
        }
    }
    println!("{}", ans.iter().map(|x| x.to_string()).collect::<Vec<_>>().join(" "));

}