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)

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 mut graph = vec![vec![]; n];
    for _ in 1..n {
        let a = sc.read::<usize>() - 1;
        let b = sc.read::<usize>() - 1;
        graph[a].push(b);
        graph[b].push(a);
    }

    let mut color = vec![0; n];
    color[0] = 1;
    color_dfs(0, &graph, &mut color);

    let count1 = color.iter().filter(|&&c| c == 1).count();
    let count2 = n - count1;

    let count_m1 = (n + 2) / 3;
    let count_m2 = (n + 1) / 3;

    let mut ans = vec![0; n];

    let (bigger_color, bigger_count) = if count1 > count2 {
        (1, count1)
    } else {
        (2, count2)
    };

    if bigger_count >= count_m1 + count_m2 {
        let mut unused = (1..(n + 1)).filter(|&i| i % 3 != 0).collect::<Vec<usize>>();
        for i in 0..n {
            if color[i] == bigger_color {
                if let Some(u) = unused.pop() {
                    ans[i] = u;
                }
            }
        }
        assert!(unused.is_empty());
        let mut unused = (1..(n + 1)).filter(|&i| i % 3 == 0).collect::<Vec<usize>>();
        for i in 0..n {
            if ans[i] == 0 {
                ans[i] = unused.pop().unwrap();
            }
        }
    } else {
        let mut unused1 = (1..(n + 1)).filter(|&i| i % 3 == 1).collect::<Vec<usize>>();
        for i in 0..n {
            if color[i] == bigger_color {
                if let Some(u) = unused1.pop() {
                    ans[i] = u;
                }
            }
        }
        let mut unused2 = (1..(n + 1)).filter(|&i| i % 3 == 2).collect::<Vec<usize>>();
        for i in 0..n {
            if color[i] != bigger_color {
                if let Some(u) = unused2.pop() {
                    ans[i] = u;
                }
            }
        }

        assert!(unused1.is_empty());
        assert!(unused2.is_empty());

        let mut unused = (1..(n + 1)).filter(|&i| i % 3 == 0).collect::<Vec<usize>>();
        for i in 0..n {
            if ans[i] == 0 {
                ans[i] = unused.pop().unwrap();
            }
        }
    }

    for i in 0..n {
        if i > 0 {
            sc.write(" ");
        }
        sc.write(ans[i].to_string());
    }
    sc.write("\n");
}

fn color_dfs(v: usize, graph: &Vec<Vec<usize>>, color: &mut Vec<usize>) {
    for &next in graph[v].iter() {
        if color[next] == 0 {
            color[next] = if color[v] == 1 { 2 } else { 1 };
            color_dfs(next, graph, color);
        } else {
            assert_ne!(color[v], color[next]);
        }
    }
}

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()
    }
}