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 C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <vector>
#include <algorithm>

struct Node {
    int id;

    std::vector<Node*> dist_1;
    std::vector<Node*> dist_2;
    std::vector<Node*> dist_3;

    explicit Node(int _id)
        : id{_id}
    {
    }
};

int main()
{
    int N;
    std::cin >> N;

    std::vector<Node> nodes;
    nodes.reserve(N);
    for (auto i = 0; i < N; i++) {
        nodes.emplace_back(i);
    }

    for (auto i = 0; i < N - 1; i++) {
        int a, b;
        std::cin >> a >> b;

        auto &node1 = nodes.at(std::min(a, b) - 1),
             &node2 = nodes.at(std::max(a, b) - 1);

        node1.dist_1.push_back(&node2);
    }
    for (auto& node : nodes) {
        std::sort(node.dist_1.begin(), node.dist_1.end());
    }


    // dist 2
    for (auto& node : nodes) {
        for (auto& dist1_node_ptr : node.dist_1) {

            std::copy_if(node.dist_1.begin(), node.dist_1.end(),
                std::back_inserter(dist1_node_ptr->dist_2),
                [&](auto const& inserted_node_ptr) {
                    return dist1_node_ptr->id < inserted_node_ptr->id;
                });


            std::copy_if(dist1_node_ptr->dist_1.begin(), dist1_node_ptr->dist_1.end(),
                std::back_inserter(node.dist_2),
                [&](auto const& inserted_node_ptr) {
                    return node.id < inserted_node_ptr->id;
                });
        }
    }

    // for (auto const& node : nodes) {
    //     std::cout << node.id << ":";
    //     for (auto const& dist2_node_ptr : node.dist_2) {
    //         std::cout << " " << dist2_node_ptr->id;
    //     }
    //     std::cout << std::endl;
    // }


    for (auto& node : nodes) {
        for (auto& dist2_node_ptr : node.dist_2) {

            std::copy_if(node.dist_1.begin(), node.dist_1.end(),
                std::back_inserter(dist2_node_ptr->dist_3),
                [&](auto const& inserted_node_ptr) {
                    return dist2_node_ptr->id < inserted_node_ptr->id;
                });


            std::copy_if(dist2_node_ptr->dist_1.begin(), dist2_node_ptr->dist_1.end(),
                std::back_inserter(node.dist_3),
                [&](auto const& inserted_node_ptr) {
                    return node.id < inserted_node_ptr->id;
                });
        }
    }
    for (auto& node : nodes) {
        std::sort(node.dist_3.begin(), node.dist_3.end());
    }
    for (auto& node : nodes) {
        // remove incorrect nodes from node.dist_3

        std::vector<Node*> tmp;
        std::set_difference(node.dist_3.begin(), node.dist_3.end(),
            node.dist_1.begin(), node.dist_1.end(),
            std::back_inserter(tmp));
        std::swap(tmp, node.dist_3);
    }

    // for (auto const& node : nodes) {
    //     std::cout << node.id << ":";
    //     for (auto const& dist2_node_ptr : node.dist_3) {
    //         std::cout << " " << dist2_node_ptr->id;
    //     }
    //     std::cout << std::endl;
    // }

    return 0;
}