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 Python (3.4.3) (judge verdict: Runtime Error)

import sys
sys.setrecursionlimit(10**6)

def dfs(v, u, d):
    if d % 2 == 0:
        even.append(v)
    else:
        odd.append(v)
    for i in g[v]:
        if i == u:
            continue
        dfs(i, v, d + 1)
    return

N = int(input())
g = [[] for _ in range(N)]
odd = []
even =[]
ans = [-1] * N
nums = [[], [], []]
for i in range(1, N + 1):
    nums[i % 3].append(i)
sanlst = []
for i in range(N - 1):
    a, b = map(int, input().split())
    a -= 1
    b -= 1
    g[a].append(b)
    g[b].append(a)

dfs(0, 0, 0)
zCnt = N // 3

if len(even) <= zCnt:
    for node in even:
        ans[node] = nums[0].pop()
    rest = nums[0] + nums[1] + nums[2]
    for node in odd:
        ans[node] = rest.pop()
elif len(even) <= zCnt:
    for node in odd:
        ans[node] = nums[0].pop()
    rest = nums[0] + nums[1] + nums[2]
    for node in even:
        ans[node] = rest.pop()
else:
    for i in nums[1]:
        ans[even.pop()] = i
    for i in nums[2]:
        ans[odd.pop()] = i
    for i in even:
        ans[i] = nums[0].pop()
    for i in odd:
        ans[i] = nums[0].pop()

print(*ans)