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 PyPy3 (2.4.0) (judge verdict: Wrong Answer)

import sys
from collections import defaultdict as dd
from collections import deque as dq
import heapq
input = sys.stdin.readline
sys.setrecursionlimit(10 ** 6)
hpop = heapq.heappop
hpush = heapq.heappush
N = int(input())
e = dd(list)
for _ in range(N - 1):
  u, v = map(int, input().split())
  e[u].append(v)
  e[v].append(u)

table = [[] for _ in range(N + 1)]
sta = []
for x in range(1, N + 1):
  Q = dq([])
  Q.append((x, 0))
  vis = set()
  while len(Q):
    y, n = Q.popleft()
    if n == 3:
      table[x].append(y)
      continue
    for q in e[y]:
      if q in vis: continue
      vis.add(q)
      Q.append((q, n + 1))
  hpush(sta, (-len(table[x]), x))

nm = [0] * 3
for x in range(1, N + 1): nm[x % 3] += 1

res = [0] * (N + 1)

while nm[0]:
  _, x = hpop(sta)
  res[x] = nm[0] * 3
  nm[0] -= 1

rest = []
while nm[1]:
  if len(sta) == 0: break
  v, x = hpop(sta)
  ng = 0
  for y in table[x]:
    if res[y] % 3 == 1:
      ng = 1
      break
  if ng:
    rest.append((v, x))
    continue
  res[x] = (nm[1] - 1) * 3 + 1
  nm[1] -= 1

for t in rest: hpush(sta, t)

while nm[2]:
  if len(sta) == 0: break
  _, x = hpop(sta)
  ng = 0
  for y in table[x]:
    if res[y] % 3 == 2:
      ng = 1
      break
  if ng: continue
  res[x] = (nm[2] - 1) * 3 + 2
  nm[2] -= 1
#print(res, table)
for i in range(1, N + 1):
  if res[i] == 0:
    print(-1)
    exit(0)
  for j in table[i]:
    if ((res[i] + res[j]) % 3) > 0 and ((res[i] * res[j]) % 3) > 0:
      print(-1)
      exit(0)
print(*res[1: ])