ThREE

AtCoder

IDhitachi2020_c

Time2000ms

Memory256MB

Difficulty—

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: $N$ $a_1$ $b_1$ $a_2$ $b_2$ $\vdots$ $a_{N-1}$ $b_{N-1}$ [samples]

Samples

Input #1

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.

API Response (JSON)

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  "problem": {
    "name": "ThREE",
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      "content": "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",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
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    "sync_url": null,
    "sign": "hitachi2020_c"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $b_i$.\nTakahashi loves the number $3$. He is seeking a permutation $p_1, p...",
      "is_translate": false,
      "language": "English"
    }
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}

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