Tree and LCS

AtCoder

IDarc156_c

Time2000ms

Memory256MB

Difficulty—

We have a tree $T$ with vertices numbered $1$ to $N$. The $i$\-th edge of $T$ connects vertex $u_i$ and vertex $v_i$. Let us use $T$ to define the **similarity** of a permutation $P = (P_1,P_2,\ldots,P_N)$ of $(1,2,\ldots,N)$ as follows. * For a simple path $x=(x_1,x_2,\ldots,x_k)$ in $T$, let $y=(P_{x_1}, P_{x_2},\ldots,P_{x_k})$. The similarity is the maximum possible length of a longest common subsequence of $x$ and $y$. Construct a permutation $P$ with the minimum similarity. What is a subsequence? A **subsequence** of a sequence is a sequence obtained by removing zero or more elements from that sequence and concatenating the remaining elements without changing the relative order. For instance, $(10,30)$ is a subsequence of $(10,20,30)$, but $(20,10)$ is not. What is a simple path? For vertices $X$ and $Y$ in a graph $G$, a **walk** from $X$ to $Y$ is a sequence of vertices $v_1,v_2, \ldots, v_k$ such that $v_1=X$, $v_k=Y$, and there is an edge connecting $v_i$ and $v_{i+1}$. A **simple path** (or simply a **path**) is a walk such that $v_1,v_2, \ldots, v_k$ are all different. ## Constraints * $2 \leq N \leq 5000$ * $1\leq u_i,v_i\leq N$ * The given graph is a tree. * All numbers in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_{N-1}$ $v_{N-1}$ [samples]

Samples

Input #1

3
1 2
2 3

Output #1

3 2 1

This permutation has a similarity of $1$, which can be computed as follows.

*   For $x=(1)$, we have $y=(P_1)=(3)$. The length of a longest common subsequence of $x$ and $y$ is $0$.
    
*   For $x=(2)$, we have $y=(P_2)=(2)$. The length of a longest common subsequence of $x$ and $y$ is $1$.
    
*   For $x=(3)$, we have $y=(P_2)=(1)$. The length of a longest common subsequence of $x$ and $y$ is $0$.
    
*   For $x=(1,2)$, we have $y=(P_1,P_2)=(3,2)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(2,1)$, the reversal of $(1,2)$.
    
*   For $x=(2,3)$, we have $y=(P_2,P_3)=(2,1)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(3,2)$, the reversal of $(2,3)$.
    
*   For $x=(1,2,3)$, we have $y=(P_1,P_2,P_3)=(3,2,1)$. The length of a longest common subsequence of $x$ and $y$ is $1$. The same goes for $x=(3,2,1)$, the reversal of $(1,2,3)$.
    

We can prove that no permutation has a similarity of $0$ or less, so this permutation is a valid answer.

Input #2

Output #2

3 4 1 2

If multiple permutations have the minimum similarity, you may print any of them. For instance, you may also print `4 3 2 1`.

API Response (JSON)

{
  "problem": {
    "name": "Tree and LCS",
    "description": {
      "content": "We have a tree $T$ with vertices numbered $1$ to $N$. The $i$\\-th edge of $T$ connects vertex $u_i$ and vertex $v_i$. Let us use $T$ to define the **similarity** of a permutation $P = (P_1,P_2,\\ldots,",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc156_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a tree $T$ with vertices numbered $1$ to $N$. The $i$\\-th edge of $T$ connects vertex $u_i$ and vertex $v_i$.\nLet us use $T$ to define the **similarity** of a permutation $P = (P_1,P_2,\\ldots,...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments