Diverge and Converge

AtCoder

IDttpc2024_1_g

Time2000ms

Memory256MB

Difficulty—

You are given a tree $A$ with $N$ vertices. The vertices of $A$ are numbered $1, 2, \dots, N$, and the $i$\-th edge ($1 \leq i \leq N-1$) connects vertices $u_i$ and $v_i$ of $A$. Additionally, you are given another tree $B$ with $N$ vertices. The vertices of $B$ are also numbered $1, 2, \dots, N$, and the $j$\-th edge ($1 \leq j \leq N-1$) connects vertices $x_j$ and $y_j$ of $B$. Your task is to find a pair of permutations $((P_1, P_2, \dots, P_{N-1}), (Q_1, Q_2, \dots, Q_{N-1}))$ that satisfies the following conditions: > **Conditions** > Perform the following operations 1. and 2. in order for $k = 1, 2, \dots, N-1$. **After completing operations 1. and 2. for each $k$**, both $A$ and $B$ must remain as trees. > > 1. In tree $A$, remove the edge connecting vertices $u_{P_k}$ and $v_{P_k}$, and add an edge connecting vertices $x_{Q_k}$ and $y_{Q_k}$. > 2. In tree $B$, remove the edge connecting vertices $x_{Q_k}$ and $y_{Q_k}$, and add an edge connecting vertices $u_{P_k}$ and $v_{P_k}$. It can be proven that under the constraints of this problem, a valid solution always exists. ## Constraints * All input values are integers. * $2 \leq N \leq 1000$ * $1 \leq u_i, v_i, x_j, y_j \leq N$ * The given $A$ and $B$ form trees. ## Input The input is given from Standard Input in the following format: $N$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_{N-1}$ $v_{N-1}$ $x_1$ $y_1$ $x_2$ $y_2$ $\vdots$ $x_{N-1}$ $y_{N-1}$ [samples]

Samples

Input #1

Output #1

3 1 2
2 1 3

Before the operation, $A$ is a path graph, and $B$ is a star graph.
After the operation for $k = 1$, $A$ becomes a star graph, and $B$ becomes a path graph.
In the operations for $k = 2$, the same edge (in terms of vertex numbers) is removed and added, so the shape of the tree remains unchanged.
After completing the operations for $k = 3$, the original shapes of trees $A$ and $B$ are completely swapped.

Input #2

2
1 2
2 1

Output #2

1
1

Input #3

Output #3

1 2 3 4 5 6
1 2 6 4 5 3

API Response (JSON)

{
  "problem": {
    "name": "Diverge and Converge",
    "description": {
      "content": "You are given a tree $A$ with $N$ vertices. The vertices of $A$ are numbered $1, 2, \\dots, N$, and the $i$\\-th edge ($1 \\leq i \\leq N-1$) connects vertices $u_i$ and $v_i$ of $A$. Additionally, you ar",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "ttpc2024_1_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a tree $A$ with $N$ vertices. The vertices of $A$ are numbered $1, 2, \\dots, N$, and the $i$\\-th edge ($1 \\leq i \\leq N-1$) connects vertices $u_i$ and $v_i$ of $A$.\nAdditionally, you ar...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments