Score : $600$ points

问题描述

我们有一个有向图 $G_S$，包含 $N$ 个顶点，编号为 $1$ 到 $N$。它有 $N-1$ 条边。第 $i$ 条边（$1\leq i \leq N-1$）从顶点 $p_i\ (1\leq p_i \leq i)$ 指向顶点 $i+1$。

我们还有一个有向图 $G$，同样包含 $N$ 个顶点，编号也是 $1$ 到 $N$。最初，$G$ 等于 $G_S$。按照给定顺序对 $G$ 进行 $Q$ 个查询操作。查询有两种类型，如下所示。

• 1 u v：在 $G$ 中添加一条从顶点 $u$ 指向顶点 $v$ 的边。保证满足以下条件：
  • $u \neq v$。
  • 在 $G_S$ 上，通过一些边可以从顶点 $v$ 达到顶点 $u$。
• 2 x：输出一个可以通过 $G$ 上的一些边（包括顶点 $x$ 自身的边）从顶点 $x$ 达到的最小顶点编号。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a directed graph $G_S$ with $N$ vertices numbered $1$ to $N$. It has $N-1$ edges. The $i$-th edge $(1\leq i \leq N-1)$ goes from vertex $p_i\ (1\leq p_i \leq i)$ to vertex $i+1$.

We have another directed graph $G$ with $N$ vertices numbered $1$ to $N$. Initially, $G$ equals $G_S$. Process $Q$ queries on $G$ in the order they are given. There are two kinds of queries as follows.

• 1 u v : Add an edge to $G$ that goes from vertex $u$ to vertex $v$. It is guaranteed that the following conditions are satisfied.
  • $u \neq v$.
  • On $G_S$, vertex $u$ is reachable from vertex $v$ via some edges.
• 2 x : Print the smallest vertex number of a vertex reachable from vertex $x$ via some edges on $G$ (including vertex $x$).

Constraints

• $2\leq N \leq 2\times 10^5$
• $1\leq Q \leq 2\times 10^5$
• $1\leq p_i\leq i$
• For each query in the first format:
  • $1\leq u,v \leq N$.
  • $u \neq v$.
  • On $G_S$, vertex $u$ is reachable from vertex $v$ via some edges.
• For each query in the second format, $1\leq x \leq N$.
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$p_1$ $p_2$ $\dots$ $p_{N-1}$

$Q$

$\mathrm{query}_1$

$\mathrm{query}_2$

$\vdots$

$\mathrm{query}_Q$

Here, $\mathrm{query}_i$ denotes the $i$-th query and is in one of the following formats:

$1$ $u$ $v$

$2$ $x$

Output

Print $q$ lines, where $q$ is the number of queries in the second format. The $i$-th line $(1\leq j \leq q)$ should contain the answer to the $j$-th query in the second format.

Sample Input 1

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

Sample Output 1

4
2
1

• At the time of the first query, only vertex $4$ is reachable from vertex $4$ via some edges on $G$.
• At the time of the third query, vertices $2$, $3$, $4$, and $5$ are reachable from vertex $4$ via some edges on $G$.
• At the time of the fifth query, vertices $1$, $2$, $3$, $4$, and $5$ are reachable from vertex $4$ via some edges on $G$.

Sample Input 2

7
1 1 2 2 3 3
10
2 5
1 5 2
2 5
1 2 1
1 7 1
1 6 3
2 5
2 6
2 1
1 7 1

Sample Output 2

5
2
1
1
1

update @ 2024/3/10 12:17:02