Yuezheng Ling and Dynamic Tree

题意翻译

给定一棵大小为 $n$ 的 $1$ 为根节点的树，树用如下方式给出：输入 $a_2,a_3,\dots,a_n$，保证 $1\leq a_i<i$，将 $a_i$ 与 $i$ 连边形成一棵树。 接下来有两种操作： - `1 l r x` 令 $a_i=\max(a_i-x,1)(l\leq i\leq r)$。 - `2 u v` 查询在当前的 $a$ 数组构成的树上 $u,v$ 的 LCA。 两种操作共有 $q$ 个。 $2\leq n,q\leq 10^5$，$2\leq l\leq r\leq n$，$1\leq x\leq 10^5$，$1\leq u,v\leq n$

题目描述

Yuezheng Ling gives Luo Tianyi a tree which has $ n $ nodes, rooted at $ 1 $ . Luo Tianyi will tell you that the parent of the $ i $ -th node is $ a_i $ ( $ 1 \leq a_i<i $ for $ 2 \le i \le n $ ), and she will ask you to perform $ q $ queries of $ 2 $ types: 1. She'll give you three integers $ l $ , $ r $ and $ x $ ( $ 2 \le l \le r \le n $ , $ 1 \le x \le 10^5 $ ). You need to replace $ a_i $ with $ \max(a_i-x,1) $ for all $ i $ with $ l \leq i \leq r $ . 2. She'll give you two integers $ u $ , $ v $ ( $ 1 \le u, v \le n $ ). You need to find the [LCA](https://en.wikipedia.org/wiki/Lowest_common_ancestor) of nodes $ u $ and $ v $ (their lowest common ancestor).

输入输出格式

输入格式

The first line contains two integers $ n $ and $ q $ ( $ 2\leq n,q \leq 10^5 $ ) — the number of nodes and the number of queries, respectively. The second line contains $ n-1 $ integers $ a_2, a_3,\dots, a_n $ ( $ 1 \le a_i < i $ ), where $ a_i $ is the parent of the node $ i $ . Next $ q $ lines contain queries. For each query, the first integer of each line is $ t $ ( $ t = 1 $ or $ 2 $ ) — the type of the query. If $ t = 1 $ , this represents the query of the first type. Then, three integers will follow: $ l $ , $ r $ , $ x $ ( $ 2 \le l \le r \le n $ , $ 1 \le x \le 10^5 $ ), meaning that you have to replace $ a_i $ with $ \max(a_i-x,1) $ for all $ i $ with $ l \leq i \leq r $ . If $ t = 2 $ , this represents the query of the second type. Then, two integers will follow: $ u $ and $ v $ ( $ 1 \le u, v \le n $ ), and you have to find the LCA of $ u $ and $ v $ . It's guaranteed that there is at least one query of the second type.

输出格式

For each query of the second type output answer on a new line.

输入输出样例

输入样例 #1

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

输出样例 #1

3
3
1

说明

The tree in example is shown below. After the query of the first type, the tree changes and is looking as shown below. 

题意翻译

给定一棵大小为 $n$ 的 $1$ 为根节点的树，树用如下方式给出：输入 $a_2,a_3,\dots,a_n$，保证 $1\leq a_i<i$，将 $a_i$ 与 $i$ 连边形成一棵树。 接下来有两种操作： - `1 l r x` 令 $a_i=\max(a_i-x,1)(l\leq i\leq r)$。 - `2 u v` 查询在当前的 $a$ 数组构成的树上 $u,v$ 的 LCA。 两种操作共有 $q$ 个。 $2\leq n,q\leq 10^5$，$2\leq l\leq r\leq n$，$1\leq x\leq 10^5$，$1\leq u,v\leq n$