A12568. The Awesomest Vertex
编程题
普及/提高-
知识点
题目描述
You are given a rooted tree on $n$ vertices. The vertices are numbered from $1$ to $n$ ; the root is the vertex number $1$ .
Each vertex has two integers associated with it: $a_i$ and $b_i$ . We denote the set of all ancestors of $v$ (including $v$ itself) by $R(v)$ . The awesomeness of a vertex $v$ is defined as
$$$$\left| \sum_{w \in R(v)} a_w\right| \cdot \left|\sum_{w \in R(v)} b_w\right|, $$ </p><p>where $|x|$ denotes the absolute value of $x$ . </p><p>Process $q$ queries of one of the following forms: </p><ul> <li> <span class="tex-font-style-tt">1 v x</span> — increase $a\_v$ by a positive integer $x$ . </li><li> <span class="tex-font-style-tt">2 v</span> — report the maximum <span class="tex-font-style-it">awesomeness</span> in the subtree of vertex $v$$$.
Each vertex has two integers associated with it: $a_i$ and $b_i$ . We denote the set of all ancestors of $v$ (including $v$ itself) by $R(v)$ . The awesomeness of a vertex $v$ is defined as
$$$$\left| \sum_{w \in R(v)} a_w\right| \cdot \left|\sum_{w \in R(v)} b_w\right|, $$ </p><p>where $|x|$ denotes the absolute value of $x$ . </p><p>Process $q$ queries of one of the following forms: </p><ul> <li> <span class="tex-font-style-tt">1 v x</span> — increase $a\_v$ by a positive integer $x$ . </li><li> <span class="tex-font-style-tt">2 v</span> — report the maximum <span class="tex-font-style-it">awesomeness</span> in the subtree of vertex $v$$$.
输入格式
The first line contains two integers $n$ and $q$ ( $1 \leq n \leq 2\cdot 10^5, 1 \leq q \leq 10^5$ ) — the number of vertices in the tree and the number of queries, respectively.
The second line contains $n - 1$ integers $p_2, p_3, \dots, p_n$ ( $1 \leq p_i < i$ ), where $p_i$ means that there is an edge between vertices $i$ and $p_i$ .
The third line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $-5000 \leq a_i \leq 5000$ ), the initial values of $a_i$ for each vertex.
The fourth line contains $n$ integers $b_1, b_2, \dots, b_n$ ( $-5000 \leq b_i \leq 5000$ ), the values of $b_i$ for each vertex.
Each of the next $q$ lines describes a query. It has one of the following forms:
- 1 v x ( $1 \leq v \leq n$ , $1\leq x \leq 5000$ ).
- 2 v ( $1 \leq v \leq n$ ).
The second line contains $n - 1$ integers $p_2, p_3, \dots, p_n$ ( $1 \leq p_i < i$ ), where $p_i$ means that there is an edge between vertices $i$ and $p_i$ .
The third line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $-5000 \leq a_i \leq 5000$ ), the initial values of $a_i$ for each vertex.
The fourth line contains $n$ integers $b_1, b_2, \dots, b_n$ ( $-5000 \leq b_i \leq 5000$ ), the values of $b_i$ for each vertex.
Each of the next $q$ lines describes a query. It has one of the following forms:
- 1 v x ( $1 \leq v \leq n$ , $1\leq x \leq 5000$ ).
- 2 v ( $1 \leq v \leq n$ ).
输出格式
For each query of the second type, print a single line with the maximum awesomeness in the respective subtree.
输入输出样例
输入 #1
5 6 1 1 2 2 10 -3 -7 -3 -10 10 3 9 3 6 2 1 2 2 1 2 6 2 1 1 2 5 2 1
输出 #1
100 91 169 240
说明/提示
The initial awesomeness of the vertices is $[100, 91, 57, 64, 57]$ . The most awesome vertex in the subtree of vertex $1$ (the first query) is $1$ , and the most awesome vertex in the subtree of vertex $2$ (the second query) is $2$ .
After the first update (the third query), the awesomeness changes to $[100, 169, 57, 160, 57]$ and thus the most awesome vertex in the whole tree (the fourth query) is now $2$ .
After the second update (the fifth query), the awesomeness becomes $[100, 234, 57, 240, 152]$ , hence the most awesome vertex (the sixth query) is now $4$ .
After the first update (the third query), the awesomeness changes to $[100, 169, 57, 160, 57]$ and thus the most awesome vertex in the whole tree (the fourth query) is now $2$ .
After the second update (the fifth query), the awesomeness becomes $[100, 234, 57, 240, 152]$ , hence the most awesome vertex (the sixth query) is now $4$ .