101753: [AtCoder]ABC175 D - Moving Piece

Problem Statement

Takahashi will play a game using a piece on an array of squares numbered $1, 2, \cdots, N$. Square $i$ has an integer $C_i$ written on it. Also, he is given a permutation of $1, 2, \cdots, N$: $P_1, P_2, \cdots, P_N$.

Now, he will choose one square and place the piece on that square. Then, he will make the following move some number of times between $1$ and $K$ (inclusive):

• In one move, if the piece is now on Square $i$ $(1 \leq i \leq N)$, move it to Square $P_i$. Here, his score increases by $C_{P_i}$.

Help him by finding the maximum possible score at the end of the game. (The score is $0$ at the beginning of the game.)

Constraints

• $2 \leq N \leq 5000$
• $1 \leq K \leq 10^9$
• $1 \leq P_i \leq N$
• $P_i \neq i$
• $P_1, P_2, \cdots, P_N$ are all different.
• $-10^9 \leq C_i \leq 10^9$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$
$P_1$ $P_2$ $\cdots$ $P_N$
$C_1$ $C_2$ $\cdots$ $C_N$

Output

Print the maximum possible score at the end of the game.

Sample Input 1

5 2
2 4 5 1 3
3 4 -10 -8 8

Sample Output 1

8

When we start at some square of our choice and make at most two moves, we have the following options:

• If we start at Square $1$, making one move sends the piece to Square $2$, after which the score is $4$. Making another move sends the piece to Square $4$, after which the score is $4 + (-8) = -4$.
• If we start at Square $2$, making one move sends the piece to Square $4$, after which the score is $-8$. Making another move sends the piece to Square $1$, after which the score is $-8 + 3 = -5$.
• If we start at Square $3$, making one move sends the piece to Square $5$, after which the score is $8$. Making another move sends the piece to Square $3$, after which the score is $8 + (-10) = -2$.
• If we start at Square $4$, making one move sends the piece to Square $1$, after which the score is $3$. Making another move sends the piece to Square $2$, after which the score is $3 + 4 = 7$.
• If we start at Square $5$, making one move sends the piece to Square $3$, after which the score is $-10$. Making another move sends the piece to Square $5$, after which the score is $-10 + 8 = -2$.

The maximum score achieved is $8$.

Sample Input 2

2 3
2 1
10 -7

Sample Output 2

13

Sample Input 3

3 3
3 1 2
-1000 -2000 -3000

Sample Output 3

-1000

We have to make at least one move.

Sample Input 4

10 58
9 1 6 7 8 4 3 2 10 5
695279662 988782657 -119067776 382975538 -151885171 -177220596 -169777795 37619092 389386780 980092719

Sample Output 4

29507023469

The absolute value of the answer may be enormous.

问题描述

高桥将使用一个棋子在一个标有数字$1, 2, \cdots, N$的方格数组上玩游戏。方格$i$上写有整数$C_i$。此外，他还得到了一个$1, 2, \cdots, N$的排列：$P_1, P_2, \cdots, P_N$。

现在，他将选择一个方格并将棋子放在那个方格上。然后，他将在$1$到$K$（包括）之间进行若干次以下移动：

• 在一次移动中，如果棋子现在位于方格$i$ $(1 \leq i \leq N)$，将其移动到方格$P_i$。这里，他的分数增加$C_{P_i}$。

帮他找到游戏结束时可能的最大分数。（游戏开始时的分数为$0$。）

约束条件

• $2 \leq N \leq 5000$
• $1 \leq K \leq 10^9$
• $1 \leq P_i \leq N$
• $P_i \neq i$
• $P_1, P_2, \cdots, P_N$都是不同的。
• $-10^9 \leq C_i \leq 10^9$
• 输入中的所有值都是整数。

输入

输入将从标准输入以以下格式给出：

$N$ $K$
$P_1$ $P_2$ $\cdots$ $P_N$
$C_1$ $C_2$ $\cdots$ $C_N$

输出

打印游戏结束时可能的最大分数。

样例输入 1

5 2
2 4 5 1 3
3 4 -10 -8 8

样例输出 1

8

当我们从某个方格开始并最多进行两次移动时，我们有以下选择：

• 如果我们从方格$1$开始，一次移动将棋子发送到方格$2$，之后分数为$4$。再次移动将棋子发送到方格$4$，之后分数为$4 + (-8) = -4$。
• 如果我们从方格$2$开始，一次移动将棋子发送到方格$4$，之后分数为$-8$。再次移动将棋子发送到方格$1$，之后分数为$-8 + 3 = -5$。
• 如果我们从方格$3$开始，一次移动将棋子发送到方格$5$，之后分数为$8$。再次移动将棋子发送到方格$3$，之后分数为$8 + (-10) = -2$。
• 如果我们从方格$4$开始，一次移动将棋子发送到方格$1$，之后分数为$3$。再次移动将棋子发送到方格$2$，之后分数为$3 + 4 = 7$。
• 如果我们从方格$5$开始，一次移动将棋子发送到方格$3$，之后分数为$-10$。再次移动将棋子发送到方格$5$，之后分数为$-10 + 8 = -2$。

达到的最大分数是$8$。

样例输入 2

2 3
2 1
10 -7

样例输出 2

13

样例输入 3

3 3
3 1 2
-1000 -2000 -3000

样例输出 3

-1000

我们必须至少进行一次移动。

样例输入 4

10 58
9 1 6 7 8 4 3 2 10 5
695279662 988782657 -119067776 382975538 -151885171 -177220596 -169777795 37619092 389386780 980092719

样例输出 4

29507023469

答案的绝对值可能非常大。