D - Moving Piece

Score : $400$ points

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.

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

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 $8$.

2 3
2 1
10 -7

13

3 3
3 1 2
-1000 -2000 -3000

-1000

We have to make at least one move.

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

29507023469

The absolute value of the answer may be enormous.