codeforces#P1968D. Permutation Game
Permutation Game
Description
Bodya and Sasha found a permutation $p_1,\dots,p_n$ and an array $a_1,\dots,a_n$. They decided to play a well-known "Permutation game".
A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).
Both of them chose a starting position in the permutation.
The game lasts $k$ turns. The players make moves simultaneously. On each turn, two things happen to each player:
- If the current position of the player is $x$, his score increases by $a_x$.
- Then the player either stays at his current position $x$ or moves from $x$ to $p_x$.
Knowing Bodya's starting position $P_B$ and Sasha's starting position $P_S$, determine who wins the game if both players are trying to win.
The first line contains a single integer $t$ ($1\le t\le 10^4$) — the number of testcases.
The first line of each testcase contains integers $n$, $k$, $P_B$, $P_S$ ($1\le P_B,P_S\le n\le 2\cdot 10^5$, $1\le k\le 10^9$) — length of the permutation, duration of the game, starting positions respectively.
The next line contains $n$ integers $p_1,\dots,p_n$ ($1 \le p_i \le n$) — elements of the permutation $p$.
The next line contains $n$ integers $a_1,\dots,a_n$ ($1\le a_i\le 10^9$) — elements of array $a$.
It is guaranteed that the sum of values of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each testcase output:
- "Bodya" if Bodya wins the game.
- "Sasha" if Sasha wins the game.
- "Draw" if the players have the same score.
Input
The first line contains a single integer $t$ ($1\le t\le 10^4$) — the number of testcases.
The first line of each testcase contains integers $n$, $k$, $P_B$, $P_S$ ($1\le P_B,P_S\le n\le 2\cdot 10^5$, $1\le k\le 10^9$) — length of the permutation, duration of the game, starting positions respectively.
The next line contains $n$ integers $p_1,\dots,p_n$ ($1 \le p_i \le n$) — elements of the permutation $p$.
The next line contains $n$ integers $a_1,\dots,a_n$ ($1\le a_i\le 10^9$) — elements of array $a$.
It is guaranteed that the sum of values of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each testcase output:
- "Bodya" if Bodya wins the game.
- "Sasha" if Sasha wins the game.
- "Draw" if the players have the same score.
10
4 2 3 2
4 1 2 3
7 2 5 6
10 8 2 10
3 1 4 5 2 7 8 10 6 9
5 10 5 1 3 7 10 15 4 3
2 1000000000 1 2
1 2
4 4
8 10 4 1
5 1 4 3 2 8 6 7
1 1 2 1 2 100 101 102
5 1 2 5
1 2 4 5 3
4 6 9 4 2
4 2 3 1
4 1 3 2
6 8 5 3
6 9 5 4
6 1 3 5 2 4
6 9 8 9 5 10
4 8 4 2
2 3 4 1
5 2 8 7
4 2 3 1
4 1 3 2
6 8 5 3
2 1000000000 1 2
1 2
1000000000 2
Bodya
Sasha
Draw
Draw
Bodya
Sasha
Sasha
Sasha
Sasha
Bodya
Note
Below you can find the explanation for the first testcase, where the game consists of $k=2$ turns.
| Turn | Bodya's position | Bodya's score | Bodya's move | Sasha's position | Sasha's score | Sasha's move |
| first | $3$ | $0 + a_3 = 0 + 5 = 5$ | stays on the same position | $2$ | $0 + a_2 = 0 + 2 = 2$ | moves to $p_2=1$ |
| second | $3$ | $5 + a_3 = 5 + 5 = 10$ | stays on the same position | $1$ | $2 + a_1 = 2 + 7 = 9$ | stays on the same position |
| final results | $3$ | $10$ | $1$ | $9$ |
As we may see, Bodya's score is greater, so he wins the game. It can be shown that Bodya always can win this game.