Score : $500$ points

问题描述

给定一个包含 $N$ 个数的序列：$A=(A_1,A_2,\ldots,A_N)$。其中，每个 $A_i$（$1\leq i\leq N$）满足 $1\leq A_i \leq N$。

高桥和青木将进行 $N$ 轮游戏。对于每一轮 $i=1,2,\ldots,N$，第 $i$ 轮游戏按以下方式进行：

1. 青木指定一个正整数 $K_i$。
2. 在得知青木指定的 $K_i$ 后，高桥从 $1$ 到 $N$（包括两端点）选择一个整数 $S_i$，并将其写在黑板上。
3. 重复以下操作 $K_i$ 次：
   • 将黑板上所写的整数 $x$ 替换为 $A_x$。

若经过 $K_i$ 次迭代后，黑板上出现数字 $i$，则高桥赢得第 $i$ 轮；否则，青木赢得第 $i$ 轮。

这里，对于每一轮 $i=1,2,\ldots,N$，$K_i$ 和 $S_i$ 可以独立选择。

假设两位玩家都采取最优策略以求获胜，请找出高桥赢得的轮数。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a sequence of $N$ numbers: $A=(A_1,A_2,\ldots,A_N)$. Here, each $A_i$ $(1\leq i\leq N)$ satisfies $1\leq A_i \leq N$.

Takahashi and Aoki will play $N$ rounds of a game. For each $i=1,2,\ldots,N$, the $i$-th game will be played as follows.

1. Aoki specifies a positive integer $K_i$.

2. After knowing $K_i$ Aoki has specified, Takahashi chooses an integer $S_i$ between $1$ and $N$, inclusive, and writes it on a blackboard.

3. Repeat the following $K_i$ times.

   • Replace the integer $x$ written on the blackboard with $A_x$.

If $i$ is written on the blackboard after the $K_i$ iterations, Takahashi wins the $i$-th round; otherwise, Aoki wins.
Here, $K_i$ and $S_i$ can be chosen independently for each $i=1,2,\ldots,N$.

Find the number of rounds Takahashi wins if both players play optimally to win.

Constraints

• $1\leq N\leq 2\times 10^5$
• $1\leq A_i\leq N$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Find the number of rounds Takahashi wins if both players play optimally to win.

Sample Input 1

3
2 2 3

Sample Output 1

2

In the first round, if Aoki specifies $K_1=2$, Takahashi cannot win whichever option he chooses for $S_1$: $1$, $2$, or $3$.

For example, if Takahashi writes $S_1=1$ on the initial blackboard, the two operations will change this number as follows: $1\to 2(=A_1)$, $2\to 2(=A_2)$. The final number on the blackboard will be $2(\neq 1)$, so Aoki wins.

On the other hand, in the second and third rounds, Takahashi can win by writing $2$ and $3$, respectively, on the initial blackboard, whatever value Aoki specifies as $K_i$.

Therefore, if both players play optimally to win, Takashi wins two rounds: the second and the third. Thus, you should print $2$.

Sample Input 2

2
2 1

Sample Output 2

2

In the first round, Takahashi can win by writing $2$ on the initial blackboard if $K_1$ specified by Aoki is odd, and $1$ if it is even.

Similarly, there is a way for Takahashi to win the second round. Thus, Takahashi can win both rounds: the answer is $2$.

update @ 2024/3/10 12:18:32