USACO 2025 February Contest, Gold
Problem 1. Bessie's Function
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Bessie has a special function $f(x)$ that takes as input an integer in $[1, N]$ and returns an integer in $[1, N]$ ($1 \le N \le 2 \cdot 10^5$). Her function $f(x)$ is defined by $N$ integers $a_1 \ldots a_N$ where $f(x) = a_x$ ($1 \le a_i \le N$).
Bessie wants this function to be idempotent. In other words, it should satisfy $f(f(x)) = f(x)$ for all integers $x \in [1, N]$.
For a cost of $c_i$, Bessie can change the value of $a_i$ to any integer in $[1, N]$ ($1 \le c_i \le 10^9$). Determine the minimum total cost Bessie needs to make $f(x)$ idempotent.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains $N$.The second line contains $N$ space-separated integers $a_1,a_2,\dots,a_N$.
The third line contains $N$ space-separated integers $c_1,c_2,\dots,c_N$.
OUTPUT FORMAT (print output to the terminal / stdout):
Output the minimum total cost Bessie needs to make $f(x)$ idempotent.SAMPLE INPUT:
5 2 4 4 5 3 1 1 1 1 1
SAMPLE OUTPUT:
3
We can change $a_1 = 4$, $a_4 = 4$, $a_5 = 4$. Since all $c_i$ equal one, the total cost is equal to $3$, the number of changes. It can be shown that there is no solution with only $2$ or fewer changes.
SAMPLE INPUT:
8 1 2 5 5 3 3 4 4 9 9 2 5 9 9 9 9
SAMPLE OUTPUT:
7
We change $a_3 = 3$ and $a_4 = 4$. The total cost is $2+5=7$.
SCORING:
Subtasks:
- Input 3: $N\le 20$
- Inputs 4-9: $a_i\ge i$
- Inputs 10-15: All $a_i$ are distinct.
- Inputs 16-21: No additional constraints.
Additionally, in each of the last three subtasks, the first half of tests will satisfy $c_i=1$ for all $i$.
Problem credits: Avnith Vijayram