Score : $500$ points

问题描述

有 $N$ 个球从左到右排列。左边第 $i$ 个球的颜色为颜色 $C_i$，并且上面写有一个整数 $X_i$。

高桥希望重新排列这些球，使得球上所写的整数从左到右非严格递增。换句话说，他的目标是达到这样一个状态：对于所有 $1\leq i\leq N-1$，左边第 $(i+1)$ 个球上所写的数字大于或等于左边第 $i$ 个球上的数字。

为了实现这一目标，高桥可以任意次数（包括零次）重复以下操作：

选择一个整数 $i$，满足 $1\leq i\leq N-1$。
若左边第 $i$ 个球和第 $(i+1)$ 个球的颜色不同，则需支付成本 $1$（如果颜色相同则不产生成本）。
将左边第 $i$ 个球与第 $(i+1)$ 个球交换位置。

请找出高桥需要支付的最小总成本以达成他的目标。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ balls arranged from left to right. The color of the $i$-th ball from the left is Color $C_i$, and an integer $X_i$ is written on it.

Takahashi wants to rearrange the balls so that the integers written on the balls are non-decreasing from left to right. In other words, his objective is to reach a situation where, for every $1\leq i\leq N-1$, the number written on the $(i+1)$-th ball from the left is greater than or equal to the number written on the $i$-th ball from the left.

For this, Takahashi can repeat the following operation any number of times (possibly zero):

Choose an integer $i$ such that $1\leq i\leq N-1$.
If the colors of the $i$-th and $(i+1)$-th balls from the left are different, pay a cost of $1$. (No cost is incurred if the colors are the same).
Swap the $i$-th and $(i+1)$-th balls from the left.

Find the minimum total cost Takahashi needs to pay to achieve his objective.

Constraints

• $2 \leq N \leq 3\times 10^5$
• $1\leq C_i\leq N$
• $1\leq X_i\leq N$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$C_1$ $C_2$ $\ldots$ $C_N$

$X_1$ $X_2$ $\ldots$ $X_N$

Output

Print the minimum total cost Takahashi needs to pay to achieve his objective, as an integer.

Sample Input 1

5
1 5 2 2 1
3 2 1 2 1

Sample Output 1

6

Let us represent a ball as $($Color$,$ Integer$)$. The initial situation is $(1,3)$, $(5,2)$, $(2,1)$, $(2,2)$, $(1,1)$. Here is a possible sequence of operations for Takahashi:

• Swap the $1$-st ball (Color $1$) and $2$-nd ball (Color $5$). Now the balls are arranged in the order $(5,2)$, $(1,3)$, $(2,1)$, $(2,2)$, $(1,1)$.
• Swap the $2$-nd ball (Color $1$) and $3$-rd ball (Color $2$). Now the balls are arranged in the order $(5,2)$, $(2,1)$, $(1,3)$, $(2,2)$, $(1,1)$.
• Swap the $3$-rd ball (Color $1$) and $4$-th ball (Color $2$). Now the balls are in the order $(5,2)$, $(2,1)$, $(2,2)$, $(1,3)$, $(1,1)$.
• Swap the $4$-th ball (Color $1$) and $5$-th ball (Color $1$). Now the balls are in the order $(5,2)$, $(2,1)$, $(2,2)$, $(1,1)$, $(1,3)$.
• Swap the $3$-rd ball (Color $2$) and $4$-th ball (Color $1$). Now the balls are in the order$(5,2)$, $(2,1)$, $(1,1)$, $(2,2)$, $(1,3)$.
• Swap the $1$-st ball (Color $5$) and $2$-nd ball (Color $2$). Now the balls are in the order $(2,1)$, $(5,2)$, $(1,1)$, $(2,2)$, $(1,3)$.
• Swap the $2$-nd ball (Color $5$) and $3$-rd ball (Color $1$). Now the balls are in the order $(2,1)$, $(1,1)$, $(5,2)$, $(2,2)$, $(1,3)$.

After the last operation, the numbers written on the balls are $1,1,2,2,3$ from left to right, which achieves Takahashi's objective.

The $1$-st, $2$-nd, $3$-rd, $5$-th, $6$-th, and $7$-th operations incur a cost of $1$ each, for a total of $6$, which is the minimum. Note that the $4$-th operation does not incur a cost since the balls are both in Color $1$.

Sample Input 2

3
1 1 1
3 2 1

Sample Output 2

0

All balls are in the same color, so no cost is incurred in swapping balls.

Sample Input 3

3
3 1 2
1 1 2

Sample Output 3

0

Takahashi's objective is already achieved without any operation.

update @ 2024/3/10 11:04:48