Score : $625$ points

问题描述

你得到一个长度为 $N$ 的序列 $A=(A_1,A_2,\ldots,A_N)$，其中包含 $1$ 到 $4$ 之间的整数。

Takahashi 可以执行以下操作任意次数（包括零次）：

• 选择一对整数 $(i,j)$，满足 $1\leq i<j\leq N$，然后交换 $A_i$ 和 $A_j$。

找出使 $A$ 变为非递减所需的最少操作次数。
若对于所有 $1\leq i\leq N-1$，均有 $A_i\leq A_{i+1}$，则称该序列为非递减序列。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a sequence $A=(A_1,A_2,\ldots,A_N)$ of length $N$, consisting of integers between $1$ and $4$.

Takahashi may perform the following operation any number of times (possibly zero):

• choose a pair of integer $(i,j)$ such that $1\leq i<j\leq N$, and swap $A_i$ and $A_j$.

Find the minimum number of operations required to make $A$ non-decreasing.
A sequence is said to be non-decreasing if and only if $A_i\leq A_{i+1}$ for all $1\leq i\leq N-1$.

Constraints

• $2 \leq N \leq 2\times 10^5$
• $1\leq A_i \leq 4$
• 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

Print the minimum number of operations required to make $A$ non-decreasing in a single line.

Sample Input 1

6
3 4 1 1 2 4

Sample Output 1

3

One can make $A$ non-decreasing with the following three operations:

• Choose $(i,j)=(2,3)$ to swap $A_2$ and $A_3$, making $A=(3,1,4,1,2,4)$.
• Choose $(i,j)=(1,4)$ to swap $A_1$ and $A_4$, making $A=(1,1,4,3,2,4)$.
• Choose $(i,j)=(3,5)$ to swap $A_3$ and $A_5$, making $A=(1,1,2,3,4,4)$.

This is the minimum number of operations because it is impossible to make $A$ non-decreasing with two or fewer operations.
Thus, $3$ should be printed.

Sample Input 2

4
2 3 4 1

Sample Output 2

3

update @ 2024/3/10 08:30:13