CF1367F2.Flying Sort (Hard Version)

This is a hard version of the problem. In this version, the given array can contain equal elements and the constraints on $n$ are greater than in the easy version of the problem.

You are given an array $a$ of $n$ integers (the given array can contain equal elements). You can perform the following operations on array elements:

1. choose any index $i$ ($1 \le i \le n$) and move the element $a[i]$ to the begin of the array;
2. choose any index $i$ ($1 \le i \le n$) and move the element $a[i]$ to the end of the array.

For example, if $n = 5$, $a = [4, 7, 2, 2, 9]$, then the following sequence of operations can be performed:

• after performing the operation of the first type to the second element, the array $a$ will become $[7, 4, 2, 2, 9]$;
• after performing the operation of the second type to the second element, the array $a$ will become $[7, 2, 2, 9, 4]$.

You can perform operations of any type any number of times in any order.

Find the minimum total number of operations of the first and second type that will make the $a$ array sorted in non-decreasing order. In other words, what is the minimum number of operations must be performed so the array satisfies the inequalities $a[1] \le a[2] \le \ldots \le a[n]$.

这是一个该问题的困难版本。在此版本中，给定数组可以包含相等的元素，且 $n$ 的约束条件比该问题的简单版本更大。

你被给定一个包含 $n$ 个整数的数组 $a$（该数组可以包含相等的元素）。你可以对数组元素执行以下两种操作：

1. 任选一个下标 $i$（$1 \le i \le n$），将元素 $a[i]$ 移动到数组开头；
2. 任选一个下标 $i$（$1 \le i \le n$），将元素 $a[i]$ 移动到数组末尾。

例如，若 $n = 5$，$a = [4, 7, 2, 2, 9]$，则可执行如下操作序列：

• 对第二个元素执行第一类操作后，数组 $a$ 变为 $[7, 4, 2, 2, 9]$；
• 对第二个元素执行第二类操作后，数组 $a$ 变为 $[7, 2, 2, 9, 4]$。

你可以以任意顺序、任意次数执行任意类型的操作。

求使数组 $a$ 变为非递减有序（即满足 $a[1] \le a[2] \le \ldots \le a[n]$）所需的最少总操作次数（即第一类与第二类操作次数之和）。

The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of test cases in the test. Then $t$ test cases follow.

Each test case starts with a line containing an integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the size of the array $a$.

Then follow $n$ integers $a_1, a_2, \ldots, a_n$ ($0 \le a_i \le 10^9$) — an array that needs to be sorted by the given operations. The given array can contain equal elements.

The sum of $n$ for all test cases in one test does not exceed $2 \cdot 10^5$.

第一行包含一个整数 $t$（$1 \le t \le 10^4$）—— 测试中测试用例的数量。随后是 $t$ 个测试用例。

每个测试用例以一行开始，该行包含一个整数 $n$（$1 \le n \le 2 \cdot 10^5$）—— 数组 $a$ 的大小。

接下来是一行 $n$ 个整数 $a_1, a_2, \ldots, a_n$（$0 \le a_i \le 10^9$）—— 需要通过给定操作进行排序的数组。该数组中可能包含相等的元素。

所有测试用例的 $n$ 值之和不超过 $2 \cdot 10^5$。

For each test case output one integer — the minimum total number of operations of the first and second type, which will make the array sorted in non-decreasing order.

对于每个测试用例，输出一个整数——使数组变为非递减顺序所需的第 1 类和第 2 类操作的总次数的最小值。

In the first test case, you first need to move two 2, to the beginning of the array. Therefore, the desired sequence of operations: $[4, 7, 2, 2, 9] \rightarrow [2, 4, 7, 2, 9] \rightarrow [2, 2, 4, 7, 9]$.

In the second test case, you need to move the 1 to the beginning of the array, and the 8 — to the end. Therefore, the desired sequence of operations: $[3, 5, 8, 1, 7] \rightarrow [1, 3, 5, 8, 7] \rightarrow [1, 3, 5, 7, 8]$.

In the third test case, the array is already sorted.

在第一个测试用例中，你需要先将两个 2 移动到数组的开头。因此，期望的操作序列为：$[4, 7, 2, 2, 9] \rightarrow [2, 4, 7, 2, 9] \rightarrow [2, 2, 4, 7, 9]$。

在第二个测试用例中，你需要将 1 移动到数组的开头，将 8 移动到数组的末尾。因此，期望的操作序列为：$[3, 5, 8, 1, 7] \rightarrow [1, 3, 5, 8, 7] \rightarrow [1, 3, 5, 7, 8]$。

在第三个测试用例中，数组已经有序。