A15134. LIS or Reverse LIS?
编程题
普及/提高-
知识点
题目描述
You are given an array $a$ of $n$ positive integers.
Let $\text{LIS}(a)$ denote the length of [longest strictly increasing subsequence](https://en.wikipedia.org/wiki/Longest_increasing_subsequence) of $a$ . For example,
- $\text{LIS}([2, \underline{1}, 1, \underline{3}])$ = $2$ .
- $\text{LIS}([\underline{3}, \underline{5}, \underline{10}, \underline{20}])$ = $4$ .
- $\text{LIS}([3, \underline{1}, \underline{2}, \underline{4}])$ = $3$ .
We define array $a'$ as the array obtained after reversing the array $a$ i.e. $a' = [a_n, a_{n-1}, \ldots , a_1]$ .
The beauty of array $a$ is defined as $min(\text{LIS}(a),\text{LIS}(a'))$ .
Your task is to determine the maximum possible beauty of the array $a$ if you can rearrange the array $a$ arbitrarily.
Let $\text{LIS}(a)$ denote the length of [longest strictly increasing subsequence](https://en.wikipedia.org/wiki/Longest_increasing_subsequence) of $a$ . For example,
- $\text{LIS}([2, \underline{1}, 1, \underline{3}])$ = $2$ .
- $\text{LIS}([\underline{3}, \underline{5}, \underline{10}, \underline{20}])$ = $4$ .
- $\text{LIS}([3, \underline{1}, \underline{2}, \underline{4}])$ = $3$ .
We define array $a'$ as the array obtained after reversing the array $a$ i.e. $a' = [a_n, a_{n-1}, \ldots , a_1]$ .
The beauty of array $a$ is defined as $min(\text{LIS}(a),\text{LIS}(a'))$ .
Your task is to determine the maximum possible beauty of the array $a$ if you can rearrange the array $a$ arbitrarily.
输入格式
The input consists of multiple test cases. The first line contains a single integer $t$ $(1 \leq t \leq 10^4)$ — the number of test cases. Description of the test cases follows.
The first line of each test case contains a single integer $n$ $(1 \leq n \leq 2\cdot 10^5)$ — the length of array $a$ .
The second line of each test case contains $n$ integers $a_1,a_2, \ldots ,a_n$ $(1 \leq a_i \leq 10^9)$ — the elements of the array $a$ .
It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$ .
The first line of each test case contains a single integer $n$ $(1 \leq n \leq 2\cdot 10^5)$ — the length of array $a$ .
The second line of each test case contains $n$ integers $a_1,a_2, \ldots ,a_n$ $(1 \leq a_i \leq 10^9)$ — the elements of the array $a$ .
It is guaranteed that the sum of $n$ over all test cases does not exceed $2\cdot 10^5$ .
输出格式
For each test case, output a single integer — the maximum possible beauty of $a$ after rearranging its elements arbitrarily.
输入输出样例
输入 #1
3 3 6 6 6 6 2 5 4 5 2 4 4 1 3 2 2
输出 #1
1 3 2
说明/提示
In the first test case, $a$ = $[6, 6, 6]$ and $a'$ = $[6, 6, 6]$ . $\text{LIS}(a) = \text{LIS}(a')$ = $1$ . Hence the beauty is $min(1, 1) = 1$ .
In the second test case, $a$ can be rearranged to $[2, 5, 4, 5, 4, 2]$ . Then $a'$ = $[2, 4, 5, 4, 5, 2]$ . $\text{LIS}(a) = \text{LIS}(a') = 3$ . Hence the beauty is $3$ and it can be shown that this is the maximum possible beauty.
In the third test case, $a$ can be rearranged to $[1, 2, 3, 2]$ . Then $a'$ = $[2, 3, 2, 1]$ . $\text{LIS}(a) = 3$ , $\text{LIS}(a') = 2$ . Hence the beauty is $min(3, 2) = 2$ and it can be shown that $2$ is the maximum possible beauty.
In the second test case, $a$ can be rearranged to $[2, 5, 4, 5, 4, 2]$ . Then $a'$ = $[2, 4, 5, 4, 5, 2]$ . $\text{LIS}(a) = \text{LIS}(a') = 3$ . Hence the beauty is $3$ and it can be shown that this is the maximum possible beauty.
In the third test case, $a$ can be rearranged to $[1, 2, 3, 2]$ . Then $a'$ = $[2, 3, 2, 1]$ . $\text{LIS}(a) = 3$ , $\text{LIS}(a') = 2$ . Hence the beauty is $min(3, 2) = 2$ and it can be shown that $2$ is the maximum possible beauty.