Nene’s Magical Matrix

题面翻译

有一个 $n\times n$ 的数组，初始时所有位置均为 $0$ ，现在要对其进行若干次操作，操作有两种：

• 选择一个正整数 $i~(1\le i \le n)$ 和一个 $1$ ~ $n$ 的排列 $p_1,p_2,\cdots,p_n$，将每个 $a_{i,j}~(1\le j \le n)$ 同时赋值为 $p_j$。
• 选择一个正整数 $i~(1 \le i \le n)$ 和一个 $1$ ~ $n$ 的排列 $p_1,p_2,\cdots,p_n$，将每个 $a_{j,i}~(1\le j \le n)$ 同时赋值为 $p_j$。

需要进行不多于 $2\times n$ 次操作，使得数组中元素的和最大。输出最终数组中元素的和、操作次数和操作顺序。

一共 $t~ (1\le t \le 500)$ 组数据，每组数据中 $1\le n\le 500$ ，$1\le \sum n^2 \le 5\times 10^5$。

题目描述

The magical girl Nene has an $n\times n$ matrix $a$ filled with zeroes. The $j$ -th element of the $i$ -th row of matrix $a$ is denoted as $a_{i, j}$ .

She can perform operations of the following two types with this matrix:

• Type $1$ operation: choose an integer $i$ between $1$ and $n$ and a permutation $p_1, p_2, \ldots, p_n$ of integers from $1$ to $n$ . Assign $a_{i, j}:=p_j$ for all $1 \le j \le n$ simultaneously.
• Type $2$ operation: choose an integer $i$ between $1$ and $n$ and a permutation $p_1, p_2, \ldots, p_n$ of integers from $1$ to $n$ . Assign $a_{j, i}:=p_j$ for all $1 \le j \le n$ simultaneously.

Nene wants to maximize the sum of all the numbers in the matrix $\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}a_{i,j}$ . She asks you to find the way to perform the operations so that this sum is maximized. As she doesn’t want to make too many operations, you should provide a solution with no more than $2n$ operations.

A permutation of length $n$ is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ( $2$ appears twice in the array), and $[1,3,4]$ is also not a permutation ( $n=3$ but there is $4$ in the array).

输入格式

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 500$ ). The description of test cases follows.

The only line of each test case contains a single integer $n$ ( $1 \le n \le 500$ ) — the size of the matrix $a$ .

It is guaranteed that the sum of $n^2$ over all test cases does not exceed $5 \cdot 10^5$ .

输出格式

For each test case, in the first line output two integers $s$ and $m$ ( $0\leq m\leq 2n$ ) — the maximum sum of the numbers in the matrix and the number of operations in your solution.

In the $k$ -th of the next $m$ lines output the description of the $k$ -th operation:

• an integer $c$ ( $c \in \{1, 2\}$ ) — the type of the $k$ -th operation;
• an integer $i$ ( $1 \le i \le n$ ) — the row or the column the $k$ -th operation is applied to;
• a permutation $p_1, p_2, \ldots, p_n$ of integers from $1$ to $n$ — the permutation used in the $k$ -th operation.

Note that you don’t need to minimize the number of operations used, you only should use no more than $2n$ operations. It can be shown that the maximum possible sum can always be obtained in no more than $2n$ operations.

样例 #1

样例输入 #1

2
1
2

样例输出 #1

1 1
1 1 1
7 3
1 1 1 2
1 2 1 2
2 1 1 2

提示

In the first test case, the maximum sum $s=1$ can be obtained in $1$ operation by setting $a_{1, 1}:=1$ .

In the second test case, the maximum sum $s=7$ can be obtained in $3$ operations as follows:

It can be shown that it is impossible to make the sum of the numbers in the matrix larger than $7$ .

最优

1 2 3 4
2 2 3 4
3 3 3 4
4 4 4 4

 const int N = 2e5 + 10;
int a[N];
void solve()
{
    int n;
    cin >> n;
    for (int i = 1; i <= n; i++) a[i] = a[i - 1] + i;
    int ans = 0;
    for (int i = 1; i <= n; i++)
    {
        ans += i * i + a[n] - a[i];
    }
    cout << ans << " " << 2 * n << '\n';
    for (int i = 1, j = n; i <= n, j >= 1; i++, j--)
    {
        cout << "1" << " " << i << " ";
        for (int k = 1; k <= n; k++) cout << k << " ";
        cout << '\n';
        cout << "2" << " " << j << " ";
        for (int k = n; k >= 1; k--) cout << k << " ";
        cout << '\n';
    }

}