Weight of the System of Nested Segments

2000ms

262144K

Description:

On the number line there are $$$m$$$ points, $$$i$$$-th of which has integer coordinate $$$x_i$$$ and integer weight $$$w_i$$$. The coordinates of all points are different, and the points are numbered from $$$1$$$ to $$$m$$$.

A sequence of $$$n$$$ segments $$$[l_1, r_1], [l_2, r_2], \dots, [l_n, r_n]$$$ is called system of nested segments if for each pair $$$i, j$$$ ($$$1 \le i < j \le n$$$) the condition $$$l_i < l_j < r_j < r_i$$$ is satisfied. In other words, the second segment is strictly inside the first one, the third segment is strictly inside the second one, and so on.

For a given number $$$n$$$, find a system of nested segments such that:

both ends of each segment are one of $$$m$$$ given points;
the sum of the weights $$$2\cdot n$$$ of the points used as ends of the segments is minimal.

For example, let $$$m = 8$$$. The given points are marked in the picture, their weights are marked in red, their coordinates are marked in blue. Make a system of three nested segments:

weight of the first segment: $$$1 + 1 = 2$$$
weight of the second segment: $$$10 + (-1) = 9$$$
weight of the third segment: $$$3 + (-2) = 1$$$
sum of the weights of all the segments in the system: $$$2 + 9 + 1 = 12$$$

$\text{[math]}$ System of three nested segments

Input:

The first line of input data contains an integer $$$t$$$ ($$$1 \le t \le 10^4$$$) —the number of input test cases.

An empty line is written before each test case.

The first line of each test case contains two positive integers $$$n$$$ ($$$1 \le n \le 10^5$$$) and $$$m$$$ ($$$2 \cdot n \le m \le 2 \cdot 10^5$$$).

The next $$$m$$$ lines contain pairs of integers $$$x_i$$$ ($$$-10^9 \le x_i \le 10^9$$$) and $$$w_i$$$ ($$$-10^4 \le w_i \le 10^4$$$) — coordinate and weight of point number $$$i$$$ ($$$1 \le i \le m$$$) respectively. All $$$x_i$$$ are different.

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

Output:

For each test case, output $$$n + 1$$$ lines: in the first of them, output the weight of the composed system, and in the next $$$n$$$ lines output exactly two numbers — the indices of the points which are the endpoints of the $$$i$$$-th segment ($$$1 \le i \le n$$$). The order in which you output the endpoints of a segment is not important — you can output the index of the left endpoint first and then the number of the right endpoint, or the other way around.

If there are several ways to make a system of nested segments with minimal weight, output any of them.

Sample Input:

Sample Output:

Note:

The first test case coincides with the example from the condition. It can be shown that the weight of the composed system is minimal.

The second test case has only $$$6$$$ points, so you need to use each of them to compose $$$3$$$ segments.

Informação

Provedor Codeforces

Código CF1650C