Vlad and the Nice Paths (hard version)

3000ms

524288K

Description:

This is hard version of the problem, it differs from the easy one only by constraints on $$$n$$$ and $$$k$$$.

Vlad found a row of $$$n$$$ tiles and the integer $$$k$$$. The tiles are indexed from left to right and the $$$i$$$-th tile has the color $$$c_i$$$. After a little thought, he decided what to do with it.

You can start from any tile and jump to any number of tiles right, forming the path $$$p$$$. Let's call the path $$$p$$$ of length $$$m$$$ nice if:

$$$p$$$ can be divided into blocks of length exactly $$$k$$$, that is, $$$m$$$ is divisible by $$$k$$$;
$$$c_{p_1} = c_{p_2} = \ldots = c_{p_k}$$$;
$$$c_{p_{k+1}} = c_{p_{k+2}} = \ldots = c_{p_{2k}}$$$;
$$$\ldots$$$
$$$c_{p_{m-k+1}} = c_{p_{m-k+2}} = \ldots = c_{p_{m}}$$$;

Your task is to find the number of nice paths of maximum length. Since this number may be too large, print it modulo $$$10^9 + 7$$$.

Input:

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

The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k \le n \le 5000$$$) — the number of tiles in a row and the length of the block.

The second line of each test case contains $$$n$$$ integers $$$c_1, c_2, c_3, \dots, c_n$$$ ($$$1 \le c_i \le n$$$) — tile colors.

It is guaranteed that the sum of $$$n^2$$$ over all test cases does not exceed $$$25 \cdot 10^6$$$.

Output:

Print $$$t$$$ numbers, each of which is the answer to the corresponding test case — the number of nice paths of maximum length modulo $$$10^9 + 7$$$.

Sample Input:

5
5 2
1 2 3 4 5
7 2
1 3 1 3 3 1 3
11 4
1 1 1 1 1 1 1 1 1 1 1
5 2
1 1 2 2 2
5 1
1 2 3 4 5

Sample Output:

Note:

In the first sample, it is impossible to make a nice path with a length greater than $$$0$$$.

In the second sample, we are interested in the following paths:

$$$1 \rightarrow 3 \rightarrow 4 \rightarrow 5$$$
$$$2 \rightarrow 4 \rightarrow 5 \rightarrow 7$$$
$$$1 \rightarrow 3 \rightarrow 5 \rightarrow 7$$$
$$$1 \rightarrow 3 \rightarrow 4 \rightarrow 7$$$

In the third example, any path of length $$$8$$$ is nice.

Informação

Provedor Codeforces

Código CF1811G2