CF1842I.Tenzing and Necklace

bright, sunny and innocent......

Tenzing has a beautiful necklace. The necklace consists of $n$ pearls numbered from $1$ to $n$ with a string connecting pearls $i$ and $(i \text{ mod } n)+1$ for all $1 \leq i \leq n$ .

One day Tenzing wants to cut the necklace into several parts by cutting some strings. But for each connected part of the necklace, there should not be more than $k$ pearls. The time needed to cut each string may not be the same. Tenzing needs to spend $a_i$ minutes cutting the string between pearls $i$ and $(i \text{ mod } n)+1$ .

Tenzing wants to know the minimum time in minutes to cut the necklace such that each connected part will not have more than $k$ pearls.

Each test contains multiple test cases. The first line of input contains a single integer $t$ ( $1 \le t \le 10^5$ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains two integers $n$ and $k$ ( $2\leq n\leq 5\cdot 10^5$ , $1\leq k <n$ ).

The second line of each test case contains $n$ integers $a_1,a_2,\ldots,a_n$ ( $1\leq a_i\leq 10^9$ ).

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

For each test case, output the minimum total time in minutes required.

In the first test case, the necklace will be cut into $3$ parts: $[1,2][3,4][5]$ , so the total time is $3$ .

In the second test case, the necklace will be cut into $3$ parts: $[5,1][2][3,4]$ , Tenzing will cut the strings connecting $(1,2), (2,3)$ and $(4,5)$ , so the total time is $a_1+a_2+a_4=7$ .