Array Stabilization (AND version)

2000ms

262144K

Description:

You are given an array $$$a[0 \ldots n - 1] = [a_0, a_1, \ldots, a_{n - 1}]$$$ of zeroes and ones only. Note that in this problem, unlike the others, the array indexes are numbered from zero, not from one.

In one step, the array $$$a$$$ is replaced by another array of length $$$n$$$ according to the following rules:

First, a new array $$$a^{\rightarrow d}$$$ is defined as a cyclic shift of the array $$$a$$$ to the right by $$$d$$$ cells. The elements of this array can be defined as $$$a^{\rightarrow d}_i = a_{(i + n - d) \bmod n}$$$, where $$$(i + n - d) \bmod n$$$ is the remainder of integer division of $$$i + n - d$$$ by $$$n$$$.
It means that the whole array $$$a^{\rightarrow d}$$$ can be represented as a sequence $$$$$$a^{\rightarrow d} = [a_{n - d}, a_{n - d + 1}, \ldots, a_{n - 1}, a_0, a_1, \ldots, a_{n - d - 1}]$$$$$$
Then each element of the array $$$a_i$$$ is replaced by $$$a_i \,\&\, a^{\rightarrow d}_i$$$, where $$$\&$$$ is a logical "AND" operator.

For example, if $$$a = [0, 0, 1, 1]$$$ and $$$d = 1$$$, then $$$a^{\rightarrow d} = [1, 0, 0, 1]$$$ and the value of $$$a$$$ after the first step will be $$$[0 \,\&\, 1, 0 \,\&\, 0, 1 \,\&\, 0, 1 \,\&\, 1]$$$, that is $$$[0, 0, 0, 1]$$$.

The process ends when the array stops changing. For a given array $$$a$$$, determine whether it will consist of only zeros at the end of the process. If yes, also find the number of steps the process will take before it finishes.

Input:

The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases.

The next $$$2t$$$ lines contain descriptions of the test cases.

The first line of each test case description contains two integers: $$$n$$$ ($$$1 \le n \le 10^6$$$) — array size and $$$d$$$ ($$$1 \le d \le n$$$) — cyclic shift offset. The second line of the description contains $$$n$$$ space-separated integers $$$a_i$$$ ($$$0 \le a_i \le 1$$$) — elements of the array.

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

Output:

Print $$$t$$$ lines, each line containing the answer to the corresponding test case. The answer to a test case should be a single integer — the number of steps after which the array will contain only zeros for the first time. If there are still elements equal to $$$1$$$ in the array after the end of the process, print -1.

Sample Input:

Sample Output:

Note:

In the third sample test case the array will change as follows:

At the beginning $$$a = [1, 1, 0, 1, 0]$$$, and $$$a^{\rightarrow 2} = [1, 0, 1, 1, 0]$$$. Their element-by-element "AND" is equal to $$$$$$[1 \,\&\, 1, 1 \,\&\, 0, 0 \,\&\, 1, 1 \,\&\, 1, 0 \,\&\, 0] = [1, 0, 0, 1, 0]$$$$$$
Now $$$a = [1, 0, 0, 1, 0]$$$, then $$$a^{\rightarrow 2} = [1, 0, 1, 0, 0]$$$. Their element-by-element "AND" equals to $$$$$$[1 \,\&\, 1, 0 \,\&\, 0, 0 \,\&\, 1, 1 \,\&\, 0, 0 \,\&\, 0] = [1, 0, 0, 0, 0]$$$$$$
And finally, when $$$a = [1, 0, 0, 0, 0]$$$ we get $$$a^{\rightarrow 2} = [0, 0, 1, 0, 0]$$$. Their element-by-element "AND" equals to $$$$$$[1 \,\&\, 0, 0 \,\&\, 0, 0 \,\&\, 1, 0 \,\&\, 0, 0 \,\&\, 0] = [0, 0, 0, 0, 0]$$$$$$

Thus, the answer is $$$3$$$ steps.

In the fourth sample test case, the array will not change as it shifts by $$$2$$$ to the right, so each element will be calculated as $$$0 \,\&\, 0$$$ or $$$1 \,\&\, 1$$$ thus not changing its value. So the answer is -1, the array will never contain only zeros.

Informação

Provedor Codeforces

Código CF1579F