A15193. Chopping Carrots (Hard Version)

This is the hard version of the problem. The only difference between the versions is the constraints on $n$ , $k$ , $a_i$ , and the sum of $n$ over all test cases. You can make hacks only if both versions of the problem are solved.

Note the unusual memory limit.

You are given an array of integers $a_1, a_2, \ldots, a_n$ of length $n$ , and an integer $k$ .

The cost of an array of integers $p_1, p_2, \ldots, p_n$ of length $n$ is $\max\limits_{1 \le i \le n}\left(\left \lfloor \frac{a_i}{p_i} \right \rfloor \right) - \min\limits_{1 \le i \le n}\left(\left \lfloor \frac{a_i}{p_i} \right \rfloor \right).$

Here, $\lfloor \frac{x}{y} \rfloor$ denotes the integer part of the division of $x$ by $y$ . Find the minimum cost of an array $p$ such that $1 \le p_i \le k$ for all $1 \le i \le n$.

The first line contains a single integer $t$ ( $1 \le t \le 100$ ) — the number of test cases.

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

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_1 \le a_2 \le \ldots \le a_n \le 10^5$ ).

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

For each test case, print a single integer — the minimum possible cost of an array $p$ satisfying the condition above.

In the first test case, the optimal array is $p = [1, 1, 1, 2, 2]$ . The resulting array of values of $\lfloor \frac{a_i}{p_i} \rfloor$ is $[4, 5, 6, 4, 5]$ . The cost of $p$ is $\max\limits_{1 \le i \le n}(\lfloor \frac{a_i}{p_i} \rfloor) - \min\limits_{1 \le i \le n}(\lfloor \frac{a_i}{p_i} \rfloor) = 6 - 4 = 2$ . We can show that there is no array (satisfying the condition from the statement) with a smaller cost.

In the second test case, one of the optimal arrays is $p = [12, 12, 12, 12, 12]$ , which results in all $\lfloor \frac{a_i}{p_i} \rfloor$ being $0$ .

In the third test case, the only possible array is $p = [1, 1, 1]$ .