2111 - Minimum Operations to Make the Array K-Increasing\.

Hard

You are given a 0-indexed array arr consisting of n positive integers, and a positive integer k.

The array arr is called K-increasing if arr[i-k] <= arr[i] holds for every index i, where k <= i <= n-1.

• For example, arr = [4, 1, 5, 2, 6, 2] is K-increasing for k = 2 because:

• However, the same arr is not K-increasing for k = 1 (because arr[0] > arr[1]) or k = 3 (because arr[0] > arr[3]).

In one operation , you can choose an index i and change arr[i] into any positive integer.

Return the minimum number of operations required to make the array K-increasing for the given k.

Example 1:

Input: arr = 5,4,3,2,1, k = 1

Output: 4

Explanation:

For k = 1, the resultant array has to be non-decreasing.

Some of the K-increasing arrays that can be formed are 5, **6** , **7** , **8** , **9** , **1** , **1** , **1** , **1** ,1, **2** , **2** ,3, **4** , **4** . All of them require 4 operations.

It is suboptimal to change the array to, for example, **6** , **7** , **8** , **9** , **10** because it would take 5 operations.

It can be shown that we cannot make the array K-increasing in less than 4 operations.

Example 2:

Input: arr = 4,1,5,2,6,2, k = 2

Output: 0

Explanation:

This is the same example as the one in the problem description.

Here, for every index i where 2 <= i <= 5, arri-2<= arri.

Since the given array is already K-increasing, we do not need to perform any operations.

Example 3:

Input: arr = 4,1,5,2,6,2, k = 3

Output: 2

Explanation:

Indices 3 and 5 are the only ones not satisfying arri-3<= arri for 3 <= i <= 5.

One of the ways we can make the array K-increasing is by changing arr3 to 4 and arr5 to 5.

The array will now be 4,1,5, **4** ,6, **5** .

Note that there can be other ways to make the array K-increasing, but none of them require less than 2 operations.

Constraints:

• <code>1 <= arr.length <= 10⁵</code>

• 1 <= arr[i], k <= arr.length