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g2101_2200.s2111_minimum_operations_to_make_the_array_k_increasing.Solution

public class Solution extends java.lang.Object

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:
- arr[0] <= arr[2] (4 <= 5)
- arr[1] <= arr[3] (1 <= 2)
- arr[2] <= arr[4] (5 <= 6)
- arr[3] <= arr[5] (2 <= 2)
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, arr[i-2] <= arr[i].

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 arr[i-3] <= arr[i] for 3 <= i <= 5.

One of the ways we can make the array K-increasing is by changing arr[3] to 4 and arr[5] 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:

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