#2111

Minimum Operations to Make the Array K-Increasing

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⚑ Longest Increasing Subsequence (LIS)

Description

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

The array is called K-increasing if arr[i-k] <= arr[i] for every index i satisfying 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 select any index i and assign arr[i] any positive integer value.

Return the minimum number of operations required to make arr 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.

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