Minimum Operations to Make the Array K-Increasing

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Problem Statement

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 .

Examples

Example 1

Input: arr = [5,4,3,2,1], k = 1
Output: 4
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
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
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

1 <= arr.length <= 10 ^5

1 <= arr[i], k <= arr.length

😤

Hard

Difficulty

Topic Info

ModuleDSA

CategoryArray

Sub-topicBinary Search