Problem Description
You are given an array nums of n integers and an integer k. For each subarray of nums, you can apply up to k operations on it. In each operation, you increment any element of the subarray by 1. Return the number of subarrays that you can make non-decreasing after performing at most k operations. An array is said to be non-decreasing if each element is greater than or equal to its previous element, if it exists.
Key Insights
- A non-decreasing subarray is defined as one where every element is less than or equal to the next.
- You can increment elements in a subarray to achieve the non-decreasing property, but the total number of increments across the subarray must not exceed
k. - The challenge is to efficiently count all subarrays that can be made non-decreasing with the allowed operations.
Space and Time Complexity
Time Complexity: O(n)
Space Complexity: O(1)
Solution
The approach to solving this problem is to use a sliding window technique, which allows us to efficiently keep track of the number of necessary operations to convert subarrays into non-decreasing ones. We will maintain a window of elements and calculate the total operations required to ensure the current window is non-decreasing. If the operations exceed k, we will adjust the starting index of the window. The algorithm operates in linear time by traversing the array once.
We will utilize a two-pointer technique to expand and contract our window based on the number of operations required.