Minimum Increment Operations to Make Array Beautiful

Problem Description

You are given a 0-indexed integer array nums having length n, and an integer k. You can perform the following increment operation any number of times (including zero): Choose an index i in the range [0, n - 1], and increase nums[i] by 1. An array is considered beautiful if, for any subarray with a size of 3 or more, its maximum element is greater than or equal to k. Return an integer denoting the minimum number of increment operations needed to make nums beautiful.

Key Insights

• To ensure the maximum of every subarray of size 3 or more is at least k, we need to check the maximum values within the relevant ranges.
• If any element is less than k, we need to calculate how much we need to increment it to reach k.
• Each element affects the maximum of multiple consecutive subarrays, hence we should prioritize increments for elements that are part of these larger subarrays.
• The number of increments required depends on the maximum of the last three elements in the array as we traverse.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we iterate through the array nums. For each element at index i, we check if it is less than k. If it is, we calculate the difference k - nums[i] to determine how many increments are needed. We then accumulate these increments. We also ensure that we only apply increments in a way that respects the condition that any subarray of size 3 or more should have a maximum of at least k. This can be efficiently managed by checking the last three elements as we iterate.

The main data structure used is an integer array for nums, and we use simple arithmetic to track the total increments needed.

Code Solutions