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):
\n\ni 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.
A subarray is a contiguous non-empty sequence of elements within an array.
\n\n\n
Example 1:
\n\n\nInput: nums = [2,3,0,0,2], k = 4\nOutput: 3\nExplanation: We can perform the following increment operations to make nums beautiful:\nChoose index i = 1 and increase nums[1] by 1 -> [2,4,0,0,2].\nChoose index i = 4 and increase nums[4] by 1 -> [2,4,0,0,3].\nChoose index i = 4 and increase nums[4] by 1 -> [2,4,0,0,4].\nThe subarrays with a size of 3 or more are: [2,4,0], [4,0,0], [0,0,4], [2,4,0,0], [4,0,0,4], [2,4,0,0,4].\nIn all the subarrays, the maximum element is equal to k = 4, so nums is now beautiful.\nIt can be shown that nums cannot be made beautiful with fewer than 3 increment operations.\nHence, the answer is 3.\n\n\n
Example 2:
\n\n\nInput: nums = [0,1,3,3], k = 5\nOutput: 2\nExplanation: We can perform the following increment operations to make nums beautiful:\nChoose index i = 2 and increase nums[2] by 1 -> [0,1,4,3].\nChoose index i = 2 and increase nums[2] by 1 -> [0,1,5,3].\nThe subarrays with a size of 3 or more are: [0,1,5], [1,5,3], [0,1,5,3].\nIn all the subarrays, the maximum element is equal to k = 5, so nums is now beautiful.\nIt can be shown that nums cannot be made beautiful with fewer than 2 increment operations.\nHence, the answer is 2.\n\n\n
Example 3:
\n\n\nInput: nums = [1,1,2], k = 1\nOutput: 0\nExplanation: The only subarray with a size of 3 or more in this example is [1,1,2].\nThe maximum element, 2, is already greater than k = 1, so we don't need any increment operation.\nHence, the answer is 0.\n\n\n
\n
Constraints:
\n\n3 <= n == nums.length <= 1050 <= nums[i] <= 1090 <= k <= 1093 consecutive values in the array that is greater than or equal to k.",
"The problem can be solved using dynamic programming.",
"Let dp[i] be the minimum number of increment operations required to make the subarray consisting of the first i values beautiful, while also having the value at nums[i] >= k.",
"dp[0] = max(0, k - nums[0]), dp[1] = max(0, k - nums[1]), and dp[2] = max(0, k - nums[2]).",
"dp[i] = max(0, k - nums[i]) + min(dp[i - 1], dp[i - 2], dp[i - 3]) for i in the range [3, n - 1].",
"The answer to the problem is min(dp[n - 1], dp[n - 2], dp[n - 3])."
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