#2919

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Increment Operations to Make Array Beautiful

Optimize the number of increment operations to make an array beautiful by ensuring subarrays of size 3 or more meet the k condition.

array dynamic programming

Problem Statement

You are given an integer array nums of length n and an integer k. You are allowed to perform any number of increment operations on the elements of the array. Each operation increases an element by 1.

The goal is to make the array beautiful by ensuring that for every subarray of size 3 or more, the maximum element is greater than or equal to k. The problem asks you to return the minimum number of increment operations required.

Examples

Example 1

Input: nums = [2,3,0,0,2], k = 4

Output: 3

We can perform the following increment operations to make nums beautiful: Choose index i = 1 and increase nums[1] by 1 -> [2,4,0,0,2]. Choose index i = 4 and increase nums[4] by 1 -> [2,4,0,0,3]. Choose index i = 4 and increase nums[4] by 1 -> [2,4,0,0,4]. The 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]. In all the subarrays, the maximum element is equal to k = 4, so nums is now beautiful. It can be shown that nums cannot be made beautiful with fewer than 3 increment operations. Hence, the answer is 3.

Example 2

Input: nums = [0,1,3,3], k = 5

Output: 2

We can perform the following increment operations to make nums beautiful: Choose index i = 2 and increase nums[2] by 1 -> [0,1,4,3]. Choose index i = 2 and increase nums[2] by 1 -> [0,1,5,3]. The subarrays with a size of 3 or more are: [0,1,5], [1,5,3], [0,1,5,3]. In all the subarrays, the maximum element is equal to k = 5, so nums is now beautiful. It can be shown that nums cannot be made beautiful with fewer than 2 increment operations. Hence, the answer is 2.

Example 3

Input: nums = [1,1,2], k = 1

Output: 0

The only subarray with a size of 3 or more in this example is [1,1,2]. The maximum element, 2, is already greater than k = 1, so we don't need any increment operation. Hence, the answer is 0.

Constraints

3 <= n == nums.length <= 105
0 <= nums[i] <= 109
0 <= k <= 109

Solution Approach

State Transition Dynamic Programming

This problem can be approached by using dynamic programming to track the minimum number of operations needed to achieve the required maximum element for each subarray of size 3 or more. A state transition approach can be applied by updating the values based on previous states and applying operations incrementally.

Greedy Strategy

In each step, we can greedily decide which element of the array to increment by considering subarrays and their maximum values. This can be optimized by focusing on the elements that are closest to reaching the threshold k and applying the necessary increments accordingly.

Sliding Window Technique

A sliding window technique can be employed to iterate over all possible subarrays efficiently. By maintaining the maximum element within the window and adjusting as necessary, we can minimize the number of operations needed to ensure the array meets the 'beautiful' condition.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the chosen approach. A dynamic programming solution may require O(n) time and space, while a greedy or sliding window approach may offer better time complexity in some cases.

What Interviewers Usually Probe

Look for understanding of dynamic programming and how state transitions can be used to minimize operations.
Evaluate the candidate's ability to optimize with a greedy or sliding window approach to handle large inputs efficiently.
Check if the candidate can explain the trade-offs between different approaches based on the constraints.

Common Pitfalls or Variants

Common pitfalls

Failing to correctly track the minimum number of operations needed for each possible state transition.
Overlooking the fact that an optimal solution requires ensuring the maximum value condition for all subarrays of size 3 or more.
Not considering edge cases where the array is already beautiful or requires no operations.

Follow-up variants

Instead of tracking the minimum operations, track the number of subarrays that meet the condition.
Modify the problem to work with a sliding window of size 4 or more instead of 3 or more.
Introduce constraints where the maximum value of k is dynamically adjusted during the process.

FAQ

What is the best approach to solve Minimum Increment Operations to Make Array Beautiful?

The best approach involves using dynamic programming with state transitions to track the minimum number of operations required for each subarray.

How does the sliding window technique help in solving this problem?

The sliding window technique helps by efficiently iterating over subarrays and adjusting the maximum element, minimizing the number of operations needed.

Can the problem be solved using a greedy approach?

Yes, a greedy approach can be used by incrementing the elements closest to meeting the k condition first, optimizing the solution.

What is the time complexity of the dynamic programming solution?

The time complexity is typically O(n), where n is the length of the array, as each element is processed once.

How can I handle edge cases where no operations are needed?

Edge cases can be handled by checking if the array already meets the beautiful condition before performing any operations.

terminal

Solution

Solution 1: Dynamic Programming

We define $f$, $g$, and $h$ as the minimum number of increment operations needed to get the maximum value from the last three items in the first $i$ items, initially $f = 0$, $g = 0$, $h = 0$.

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class Solution:
    def minIncrementOperations(self, nums: List[int], k: int) -> int:
        f = g = h = 0
        for x in nums:
            f, g, h = g, h, min(f, g, h) + max(k - x, 0)
        return min(f, g, h)

Continue Practicing

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