#2779

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Maximum Beauty of an Array After Applying Operation

Find the maximum beauty of an array by adjusting elements within a range using binary search and sliding window techniques efficiently.

array binary search sliding window sorting

Problem Statement

You are given a 0-indexed array nums and a non-negative integer k. In one operation, you can increase or decrease any element of nums by at most k.

The beauty of an array is defined as the length of the longest subsequence where all elements are equal. Determine the maximum beauty obtainable after performing any number of allowed operations.

Examples

Example 1

Input: nums = [4,6,1,2], k = 2

Output: 3

In this example, we apply the following operations:

Choose index 1, replace it with 4 (from range [4,8]), nums = [4,4,1,2].
Choose index 3, replace it with 4 (from range [0,4]), nums = [4,4,1,4]. After the applied operations, the beauty of the array nums is 3 (subsequence consisting of indices 0, 1, and 3). It can be proven that 3 is the maximum possible length we can achieve.

Example 2

Input: nums = [1,1,1,1], k = 10

Output: 4

In this example we don't have to apply any operations. The beauty of the array nums is 4 (whole array).

Constraints

1 <= nums.length <= 105
0 <= nums[i], k <= 105

Solution Approach

Sort the Array

Begin by sorting nums to simplify checking contiguous subsequences. Sorting allows sliding window evaluation to efficiently measure how many elements can be made equal within k operations.

Binary Search Over Beauty

Apply binary search on the possible length of the subsequence (beauty). For each candidate length, verify if a subsequence can be adjusted to all equal elements within k, leveraging prefix sums or sliding window sums.

Sliding Window Validation

Use a sliding window to check if the current subsequence can be made equal with allowed operations. Maintain a running sum and compare with the target to quickly decide feasibility for each window.

Complexity Analysis

Metric	Value
Time	O(n + \text{maxValue})
Space	O(\text{maxValue})

Sorting takes O(n log n). The sliding window validation for each binary search candidate runs in O(n). Overall, the algorithm is O(n log n) time and O(1) extra space if prefix sums are not stored, otherwise O(n).

What Interviewers Usually Probe

Check if you can use a sliding window after sorting to reduce repeated computations.
Consider whether binary search on the answer space simplifies verification of maximum beauty.
Ask about constraints on element modification and how k affects feasibility checks.

Common Pitfalls or Variants

Common pitfalls

Failing to sort the array first, which leads to incorrect sliding window ranges.
Misapplying the binary search by checking raw element values instead of subsequence lengths.
Not accounting for operation limit k when adjusting elements in a subsequence.

Follow-up variants

Find maximum beauty when only increases are allowed, not decreases.
Determine maximum beauty using a fixed number of operations rather than unlimited within k.
Compute maximum beauty for multi-dimensional arrays with similar adjustment rules.

FAQ

What is the best approach to maximize array beauty in this problem?

Use a combination of sorting, binary search over possible beauty values, and sliding window checks to validate subsequences within k modifications.

Why is binary search over the answer space effective here?

It allows checking feasible subsequence lengths efficiently instead of iterating all possibilities, reducing time complexity.

How does k influence the solution?

k sets the limit for adjusting each element, so sliding window sums must consider k to ensure subsequences can be equalized.

Can we skip sorting the array?

No, sorting is critical because it enables efficient sliding window validation and prevents incorrect subsequence evaluation.

What pattern does this problem follow in interviews?

It follows the 'binary search over valid answer space' pattern combined with sliding window for subsequence validation.

terminal

Solution

Solution 1: Difference Array

We notice that for each operation, all elements within the interval $[nums[i]-k, nums[i]+k]$ will increase by $1$. Therefore, we can use a difference array to record the contributions of these operations to the beauty value.

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class Solution:
    def maximumBeauty(self, nums: List[int], k: int) -> int:
        m = max(nums) + k * 2 + 2
        d = [0] * m
        for x in nums:
            d[x] += 1
            d[x + k * 2 + 1] -= 1
        return max(accumulate(d))

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