#3346

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Maximum Frequency of an Element After Performing Operations I

Maximize the frequency of an element in an array after performing a series of operations to find the best possible result.

array binary search sliding window sorting prefix sum

Problem Statement

You are given an integer array nums and two integers k and numOperations. The goal is to perform numOperations operations on the array, where in each operation, you can increase any element in the array by k. After performing the operations, return the maximum possible frequency of any element in the array.

To solve this problem, you need to determine the maximum frequency that can be achieved by applying up to numOperations operations on the array. The challenge involves finding an efficient method to apply the operations to increase the frequency of elements while considering constraints and the most optimal strategy for achieving the highest frequency.

Examples

Example 1

Input: nums = [1,4,5], k = 1, numOperations = 2

Output: 2

We can achieve a maximum frequency of two by:

Example 2

Input: nums = [5,11,20,20], k = 5, numOperations = 1

Output: 2

We can achieve a maximum frequency of two by:

Constraints

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

Solution Approach

Binary Search over the Frequency Space

Use binary search to explore possible frequencies and check if a given frequency can be achieved with the available numOperations. This approach helps narrow down the search space efficiently.

Sliding Window Technique

Apply the sliding window technique to determine how many operations are needed to achieve a specific frequency. By maintaining a window of valid elements, the algorithm can compute the required number of operations and check whether it’s feasible to achieve the target frequency.

Sorting and Prefix Sum Optimization

Sort the array and use prefix sums to efficiently calculate the total number of operations required to increase elements to match the target frequency. This allows for quick checks on whether a frequency is achievable.

Complexity Analysis

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

The time complexity depends on the binary search for the target frequency and the sliding window approach for each check. The sorting step adds an additional complexity of O(n log n), where n is the size of the array. The space complexity is O(n) due to the array manipulations and prefix sum calculations.

What Interviewers Usually Probe

Ability to implement binary search over a dynamic range of values.
Understanding the sliding window technique to optimize operations.
Proficiency in combining sorting and prefix sums for optimization.

Common Pitfalls or Variants

Common pitfalls

Failing to account for edge cases such as small arrays or zero operations.
Not efficiently handling the sliding window bounds while calculating required operations.
Overlooking the sorting step which may impact the overall efficiency of the solution.

Follow-up variants

Introducing constraints on the number of operations for different elements.
Handling multiple arrays with different lengths in the same problem.
Changing the operation from addition to subtraction, making the problem more complex.

FAQ

What is the main approach to solving the Maximum Frequency of an Element After Performing Operations I?

The problem is solved by using binary search over the valid frequency space combined with sliding window and sorting techniques.

How does the sliding window technique help in this problem?

The sliding window technique is used to track the operations required to increase the frequency of elements, allowing for an efficient check on whether a certain frequency can be achieved.

What are the time and space complexities of this solution?

The time complexity is O(n log n) due to the sorting and binary search. The space complexity is O(n) for managing the array and prefix sums.

Can the problem be solved using dynamic programming?

While dynamic programming is not required, it can be used for subproblems like determining the number of operations needed to achieve a target frequency.

How can GhostInterview help with the Maximum Frequency of an Element After Performing Operations I?

GhostInterview helps by offering an in-depth understanding of binary search combined with sliding windows and sorting for solving array problems optimally.

terminal

Solution

Solution 1: Difference Array

According to the problem description, for each element $x$ in the array $\textit{nums}$, we can change it to any integer within the range $[x-k, x+k]$. We want to perform operations on some elements in $\textit{nums}$ to maximize the frequency of a certain integer in the array.

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class Solution:
    def maxFrequency(self, nums: List[int], k: int, numOperations: int) -> int:
        cnt = defaultdict(int)
        d = defaultdict(int)
        for x in nums:
            cnt[x] += 1
            d[x] += 0
            d[x - k] += 1
            d[x + k + 1] -= 1
        ans = s = 0
        for x, t in sorted(d.items()):
            s += t
            ans = max(ans, min(s, cnt[x] + numOperations))
        return ans

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