This article delves into the LeetCode problem 3347, “Maximum Frequency of an Element After Performing Operations II,” providing a detailed explanation of its approach and solution strategy.

Problem Statement:

You are provided with an integer array nums, an integer k, and an integer numOperations. The task is to execute numOperations on the nums array. For each operation, you must select an index i that has not been previously chosen and modify nums[i] by adding an integer x where x is within the range [-k, k]. Your goal is to determine the highest possible frequency any single element can achieve in nums after all numOperations have been performed.

Illustrative Examples:

• Example 1:
  • Input: nums = [1, 4, 5], k = 1, numOperations = 2
  • Output: 2
  • Explanation: One way to obtain a frequency of 2 is to first add 0 to nums[1] (array remains [1, 4, 5]), and then add -1 to nums[2] (array becomes [1, 4, 4]). In the final array, the element 4 appears twice.
• Example 2:
  • Input: nums = [5, 11, 20, 20], k = 5, numOperations = 1
  • Output: 2
  • Explanation: By performing one operation, for instance, adding 0 to nums[1], the array remains [5, 11, 20, 20]. The number 20 has a frequency of 2.

Constraints:

• The length of nums is between 1 and 10^5.
• Each element nums[i] is between 1 and 10^9.
• k is between 0 and 10^9.
• numOperations is between 0 and nums.length.

Key Hint:

Optimal target values for analysis often include nums[i] - k, nums[i], and nums[i] + k.

Solution Strategy:

The core of solving this problem lies in identifying a target value T such that the maximum possible number of elements in nums can be converted into T by utilizing the allowed operations. Remember that an element nums[i] can be adjusted to any value within the range [nums[i] - k, nums[i] + k].

Here’s a step-by-step approach:

1. Sorting the Array: The first crucial step is to sort the input array nums in non-decreasing order. This organization is fundamental for efficiently performing range queries and binary search operations in subsequent steps.

2. Generating Candidate Target Values: Consider potential target values T. The hint suggests focusing on nums[i] - k, nums[i], and nums[i] + k for each element nums[i] in the array. These values represent critical points at the boundaries or center of the modification ranges.

3. Evaluating Each Candidate Target: For every chosen candidate target T:

   • Identify Modifiable Elements: Determine all elements nums[j] in the sorted array that could potentially be transformed into T. An element nums[j] can become T if T falls within its modification range, i.e., nums[j] - k <= T <= nums[j] + k. Equivalently, this means nums[j] must be within the range [T - k, T + k].
   • Efficient Counting: Utilize binary search (e.g., lower_bound and upper_bound if available in your language’s standard library) on the sorted array to quickly count how many elements fall within the range [T - k, T + k].
   • Count Exact Matches: Separately, count how many elements are already precisely equal to T.
4. Calculating Possible Frequency: For a given target T, the achievable frequency is calculated as follows:

   possibleFrequency = (Number of elements exactly equal to T) + min(numOperations, Number of elements within [T-k, T+k] that are NOT equal to T)
   

   This formula accounts for elements that already match the target and then adds as many additional elements as possible using the available numOperations, up to the total number of elements that can be adjusted to T.

5. Tracking the Maximum: Maintain a variable to store the maximum possibleFrequency encountered across all evaluated candidate targets. This maximum value will represent the final answer.

Underlying Principle:

The effectiveness of this approach stems from the observation that if we aim to make a group of numbers equal to T, each number nums[j] in that group must satisfy the condition that T is reachable from nums[j]. This translates to nums[j] being within [T - k, T + k]. Sorting nums streamlines the process of finding such elements. By considering boundary-related targets (nums[i] ± k) and the original values (nums[i]), we cover the most probable candidates for achieving maximum frequency. The constraint numOperations limits how many distinct elements we can change, which is incorporated by taking the minimum in the frequency calculation.