3397. Maximum Number of Distinct Elements After Operations

3397. Maximum Number of Distinct Elements After Operations

Difficulty: Medium

Topics: Array, Greedy, Sorting, Weekly Contest 429

You are given an integer array nums and an integer k.

You are allowed to perform the following operation on each element of the array at most once:

• Add an integer in the range [-k, k] to the element.

Return the maximum possible number of distinct elements in nums after performing the operations.

Example 1:

• Input: nums = [1,2,2,3,3,4], k = 2
• Output: 6
• Explanation: nums changes to [-1, 0, 1, 2, 3, 4] after performing operations on the first four elements.

Example 2:

• Input: nums = [4,4,4,4], k = 1
• Output: 3
• Explanation: By adding -1 to nums[0] and 1 to nums[1], nums changes to [3, 5, 4, 4].

Constraints:

• 1 <= nums.length <= 10⁵
• 1 <= nums[i] <= 10⁹
• 0 <= k <= 10⁹

Hint:

1. Can we use sorting here?
2. Find the minimum element which is not used for each element.

Solution:

We need to find the maximum number of distinct elements after being allowed to add any integer in the range [-k, k] to each element at most once.

The key insight is that for each number, we can transform it to any value in the range [num-k, num+k]. To maximize distinct elements, we should assign the smallest possible distinct value to each number while ensuring it stays within its allowed range.

Approach:

1. Sort the array to process numbers in order
2. For each number, try to assign the smallest possible value that:
   • Is greater than the previously assigned value
   • Is within the number's allowed range [num-k, num+k]
3. Keep track of the last assigned value and count how many distinct values we can assign

Let's implement this solution in PHP: 3397. Maximum Number of Distinct Elements After Operations

<?php
/**
 * @param Integer[] $nums
 * @param Integer $k
 * @return Integer
 */
function maxDistinctElements($nums, $k) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Test cases
$nums1 = array(1,2,2,3,3,4);
$k1 = 2;
echo "Output 1: " . maxDistinctElements($nums1, $k1) . "\n"; // Expected: 6

$nums2 = array(4,4,4,4);
$k2 = 1;
echo "Output 2: " . maxDistinctElements($nums2, $k2) . "\n"; // Expected: 3
?>

Explanation:

• Sorting: We sort the array to process numbers from smallest to largest, which helps us systematically assign the smallest possible distinct values.
• Minimum Candidate: For each number, we calculate the smallest value we can assign that maintains distinctness (lastAssigned + 1) while staying within the allowed range (num - k).
• Validation: We check if this candidate value is within the upper bound (num + k). If yes, we assign it and increment our count.
• Tracking: We keep track of the last assigned value to ensure all assigned values are distinct.

Examples:

• nums = [1,2,2,3,3,4], k = 2: We can assign [-1, 0, 1, 2, 3, 4] giving 6 distinct elements
• nums = [4,4,4,4], k = 1: We can assign [3, 4, 5] giving 3 distinct elements

Complexity:

• Time: O(n log n) due to sorting
• Space: O(1) excluding the space used by sorting

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