Solution Explanation:

This problem asks to find the k-th missing element in a sorted array. A naive approach would iterate through the array, counting missing elements until the k-th missing element is found. However, this approach has a time complexity of O(n), where n is the length of the array. The problem suggests a logarithmic time complexity solution, indicating the use of binary search.

The core idea is to leverage the sorted nature of the input array. We can efficiently determine the number of missing elements up to a given index using a helper function missing(i). This function calculates the difference between the expected value at index i (assuming a sequence starting from nums[0]) and the actual value nums[i].

The main function then performs a binary search to find the index l where the number of missing elements is equal to or greater than k. If the number of missing elements at index l-1 is less than k, the k-th missing number is calculated by adding k - missing(l-1) to nums[l-1].

Algorithm:

1. missing(i) function: This helper function calculates the number of missing elements up to index i. It's simply nums[i] - nums[0] - i.

2. Binary Search: A binary search is performed on the array nums.

   • The left pointer starts at index 0 and the right pointer at n-1.
   • In each iteration, the mid index is calculated.
   • If missing(mid) is greater than or equal to k, it means the k-th missing element is at or before mid. The right pointer is updated to mid.
   • Otherwise, the k-th missing element is after mid, and the left pointer is updated to mid + 1.
   • The loop continues until left equals right.

3. Result Calculation: After the binary search, left points to the index where the k-th missing number is found or would be inserted. The k-th missing element is then calculated as nums[left - 1] + k - missing(left - 1). This handles the case where k exceeds the number of missing elements before nums[n-1].

Time Complexity Analysis:

• The missing(i) function takes O(1) time.
• The binary search takes O(log n) time.
• The result calculation takes O(1) time.

Therefore, the overall time complexity of the solution is O(log n).

Space Complexity Analysis:

The space complexity is O(1) because the solution uses a constant amount of extra space regardless of the input size.

Code Explanation (Python):

The Python code implements the algorithm described above. The missing function is defined as a lambda function for conciseness. The binary search is straightforward, and the result is calculated based on the index found by the binary search.

Code Explanation (Java, C++, Go):

The Java, C++, and Go codes are analogous to the Python code. They all implement the same algorithm using the respective language syntax. The missing function is a separate helper method for better readability. The binary search logic and result calculation remain the same.