Naive Solution

The naive solution would involve iterating through all possible pairs of indices (i, j) in the array and checking if nums[i] == nums[j] and abs(i - j) <= k. This would involve nested loops.

Code (Python)

def containsNearbyDuplicate_naive(nums, k):
    for i in range(len(nums)):  
        for j in range(i + 1, len(nums)):  
            if nums[i] == nums[j] and abs(i - j) <= k:
                return True
    return False

Time Complexity

O(n^2), where n is the length of the nums array, due to the nested loops.

Space Complexity

O(1), as we are using only a constant amount of extra space.

Optimal Solution

A more efficient solution involves using a hash set (or dictionary) to store the elements encountered so far, along with their indices. For each element, check if it exists in the set. If it does, check if the absolute difference between the current index and the stored index is less than or equal to k. If it is, return true. Otherwise, update the index of the element in the hash set. If the element is not in the hash set, add it along with its index. This way you only need to iterate once.

Code (Python)

def containsNearbyDuplicate(nums, k):
    num_map = {}
    for i, num in enumerate(nums):
        if num in num_map and abs(i - num_map[num]) <= k:
            return True
        num_map[num] = i
    return False

Time Complexity

O(n), where n is the length of the nums array, as we iterate through the array only once.

Space Complexity

O(n) in the worst case, where n is the length of the nums array. This is because, in the worst-case scenario, all elements in the array are distinct, and we would store all of them in the hash set.

Edge Cases

1. Empty Array: If the input array nums is empty, the function should return false because there are no elements to compare.
2. k = 0: If k is 0, the function should return true if there are duplicate elements at the same index, which is impossible given the distinct indices constraint. Thus, when k is 0, it will return false.
3. Large k: If k is very large (e.g., greater than or equal to the length of the array), the function should still work correctly, comparing all possible pairs of indices.
4. Duplicate Values: The array may contain duplicate values at various indices. The function should correctly identify if any two duplicates are within the distance k of each other.
5. No Duplicates: The function should return false if the array contains no duplicate values.