Problem Overview

LeetCode's "Contains Duplicate III" is part of a series of algorithmic challenges that test one's ability to handle complex data structures and implement efficient algorithms. This problem asks you to determine if there are two distinct indices i and j in an array such that the absolute difference between numbers at these indices is at most t and the absolute difference between i and j is at most k.

In mathematical terms, the problem is to find if there exist indices i and j such that:

• $|nums[i] - nums[j]| \leq t$
• $|i - j| \leq k$

Understanding these constraints is critical: k limits the index distance, while t limits the value difference.

Approach

Solving this problem efficiently involves using data structures that can manage the current "window" of elements. The brute-force method is inefficient with a time complexity of $O(n^2)$. Instead, leveraging a data structure that supports logarithmic time complexity operations for insert, delete, and search is crucial.

Sliding Window with Balanced Trees

We use a sliding window approach combined with a self-balancing tree, such as a SortedList in Python or TreeSet in Java. This approach effectively checks recent k elements for the t condition.

Steps

1. Initialize: Create a list (or set) to keep track of elements in the current window.
2. Iterate: For each element in nums, slide over the window.
3. Check Conditions:
   • For each element, calculate the range of permissible values.
   • Use binary search to check if such an element exists within the current window.
4. Maintain Window Size: Ensure that the window size does not exceed k by continuously removing the oldest element.
5. Return: If any viable pair is found, return True. If no such pairs exist after processing the entire list, return False.

Implementation

Here's a Python implementation using SortedList from the sortedcontainers library:

1from sortedcontainers import SortedList
2
3def containsNearbyAlmostDuplicate(nums, k, t):
4    if k <= 0 or t < 0:
5        return False
6
7    sorted_list = SortedList()
8    for i, num in enumerate(nums):
9        # Maintain the sliding window of size k
10        if i > k:
11            sorted_list.remove(nums[i - k - 1])
12        
13        # Use binary search to find the insertion point
14        pos = SortedList.bisect_left(sorted_list, num)
15
16        # Check if there's an element within the required range
17        if (pos < len(sorted_list) and sorted_list[pos] - num <= t) or \
18           (pos > 0 and num - sorted_list[pos - 1] <= t):
19            return True
20        
21        # Add the current number to the set
22        sorted_list.add(num)
23    
24    return False

Explanation

• Boundary Conditions: Check if k is non-positive or t is negative, immediately return False as it's impossible to fulfill the constraints.
• Sliding Window: Maintain the window with a maximum size of k. Use a SortedList to maintain order, allowing for an efficient check of the nearest elements.
• Binary Search: Use bisect_left to determine where num would be inserted in sorted_list, allowing us to find potential matching elements quickly.

Complexity Analysis

• Time Complexity: $O(n \log k)$, where $n$`` is the number of elements in numsandkis the maximum number of elements in the sliding window. Operations withSortedList like insert, delete and position finding happen in ``$O(\log k)$.
• Space Complexity: $O(\min(n, k))$, since the SortedList holds at most k elements.

Table Summarizing the Solution

Additional Considerations

Edge Cases

Handling edge cases is vital, particularly when dealing with constraints and data types:

• Large Values of t: In some languages, integer overflow might need consideration. Python handles large integers gracefully, but in languages like C++, checks are necessary.
• Negatives and Zeros: Correct handling of negative numbers and zeros is generally provided by the chosen data structure but should be verified.

Alternative Approaches

While the sliding window with a self-balancing tree is efficient, there are other methods like using hashing with bucket sort methods in specific scenarios. These might offer performance benefits with various inputs.

Understanding such problems deepens one's grasp of algorithm design, particularly in optimizing time and space complexity for constrained conditions.