Array Diff in O(n log n) Time Complexity using Python

Given two arrays of integers, find a pair of integers, one from each array, whose absolute difference is the closest to zero.

Example:

Input: first =[4, 1, 8],
       second = [12, 3, 17]

Output: [4, 3]
Explanation: abs(4-3) = 1 which is the closest to 0

Note:

Your algorithm should run in O(n log n) time and use O(1) extra space.

Understanding the Problem

The core challenge of this problem is to find a pair of integers, one from each array, such that their absolute difference is minimized. This problem is significant in various applications such as minimizing error in measurements, finding closest points in computational geometry, and more.

Potential pitfalls include not considering all pairs or not optimizing the solution to meet the required time complexity.

Approach

To solve this problem, we can use the following approach:

Sort both arrays. Sorting helps in efficiently finding the closest pairs by allowing a linear scan.
Use two pointers to traverse both arrays and find the pair with the smallest absolute difference.

Let's discuss why this approach is optimal:

Sorting both arrays takes O(n log n) time.
Using two pointers to find the closest pair takes O(n) time.

Thus, the overall time complexity is O(n log n), which meets the problem's requirements.

Algorithm

Here is a step-by-step breakdown of the algorithm:

Sort both arrays.
Initialize two pointers, one for each array.
Initialize a variable to keep track of the minimum difference and the corresponding pair.
While both pointers are within the bounds of their respective arrays:
- Calculate the absolute difference between the elements pointed to by the two pointers.
- If this difference is smaller than the current minimum difference, update the minimum difference and the pair.
- Move the pointer that points to the smaller element to the next position.
Return the pair with the smallest absolute difference.

Code Implementation

def find_closest_pair(first, second):
    # Sort both arrays
    first.sort()
    second.sort()
    
    # Initialize pointers for both arrays
    i, j = 0, 0
    
    # Initialize variables to track the minimum difference and the pair
    min_diff = float('inf')
    closest_pair = []
    
    # Traverse both arrays using two pointers
    while i < len(first) and j < len(second):
        # Calculate the absolute difference
        diff = abs(first[i] - second[j])
        
        # Update the minimum difference and the pair if a smaller difference is found
        if diff < min_diff:
            min_diff = diff
            closest_pair = [first[i], second[j]]
        
        # Move the pointer that points to the smaller element
        if first[i] < second[j]:
            i += 1
        else:
            j += 1
    
    return closest_pair

# Example usage
first = [4, 1, 8]
second = [12, 3, 17]
print(find_closest_pair(first, second))  # Output: [4, 3]

Complexity Analysis

The time complexity of this approach is O(n log n) due to the sorting step. The space complexity is O(1) as we are using a constant amount of extra space.

Edge Cases

Consider the following edge cases:

Arrays with only one element each.
Arrays with negative numbers.
Arrays with duplicate elements.

Our algorithm handles these cases effectively by sorting the arrays and using two pointers to find the closest pair.

Testing

To test the solution comprehensively, consider the following test cases:

# Test case 1: Simple case
assert find_closest_pair([4, 1, 8], [12, 3, 17]) == [4, 3]

# Test case 2: Single element arrays
assert find_closest_pair([1], [2]) == [1, 2]

# Test case 3: Arrays with negative numbers
assert find_closest_pair([-1, -3, -5], [2, 4, 6]) == [-1, 2]

# Test case 4: Arrays with duplicate elements
assert find_closest_pair([1, 1, 1], [2, 2, 2]) == [1, 2]

Thinking and Problem-Solving Tips

When approaching such problems, consider the following tips:

Break down the problem into smaller, manageable parts.
Think about how sorting can help simplify the problem.
Use two-pointer techniques for problems involving pairs or comparisons between two arrays.

Practice solving similar problems to improve your problem-solving skills and understanding of algorithms.

Conclusion

In this blog post, we discussed how to find a pair of integers from two arrays whose absolute difference is the closest to zero. We explored the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is crucial for developing strong problem-solving skills and algorithmic thinking.

Additional Resources

For further reading and practice, consider the following resources: