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Minimum Absolute Difference Between Elements With Constraint

Difficulty: Medium

Problem Description

You are given a 0-indexed integer array nums and an integer x. Find the minimum absolute difference between two elements in the array that are at least x indices apart. In other words, find two indices i and j such that abs(i - j) >= x and abs(nums[i] - nums[j]) is minimized. Return an integer denoting the minimum absolute difference between two elements that are at least x indices apart.

Key Insights

  • The problem requires checking pairs of elements in the array while ensuring they are separated by at least x indices.
  • The brute-force approach would involve checking all pairs which can be inefficient for large arrays.
  • Using a sorted data structure allows for efficient searching of the closest elements.
  • The sliding window technique can be used to maintain a set of valid candidates as we iterate through the array.

Space and Time Complexity

Time Complexity: O(n log n) - due to sorting and potentially log n operations for searching in a sorted structure. Space Complexity: O(n) - for storing elements in a sorted data structure.

Solution

To solve the problem, we can use a sliding window approach combined with a sorted data structure (like a balanced binary search tree or an ordered set) to efficiently find the minimum absolute difference between elements that are at least x indices apart. As we iterate through the array, we maintain a window of valid candidates and update our minimum difference accordingly.

Code Solutions

def minimum_abs_difference(nums, x):
    from sortedcontainers import SortedList

    sorted_list = SortedList()
    min_diff = float('inf')
    
    for i in range(len(nums)):
        if i >= x:
            # Remove the element that is out of the window
            sorted_list.remove(nums[i - x])
        
        # Add the current element to the sorted list
        sorted_list.add(nums[i])
        
        # Find the position to insert nums[i] and check neighbors
        pos = sorted_list.bisect_left(nums[i])
        
        # Check the previous element if it exists
        if pos > 0:
            min_diff = min(min_diff, abs(nums[i] - sorted_list[pos - 1]))
        
        # Check the next element if it exists
        if pos < len(sorted_list):
            min_diff = min(min_diff, abs(nums[i] - sorted_list[pos]))

    return min_diff
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