#2817

Medium

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Minimum Absolute Difference Between Elements With Constraint

Find the minimum absolute difference between two array elements that are at least x indices apart using binary search.

Problem Statement

You are given a 0-indexed integer array nums and an integer x. Your goal is to find the minimum absolute difference between two elements in nums that are at least x indices apart. In other words, you need to select two indices i and j such that abs(i - j) >= x and minimize abs(nums[i] - nums[j]).

The problem is asking you to identify an efficient way to find the optimal absolute difference given the constraint of indices being separated by at least x. This involves applying binary search techniques and ensuring the solution handles large arrays and values.

Examples

Example 1

Input: nums = [4,3,2,4], x = 2

Output: 0

We can select nums[0] = 4 and nums[3] = 4. They are at least 2 indices apart, and their absolute difference is the minimum, 0. It can be shown that 0 is the optimal answer.

Example 2

Input: nums = [5,3,2,10,15], x = 1

Output: 1

We can select nums[1] = 3 and nums[2] = 2. They are at least 1 index apart, and their absolute difference is the minimum, 1. It can be shown that 1 is the optimal answer.

Example 3

Input: nums = [1,2,3,4], x = 3

Output: 3

We can select nums[0] = 1 and nums[3] = 4. They are at least 3 indices apart, and their absolute difference is the minimum, 3. It can be shown that 3 is the optimal answer.

Constraints

1 <= nums.length <= 105
1 <= nums[i] <= 109
0 <= x < nums.length

Solution Approach

Binary Search on the Difference

One possible approach is to apply binary search on the difference between numbers in the array. By searching over possible differences in a sorted array, we can reduce the search space effectively and find the smallest possible absolute difference that satisfies the index constraint.

Sliding Window or Two Pointers

Another approach involves using a sliding window or two-pointer technique to traverse the array while checking the distance constraint. This method can help in optimizing the selection of pairs with a minimal difference by adjusting the window size dynamically.

Efficient Ordered Set Usage

Leveraging an ordered set or balanced binary search tree can optimize the process of finding the minimum difference. By maintaining a set of values within a specific index range, we can efficiently query for the closest number to any given element.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the approach. A binary search on the difference will generally lead to O(n log(max(nums))) time complexity, where n is the number of elements and max(nums) is the range of values. Using sliding window or two-pointer techniques may result in O(n) time complexity for simpler cases, while using ordered sets may also lead to O(n log n).

What Interviewers Usually Probe

Evaluate the candidate's understanding of binary search and its application to constrained problems.
Look for clarity in the explanation of how to handle the index separation constraint efficiently.
Check for familiarity with ordered set or tree-based data structures for optimizing search operations.

Common Pitfalls or Variants

Common pitfalls

Failing to efficiently manage the index separation constraint and checking all pairs directly can lead to time-limit exceeded errors.
Overlooking the impact of large input sizes and failing to optimize the solution with binary search or an ordered set.
Confusing the problem with simpler pair difference problems that do not have the index separation requirement.

Follow-up variants

Adjusting the index separation constraint (e.g., making it larger) can change the approach and affect performance.
Solving the problem for an array with a fixed length or pre-sorted array may require different optimizations.
Extending the problem to multidimensional arrays or higher-dimensional constraints would introduce added complexity.

FAQ

What is the best way to approach the Minimum Absolute Difference Between Elements With Constraint problem?

A binary search approach is often the most efficient, especially when combined with ordered sets to maintain values in a constrained index range.

How can I handle the index separation constraint efficiently?

You can utilize binary search or a sliding window approach to ensure the indices are separated by at least x, while minimizing the absolute difference.

What is the time complexity of solving this problem with binary search?

The time complexity is generally O(n log(max(nums))) when using binary search on the difference, where n is the number of elements and max(nums) is the range of values.

Are there alternative methods to solve the Minimum Absolute Difference Between Elements With Constraint?

Yes, you can also solve this problem using sliding window, two-pointer techniques, or ordered sets for more efficient searching of minimum differences.

How does GhostInterview assist with solving the Minimum Absolute Difference problem?

GhostInterview provides step-by-step assistance with understanding the binary search approach, optimizing for constraints, and refining the solution for large inputs.

terminal

Solution

Solution 1: Ordered Set

We create an ordered set to store the elements whose distance to the current index is at least $x$.

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class Solution:
    def minAbsoluteDifference(self, nums: List[int], x: int) -> int:
        sl = SortedList()
        ans = inf
        for i in range(x, len(nums)):
            sl.add(nums[i - x])
            j = bisect_left(sl, nums[i])
            if j < len(sl):
                ans = min(ans, sl[j] - nums[i])
            if j:
                ans = min(ans, nums[i] - sl[j - 1])
        return ans

Continue Practicing

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