What is the optimal time complexity of solving Closest Pair?

The optimized solution achieves a time complexity of O(n log n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Closest Pair?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Closest Pair?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Closest Pair?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Competitive ProgrammingEasy

Closest Pair

Learn how to solve the 'Closest Pair' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function closest_pair(arr) that takes an unsorted array of integers arr and returns a tuple of two integers (x, y) from the array that have the minimum absolute difference. The returned tuple must be sorted such that x <= y. If there are multiple pairs with the same minimum difference, return the one with the smaller x value (and if those are the same, the smaller y value).

Constraints

•2 <= len(arr) <= 10^5
•-10^9 <= arr[i] <= 10^9

Examples

Example 1

Input

closest_pair([4, 9, 1, 32, 13])

Output

(1, 4)

Explanation

The difference between 1 and 4 is 3, which is the minimum difference possible between any two elements in the array.

Example 2

Input

closest_pair([12, 15, 17, 20, 25])

Output

(15, 17)

Explanation

The difference between 15 and 17 is 2, which is the minimum difference possible.

Need a Hint?

Analyze the input constraints. Try sorting first (O(n log n)) or using a hash map/set to track seen elements in O(n) time.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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