Mastering the Max Number of K-Sum Pairs Problem in Python

The Max Number of K-Sum Pairs problem from LeetCode 75 is a common interview question that tests your understanding of sorting, two-pointer techniques, and pair sum calculations. The challenge is simple to describe: given an array of numbers and a target sum k, determine the maximum number of pairs whose sum equals k. However, solving it efficiently for large arrays requires a smart approach.

In this blog post, we’ll break down the problem, present a clean Python solution, and explain why it’s optimal for interviews and real-world applications.

Problem Statement

Given an integer array nums and an integer k, return the maximum number of operations you can perform on the array to form pairs whose sum equals k. Each element can be used at most once.

Understanding the Problem

The key idea is to form pairs (nums[i], nums[j]) such that nums[i] + nums[j] = k. Each number can only belong to one pair, so we need to be careful about counting.

Examples

Example 1:

Input: nums = [1,2,3,4], k = 5  
Output: 2  
Explanation: Pairs are (1,4) and (2,3)

Example 2:

Input: nums = [3,1,3,4,3], k = 6  
Output: 1  
Explanation: Only one pair sums to 6, e.g., (3,3)

Python Solution – Step-by-Step Explanation

Here’s a clean Python solution using the two-pointer technique:

class Solution:
    def maxOperations(self, nums: List[int], k: int) -> int:
        nums.sort()

        if not nums:
            return 0

        start, end, result = 0, len(nums) - 1, 0

        while start < end:
            sum_pair = nums[start] + nums[end]
            if sum_pair < k:
                start += 1
            elif sum_pair > k:
                end -= 1
            else:
                start += 1
                end -= 1
                result += 1

        return result

Step 1: Sort the Array

nums.sort()

Sorting helps efficiently check sums from both ends.
The smallest number is at start and the largest at end.

Step 2: Initialize Two Pointers

start, end = 0, len(nums) - 1
result = 0

start points to the smallest element.
end points to the largest element.
result keeps track of valid pairs found.

Step 3: Traverse and Count Pairs

while start < end:
    sum_pair = nums[start] + nums[end]
    if sum_pair < k:
        start += 1
    elif sum_pair > k:
        end -= 1
    else:
        start += 1
        end -= 1
        result += 1

If the sum is less than k, move start forward to increase the sum.
If the sum is greater than k, move end backward to decrease the sum.
If the sum equals k, a pair is found: increment result and move both pointers.

Step 4: Return Maximum Pairs

return result

After the pointers cross, result contains the maximum number of valid pairs.

Why This Solution Works

Uses a two-pointer strategy to efficiently scan the sorted array.
Ensures that each element is used at most once.
Avoids nested loops, giving a linear time complexity after sorting.

Time and Space Complexity

Metric	Value
Time Complexity	O(n log n) – due to sorting
Space Complexity	O(1) – only pointers and counters

Edge Cases to Consider

Case	Output	Explanation
nums = []	0	No pairs exist
nums = [1,1], k = 2	1	One valid pair
nums = [1,2,3], k = 7	0	No valid pair
nums = [3,3,3,3], k = 6	2	Handles duplicates correctly

Real-World Applications

Transaction matching: Pairing purchases and refunds efficiently.
Resource allocation: Grouping items to meet target constraints.
Interview prep: Tests understanding of greedy strategies and two-pointer techniques.

Conclusion

The Max Number of K-Sum Pairs problem is a classic example of how sorting combined with two-pointer techniques leads to efficient solutions. By carefully moving pointers and counting pairs, you can achieve optimal performance even on large arrays.

Mastering this problem not only strengthens your problem-solving skills for array-based challenges but also provides a strong foundation for tackling more complex pairing and summation problems in coding interviews and real-world tasks.

External Links

🔗 LeetCode Problem Page – Max Number of K-Sum Pairs

🔗 LeetCode Discuss – Community solutions & strategies

🔗 GeeksforGeeks – Pair sum problems explained

Mastering the Max Number of K-Sum Pairs Problem in Python – LeetCode 75 Explained

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