Determine the Minimum Sum of a k-avoiding Array - Video Solutions

Problem Overview: You need to build an array of n positive integers such that no two elements add up to k. Among all valid arrays, return the minimum possible sum of its elements.

Approach 1: Incremental Element Selection (Greedy with Set) (Time: O(n), Space: O(n))

Build the array one number at a time starting from 1. For every candidate number x, check whether its complement k - x already exists in the chosen set. If it does, adding x would create a pair whose sum equals k, so skip it. Otherwise include x and continue until you collect n numbers.

This works because choosing the smallest valid numbers always minimizes the total sum. A hash set allows constant‑time complement checks while iterating through natural numbers. The algorithm performs at most a few extra skips, so it remains linear in practice. This approach directly models the greedy rule and is easy to reason about during interviews involving greedy decisions and simple math observations.

Approach 2: Direct Number Generation (Math + Greedy Insight) (Time: O(n), Space: O(1))

Observe the structure of forbidden pairs: x conflicts with k - x. To minimize the sum, you want the smallest possible integers. All numbers from 1 up to k / 2 can safely be chosen because their complements lie in the larger range near k. After selecting those values, the next safe numbers must start from k, since any number between k/2 and k-1 would pair with something already chosen.

So the construction becomes deterministic: first take numbers from 1 upward until reaching k/2 (or until the array already has n elements). If more elements are needed, continue from k, k+1, and so on. This avoids complement conflicts entirely and removes the need for a hash set. The result is a clean arithmetic construction based purely on math reasoning.

Recommended for interviews: The incremental greedy approach demonstrates clear understanding of the constraint by explicitly checking complements. Interviewers often accept it first because it is intuitive and easy to verify. The direct generation method shows deeper pattern recognition and stronger greedy reasoning, making it the most polished final solution.