Determine the Minimum Sum of a k-avoiding Array

Determine the Minimum Sum of a k-avoiding Array

What is this problem about?

In the Determine the Minimum Sum of a k-avoiding Array coding problem, you need to construct an array of $n$ distinct positive integers such that no two elements in the array sum up to exactly $k$. You want to minimize the total sum of this array.

Why is this asked in interviews?

Infosys and similar IT services giants ask this to test your ability to apply greedy strategies to mathematical constraints. It evaluations whether you can identify which numbers to "avoid" to keep the sum as small as possible. It’s a test of greedy interview patterns and basic understanding of set exclusion.

Algorithmic pattern used

This problem uses a Greedy approach with a Hash Set. To minimize the sum, you should always try to include the smallest possible positive integers (starting from 1).

1. Initialize an empty set and a current number $i = 1$.
2. While the array size is less than $n$:
   • Check if $(k - i)$ is already in the set.
   • If not, adding $i$ is safe. Add $i$ to the set and increment the total sum.
   • If $(k - i)$ is in the set, skip $i$.
3. Increment $i$ and repeat.

Example explanation

$n = 5, k = 4$.

1. Start with 1. (4-1=3) not in set. Add 1. Set: {1}.
2. Try 2. (4-2=2). Self-pair check: if we add 2, it can't pair with itself to make 4 unless the problem allows duplicates (which it doesn't). So 2 is safe. Set: {1, 2}.
3. Try 3. (4-3=1). 1 is in set! Skip 3.
4. Try 4. (4-4=0). 0 is not a positive integer. Add 4. Set: {1, 2, 4}.
5. Try 5. Add 5. Set: {1, 2, 4, 5}.
6. Try 6. Add 6. Set: {1, 2, 4, 5, 6}. Total Sum: $1+2+4+5+6 = 18$.

Common mistakes candidates make

• Missing the Complement: Forgetting to check if $k - i$ was already picked.
• Duplicates: Adding the same number twice.
• Inefficiency: Using a nested loop to check for the sum instead of a Hash Set for $O(1)$ lookup.

Interview preparation tip

When minimizing a sum under constraints, always start from the smallest available units (1, 2, 3...) and only skip those that violate the rules. This greedy logic is very common in "Minimum Sum" problems.