Apply Operations to Make Sum of Array Greater Than or Equal to k

Apply Operations to Make Sum of Array Greater Than or Equal to k

What is this problem about?

In the Apply Operations to Make Sum of Array Greater Than or Equal to k interview question, you start with an array containing a single element (usually 1). You can perform two types of operations: incrementing an existing element or duplicating an element. The objective is to find the minimum number of operations required to make the sum of all elements in the array at least k. This Apply Operations to Make Sum of Array Greater Than or Equal to k coding problem challenges your ability to balance two different types of growth.

Why is this asked in interviews?

Companies like Apple and ZScaler use this question to test a candidate's mathematical intuition and optimization skills. It requires moving beyond simple loops to find a more efficient solution, often involving greedy logic or square root decomposition. It’s a classic "min-max" optimization problem that tests if you can identify the most efficient "bang for your buck" between increasing a value and increasing the quantity of values.

Algorithmic pattern used

The Math, Enumeration, Greedy interview pattern is the core of this solution. The strategy is to increase the value of your initial element to some value x, and then duplicate that x enough times to reach the sum k. The problem becomes finding the optimal x that minimizes (increment operations) + (duplication operations).

Example explanation

Suppose k = 11.

1. Option A: Keep the value 1. You need 11 ones. Operations: 0 increments + 10 duplications = 10.
2. Option B: Increment 1 to 3. (2 operations). Now you have [3]. To reach 11, you need four 3s (3+3+3+3=12). That's 3 duplications. Total: 2 + 3 = 5.
3. Option C: Increment 1 to 4. (3 operations). Now you have [4]. You need three 4s (4+4+4=12). That's 2 duplications. Total: 3 + 2 = 5. By iterating through possible values or using the square root of k, you can find the minimum.

Common mistakes candidates make

• Over-incrementing: Spending too many operations increasing a single number when duplicating would be faster.
• Over-duplicating: Creating many small numbers instead of a few larger ones.
• Off-by-one errors: Miscounting the number of duplications needed (it's ceil(k/x) - 1).

Interview preparation tip

For problems involving "minimum operations" to reach a target sum via growth, check if the growth is quadratic or related to square roots. Often, the optimal point is near sqrt(k), where the cost of increasing the base value roughly equals the cost of duplicating it.