Minimum Numbers of Function Calls to Make Target Array

Minimum Numbers of Function Calls to Make Target Array

1. What is this problem about?

In this problem, you start with an array of zeros and want to transform it into a given target array using only two types of operations: incrementing a single element by 1 or multiplying the entire array by 2. The goal is to find the minimum number of total operations required to reach the target.

2. Why is this asked in interviews?

Amazon often asks this "Minimum Numbers of Function Calls to Make Target Array coding problem" to test a candidate's ability to work backwards. Instead of figuring out how to build the target, it's often easier to figure out how to reduce the target back to zero. It tests bit manipulation skills and the ability to recognize that the "multiply by 2" operation is the same as a bit shift.

3. Algorithmic pattern used

The algorithmic pattern is Greedy combined with Bit Manipulation. By working backward from the target to zero:

• If an element is odd, decrement it (counts as an addition).
• If all elements are even (and not all are zero), divide the entire array by 2 (counts as one multiplication). Essentially, you are counting the number of set bits (for additions) and the maximum number of times any element can be divided by 2 (for multiplications).

4. Example explanation

Target: [3, 5]

• Work backward:
• 3 and 5 are odd. Decrement both: [2, 4]. (2 operations)
• All even: Divide all by 2: [1, 2]. (1 operation)
• 1 is odd. Decrement: [0, 2]. (1 operation)
• All even: Divide all by 2: [0, 1]. (1 operation)
• 1 is odd. Decrement: [0, 0]. (1 operation) Total: 2 + 1 + 1 + 1 + 1 = 6.

5. Common mistakes candidates make

A common error is counting the "multiply by 2" operation multiple times for each element. Remember, multiplying by 2 affects the entire array, so you only count it once per "round" for all elements. Another mistake is forgetting to handle the case where some elements reach zero before others.

6. Interview preparation tip

When a problem involves multiplying an entire set by a constant or incrementing individual items, always try the reverse process. Reducing a number to zero using division and subtraction is the same as analyzing its binary representation.