Make Array Elements Equal to Zero

Make Array Elements Equal to Zero

What is this problem about?

The Make Array Elements Equal to Zero problem presents an array of integers where some elements are positive and some are 0. You start at an index containing 0 and choose a direction (left or right). As you move, if you step on a positive number, you decrement it by 1 and reverse your direction. Your goal is to find the number of starting indices (containing 0) that will eventually reduce all positive numbers in the entire array to exactly 0.

Why is this asked in interviews?

This is a logic and array traversal puzzle. Interviewers ask it to test if a candidate can optimize a seemingly complex simulation. Simulating the bouncing back-and-forth for every single zero index would take massive amounts of time ($O(N^2 \times \text{MaxVal})$). The prompt tests your ability to abstract the bouncing logic into a simple mathematical property regarding prefix and suffix sums.

Algorithmic pattern used

This problem relies on Prefix Sums and mathematical deduction. If you start at a zero, you will bounce left and right, decrementing numbers. For the bounces to completely clear the array, the total sum of numbers to the left of your starting zero must either be exactly equal to the total sum of numbers to its right, or differ by exactly 1. By precalculating the total sum of the array, you can keep a running left_sum as you iterate, deriving the right_sum instantly.

Example explanation

Array: [1, 0, 2, 0, 3] Total sum of positive elements = $1 + 2 + 3 = 6$. Iterate through the array looking for 0s. We maintain left_sum.

• Index 1 (0):
  • left_sum (elements before index 1) = 1.
  • right_sum (elements after index 1) = $6 - 1 = 5$.
  • They differ by 4. If we start here, we will get trapped on the right side. Invalid.
• Index 3 (0):
  • left_sum (elements before index 3) = $1 + 2 = 3$.
  • right_sum (elements after index 3) = $6 - 3 = 3$.
  • left_sum == right_sum. Perfect! We can start here going left, or start here going right, and both will clear the array. This index gives us 2 valid starting configurations.

Common mistakes candidates make

The absolute biggest mistake is writing a while loop that actually simulates moving pointers, decrementing array values, and changing directions. This brute-force approach completely misses the core mathematical relationship and will immediately result in a Time Limit Exceeded error. Another mistake is ignoring that a zero index can offer two valid configurations if left_sum == right_sum (starting left OR starting right).

Interview preparation tip

When an interview problem involves "bouncing" or "reversing direction" upon hitting obstacles or values, look for balance. Bouncing inherently means you are alternating between two sides. If the problem asks for full depletion, the "weight" or "sum" of both sides must be balanced. Recognizing this transforms messy simulations into elegant $O(N)$ prefix sum solutions.