How would you solve the 'Product of Array Except Self' problem: return an array such that each element is the product of all others, omitting division?

Overview

This problem is a staple in technical interviews, particularly for Software Engineer roles at top-tier companies. It effectively gauges a candidate's ability to think algorithmically, optimize for time and space complexity, and handle constraints. Demonstrating a clear, efficient, and robust solution is crucial for showcasing strong problem-solving skills.

Interview Question:

How would you solve the 'Product of Array Except Self' problem: return an array such that each element is the product of all others, omitting division?

Why Interviewers Ask This:

Interviewers pose this question to evaluate several key competencies. First, it assesses a candidate's ability to navigate constraints; the prohibition of division forces a more creative and often more performant approach than a naive solution. Second, it tests understanding of array manipulation, particularly the concept of prefix and suffix products, which are fundamental in many algorithmic problems. Finally, it's a strong indicator of a candidate's capacity to optimize for both time and space complexity, aiming for an O(N) time complexity and O(1) extra space (excluding the output array), which are hallmarks of production-grade code.

Beyond technical prowess, this question also reveals a candidate's thought process. Can they break down a complex problem into simpler, manageable steps? Can they identify edge cases, such as arrays containing zeros, and handle them gracefully? The discussion around these aspects demonstrates a candidate's comprehensive problem-solving approach and their commitment to writing robust, maintainable software.

Expert Answer:

The core challenge of the Product of Array Except Self problem lies in achieving an optimal solution without using division. A naive approach might involve calculating the total product of all elements and then dividing by each element, but this is explicitly forbidden. The most efficient and widely accepted solution involves a two-pass approach using prefix and suffix products.

First, we can initialize a result array, say answer, of the same size as the input array nums. We will populate answer with the product of all elements to the left of each index. We can iterate from left to right, maintaining a running product. For answer[i], we store the product of nums[0] through nums[i-1]. The first element answer[0] will be 1, as there are no elements to its left. This pass takes O(N) time.

Next, we perform a second pass, iterating from right to left. During this pass, we maintain another running product, this time for elements to the right of the current index. For each answer[i], which currently holds the left product, we multiply it by the running right product. The right product starts at 1 and accumulates nums[i+1], nums[i+2], and so on. This effectively combines the left and right products for each index, yielding the desired result. This second pass also takes O(N) time.

This two-pass strategy results in an overall time complexity of O(N) because we traverse the array a constant number of times. Crucially, by performing the calculations directly into the answer array and using a single variable for the right product, we achieve an auxiliary space complexity of O(1), excluding the output array itself. This satisfies the strict optimality requirements for FAANG-style interviews. Edge cases involving zeros must also be considered: if there's one zero, all elements except the one at the zero's index will be zero; if there are two or more zeros, all elements in the result array will be zero.

Speaking Blueprint:

[The Hook] This is a classic problem that tests our ability to compute products efficiently under specific constraints, particularly the prohibition of division. The goal is an optimal solution with O(N) time complexity and O(1) auxiliary space.

[The Core Execution] My approach involves a two-pass iteration. First, I'd initialize a result array. I'd iterate from left to right, populating each element of the result array with the product of all numbers to its left. I'd start with a running product of 1. So, result[i] would store the product of nums[0] to nums[i-1]. After this first pass, result[i] holds the prefix product. Next, I'd iterate from right to left. I'd maintain another running product, this time for elements to the right. For each result[i], I'd multiply its current value (the left product) by this running right product. This combines both sides. I'd also consider edge cases like arrays with one or multiple zeros, which simplify the logic significantly. This method ensures we compute the product of all elements except self without division, achieving optimal time and space complexity.

[The Punchline] This two-pass strategy provides an elegant solution with O(N) time complexity and O(1) auxiliary space complexity, excluding the output array. It's robust, handles edge cases effectively, and demonstrates a strong understanding of algorithmic optimization, making it suitable for high-performance production environments.

Common Mistakes:

• Using division: Failing to respect the explicit constraint of not using the division operator, which is often the first thought for a simpler but non-compliant solution.
• O(N^2) brute force: Implementing a nested loop solution where for each element, the product of all other elements is calculated. This is inefficient and does not meet the optimal time complexity requirement.
• O(N) extra space without justification: While using two auxiliary arrays (one for left products, one for right products) is a valid O(N) time solution, it uses O(N) extra space. Not optimizing to O(1) auxiliary space (by reusing the output array) is a missed opportunity for optimality.
• Incorrectly handling zeros: Overlooking the specific logic required when the input array contains one or more zeros. A single zero means only one element in the output array will be non-zero; two or more zeros mean all output elements will be zero.
• Off-by-one errors in loops: Common mistakes in setting loop boundaries or initial values for running products, leading to incorrect calculations, especially at the array's edges.
• Unclean or unreadable code: Presenting a solution that, while functional, lacks clarity, proper variable naming, or adherence to standard coding practices, indicating a lack of attention to maintainability and code quality.