Array Of Product Except Self

How to Solve “Product of Array Except Self in Simple StepsIn programming interviews and coding challenges, one commonly asked problem is the “Product of Array Except Self. It sounds complex at first, but with a clear understanding, it becomes a great way to practice array manipulation and algorithm optimization.

This topic breaks down the concept, explains different approaches, and walks through an efficient solution all in simple language.

What Is “Product of Array Except Self?

Given an array of integers, you are asked to return a new array such that each element at index i of the new array is the product of all the numbers in the original array except the one at index i.

For example

Input [1, 2, 3, 4] Output [24, 12, 8, 6] Explanation

• 24 = 2 × 3 × 4

• 12 = 1 × 3 × 4

• 8 = 1 × 2 × 4

• 6 = 1 × 2 × 3

The key rule do not use division and aim for O(n) time complexity if possible.

Understanding the Constraints

Before jumping to the solution, consider these rules

• The array must not include division to calculate the result.

• The solution should be space-efficient.

• You may be required to solve it in one pass or with minimal extra memory.

Brute Force Approach

The most straightforward way to solve this is to use two nested loops. For each element in the array, calculate the product of all other elements.

Example in pseudocode

for i from 0 to n-1product = 1for j from 0 to n-1if i != jproduct *= array[j]result[i] = product

This works but is inefficient. Time complexity is O(n²), which is not suitable for large datasets.

Optimized Solution Without Division

To improve performance, we can use two auxiliary arrays or variables to store the prefix product and suffix product.

Step-by-step Explanation

Let’s define

• prefix[i] the product of all elements before index i

• suffix[i] the product of all elements after index i

Finally, the result at each index is result[i] = prefix[i] * suffix[i]

Example

Input [1, 2, 3, 4]

• prefix

  • index 0 1 (no elements before)

  • index 1 1 × 1 = 1

  • index 2 1 × 2 = 2

  • index 3 1 × 2 × 3 = 6 → [1, 1, 2, 6]

• suffix

  • index 3 1 (no elements after)

  • index 2 1 × 4 = 4

  • index 1 4 × 3 = 12

  • index 0 12 × 2 = 24 → [24, 12, 4, 1]

• result → [24, 12, 8, 6] (prefix[i] * suffix[i] at each index)

In-Place Optimization

To save space, we can calculate the result in one array

1. Traverse from left to right to store prefix product.

2. Then, traverse from right to left and multiply the current result with the suffix product.

def product_except_self(nums)length = len(nums)result = [1] * length# Step 1 Calculate prefix productsprefix = 1for i in range(length)result[i] = prefixprefix *= nums[i]# Step 2 Multiply by suffix productssuffix = 1for i in reversed(range(length))result[i] *= suffixsuffix *= nums[i]return result

This solution runs in O(n) time and uses O(1) space (excluding output), which is optimal.

Edge Cases to Consider

1. Array with zeros If the array has more than one zero, all products will be zero. If it has one zero, only the position of the zero will have a non-zero product.

2. Empty array or single element For an empty array, return empty. For one element, the result is undefined or usually set to 1 by convention.

3. Negative numbers The logic works the same; no change needed. But be aware of sign flips.

Why Not Use Division?

At first, it might seem easier to calculate the total product of all elements and divide it by each element to get the result.

Example

total = 1for num in numstotal *= numfor i in range(len(nums))result[i] = total // nums[i]

But this method

• Fails when any element is zero (division by zero error).

• Violates the common constraint in interviews don’t use division.

• Might not work for float arrays or languages without integer division handling.

Real-World Applications

This type of array transformation is useful in

• Data processing pipelines where each element’s output depends on the rest of the dataset.

• Mathematical modeling to remove the influence of a specific variable from aggregate data.

• Parallel computing for operations that require exclusion of the current process’s input.

Summary

• The “product of array except self problem is a classic array manipulation question.

• It challenges you to think beyond simple loops and understand how to use cumulative products efficiently.

• Avoid using division and aim for O(n) time and space.

• Use prefix and suffix products to compute the result with minimal memory.

• Practice with variations to solidify your understanding.

By mastering this problem, you improve your skills in algorithm design, space optimization, and thinking in terms of array patterns.