Problem Overview
The Product of Array Except Self is a classic problem that tests your understanding of array traversal and optimization. The task is simple to state but tricky to implement efficiently.
Given an integer array nums, you need to return an array such that each element at index i is equal to the product of all the elements in nums except nums[i].
The challenge is that:
- Division is not allowed
- The solution must run in O(n) time
Initial Thoughts
At first glance, it feels natural to compute the total product of the array and divide it by the current element. However, this approach fails because division is forbidden and handling zeroes becomes messy.
This pushed me to think differently — instead of excluding the current element, why not multiply everything around it? That’s where the prefix and suffix product pattern comes in.
Key Insight: Prefix & Suffix Products
For every index i:
- Prefix product → product of all elements to the left of
i - Suffix product → product of all elements to the right of
i
If we multiply both, we get the product of all elements except the current one.
Step 1: Prefix Pass (Left to Right)
We traverse the array from left to right and store the product of elements before the current index.
For the first index, this value is 1 because there are no elements on the left.
Step 2: Suffix Pass (Right to Left)
Next, we traverse from right to left while keeping track of the product of elements on the right. We multiply this suffix product with the already stored prefix product.
This way, every index gets the correct product without using division.
Why This Approach Is Optimal
- No division is used ✔️
- Handles zeroes automatically ✔️
- Runs in linear time ✔️
- Uses constant extra space ✔️
How Zeroes Are Handled
One of the best parts of this approach is that it naturally handles zeroes:
- If there is one zero, only that index will have a non-zero product
- If there are multiple zeroes, the entire result becomes zero
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