What is the brute force approach and complexity for Product of Array Except Self?

For each index, loop over the array and multiply all other numbers with time complexity O(N²) and space complexity O(1).

What is the optimal approach and complexity for Product of Array Except Self?

Fill answer with prefix products, then sweep from the right with one suffix variable with optimal time complexity O(N) and space complexity O(1).

How do you identify the pattern for Product of Array Except Self?

Look for key signals in the prompt: "product of every number except nums[i]" indicates Except i = everything before i and everything after i → a left pass and a right pass.; "without division" indicates You can't take the total and remove nums[i]. You must build the two sides separately.; "O(1) extra space, output doesn't count" indicates Store one pass in the output array and carry the other in a variable..

Product of Array Except Self

Medium·LeetCode #238 Pro OnlyMedium

LeetCode

Problem Statement

You are given an integer array nums.

Build an array answer where answer[i] is the product of every number in nums except nums[i].

Rules: you may not use division, and your solution must run in O(N) time. It is guaranteed that every prefix product and suffix product fits in a 32-bit integer.

Follow-up: can you do it with O(1) extra space? The output array does not count as extra space.

Input

An integer array nums of length n.

Output

An array of n integers: answer[i] = product of all nums[j] with j ≠ i.

Example 1

Input: nums = [1, 2, 3, 4]

Output: [24, 12, 8, 6]

answer[0] = 2·3·4 = 24, answer[1] = 1·3·4 = 12, answer[2] = 1·2·4 = 8, answer[3] = 1·2·3 = 6.

Example 2

Input: nums = [-1, 1, 0, -3, 3]

Output: [0, 0, 9, 0, 0]

Every product that includes the 0 is 0. Only index 2 skips it: (−1)·1·(−3)·3 = 9.

Constraints

• 2 ≤ n ≤ 10^5
• −30 ≤ nums[i] ≤ 30
• Every prefix and suffix product fits in a 32-bit integer.
• No division allowed.

Pattern Recognition Clues

Recognising these key signals in the problem description immediately identifies the pattern:

“product of every number except nums[i]”

→ Except i = everything before i and everything after i → a left pass and a right pass.

“without division”

→ You can't take the total and remove nums[i]. You must build the two sides separately.

“O(1) extra space, output doesn't count”

→ Store one pass in the output array and carry the other in a variable.

Approach Complexity Ladder

Brute Force — multiply everything else for each indexO(N²)O(1)

For each index, loop over the array and multiply all other numbers

Better — prefix-product and suffix-product arraysO(N)O(N)

Store the product before each index and after each index, then multiply

Optimal — prefix in the output, running suffixO(N)O(1)

Fill answer with prefix products, then sweep from the right with one suffix variable

Brute Force Intuition

For each i, start with 1 and multiply every nums[j] with j ≠ i. Store the result.

Recognition rule for this sub-pattern:

Each index needs information from its left and its right. Build a left pass and a right pass, then combine.

Remember this · Prefix Sum

Trigger

Range sums, "subarray sum equals K" (especially with negatives), equal counts, divisible by K.

Think

If I knew the total up to every index, would the answer become a difference of two totals?

Key question

Which earlier prefix value am I looking for when I stand at index i?

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