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LeetCode Pattern/Prefix Sum/Prefix Product Pitfalls: Handling Zeros and Division

Prefix Product Pitfalls: Handling Zeros and Division

LeetCopilot Team
Dec 22, 2025
8 min read
Prefix SumProductEdge CasesInterview Prep
Zeros break naive prefix product solutions. Learn why division fails, how to use prefix/suffix products, and the correct approach for Product of Array Except Self.
Complete Guide

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The ultimate comprehensive guide to prefix sum. Learn all variants (1D, 2D, hash map combinations), when to use each pattern, complete templates in multiple languages, and a systematic approach to solve any prefix sum problem.

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You're solving Product of Array Except Self. You think: "Easy, just divide total product by each element." You submit.

Wrong Answer.

The test case has a zero in the array, and your solution crashes or gives incorrect results.

Sound familiar? Prefix product has unique challenges that don't exist in prefix sum. Zeros break division, and handling them correctly requires a different approach.

In this guide, you'll learn why zeros are problematic, the prefix/suffix technique, and how to handle all edge cases.

TL;DR

Common pitfalls:

  • Using division (breaks with zeros)
  • Not handling multiple zeros
  • Forgetting negative numbers affect sign

Correct approach:

python
# Use prefix and suffix products (no division)
result = [1] * n

# Prefix pass
prefix = 1
for i in range(n):
    result[i] = prefix
    prefix *= nums[i]

# Suffix pass
suffix = 1
for i in range(n-1, -1, -1):
    result[i] *= suffix
    suffix *= nums[i]

The Problem: Zeros Break Division

Naive Approach (WRONG)

python
def productExceptSelf(nums):
    # Calculate total product
    total = 1
    for num in nums:
        total *= num
    
    # Divide by each element
    return [total // num for num in nums]

Why It Fails

Test case 1: Single zero

python
nums = [1, 2, 0, 4]
total = 1 * 2 * 0 * 4 = 0

result = [0//1, 0//2, 0//0, 0//4]
         = [0, 0, ???, 0]  # Division by zero!

Test case 2: Multiple zeros

python
nums = [0, 0, 1]
total = 0

result = [0//0, 0//0, 0//1]  # Multiple divisions by zero!

The Fundamental Issue

Division assumes you can "undo" multiplication. But you can't divide by zero, so this approach fundamentally breaks.

The Solution: Prefix and Suffix Products

The Approach

Instead of dividing, build products from both directions:

  1. Prefix pass: Product of all elements to the left
  2. Suffix pass: Product of all elements to the right
  3. Multiply them: result[i] = prefix[i] * suffix[i]

The Template

python
def productExceptSelf(nums: List[int]) -> List[int]:
    """
    LeetCode #238: Product of Array Except Self
    Time: O(n), Space: O(1) excluding output
    """
    n = len(nums)
    result = [1] * n
    
    # Build prefix products (left to right)
    prefix = 1
    for i in range(n):
        result[i] = prefix  # Product of elements to the left
        prefix *= nums[i]
    
    # Build suffix products (right to left) and multiply
    suffix = 1
    for i in range(n - 1, -1, -1):
        result[i] *= suffix  # Multiply by product of elements to the right
        suffix *= nums[i]
    
    return result

Why It Works

For each position i:

  • prefix = product of nums[0] * nums[1] * ... * nums[i-1]
  • suffix = product of nums[i+1] * nums[i+2] * ... * nums[n-1]
  • result[i] = prefix * suffix = product of all except nums[i]

No division needed!

Edge Case 1: Single Zero

Example

python
nums = [1, 2, 0, 4]

Trace

Prefix pass:

code
i=0: result[0] = 1, prefix = 1*1 = 1
i=1: result[1] = 1, prefix = 1*2 = 2
i=2: result[2] = 2, prefix = 2*0 = 0
i=3: result[3] = 0, prefix = 0*4 = 0

result = [1, 1, 2, 0]

Suffix pass:

code
i=3: result[3] *= 1 = 0*1 = 0, suffix = 1*4 = 4
i=2: result[2] *= 4 = 2*4 = 8, suffix = 4*0 = 0
i=1: result[1] *= 0 = 1*0 = 0, suffix = 0*2 = 0
i=0: result[0] *= 0 = 1*0 = 0, suffix = 0*1 = 0

result = [0, 0, 8, 0]

Verify:

code
result[0] = 2*0*4 = 0 ✓
result[1] = 1*0*4 = 0 ✓
result[2] = 1*2*4 = 8 ✓ (only non-zero!)
result[3] = 1*2*0 = 0 ✓

Key insight: Only the position with zero has a non-zero product (product of all other elements).

Edge Case 2: Multiple Zeros

Example

python
nums = [0, 0, 1, 2]

Trace

Prefix pass:

code
i=0: result[0] = 1, prefix = 1*0 = 0
i=1: result[1] = 0, prefix = 0*0 = 0
i=2: result[2] = 0, prefix = 0*1 = 0
i=3: result[3] = 0, prefix = 0*2 = 0

result = [1, 0, 0, 0]

Suffix pass:

code
i=3: result[3] *= 1 = 0*1 = 0, suffix = 1*2 = 2
i=2: result[2] *= 2 = 0*2 = 0, suffix = 2*1 = 2
i=1: result[1] *= 2 = 0*2 = 0, suffix = 2*0 = 0
i=0: result[0] *= 0 = 1*0 = 0, suffix = 0*0 = 0

result = [0, 0, 0, 0]

Verify:

code
result[0] = 0*1*2 = 0 ✓
result[1] = 0*1*2 = 0 ✓
result[2] = 0*0*2 = 0 ✓
result[3] = 0*0*1 = 0 ✓

Key insight: With multiple zeros, all products are zero (every position includes at least one zero).

Edge Case 3: Negative Numbers

Example

python
nums = [-1, 1, 0, -3, 3]

Trace

Prefix pass:

code
result = [1, -1, -1, 0, 0]

Suffix pass:

code
result = [0, 0, 9, 0, 0]

Verify:

code
result[0] = 1*0*(-3)*3 = 0 ✓
result[1] = (-1)*0*(-3)*3 = 0 ✓
result[2] = (-1)*1*(-3)*3 = 9 ✓
result[3] = (-1)*1*0*3 = 0 ✓
result[4] = (-1)*1*0*(-3) = 0 ✓

Key insight: Negatives work naturally—no special handling needed.

Edge Case 4: All Zeros

Example

python
nums = [0, 0, 0]

Result

python
result = [0, 0, 0]

All products are zero (every position includes at least one zero).

Edge Case 5: No Zeros

Example

python
nums = [1, 2, 3, 4]

Trace

Prefix pass:

code
result = [1, 1, 2, 6]

Suffix pass:

code
result = [24, 12, 8, 6]

Verify:

code
result[0] = 2*3*4 = 24 ✓
result[1] = 1*3*4 = 12 ✓
result[2] = 1*2*4 = 8 ✓
result[3] = 1*2*3 = 6 ✓

Works perfectly!

Alternative: Count Zeros

When to Use

If you're allowed to use division and just need to handle zeros specially.

The Approach

python
def productExceptSelf(nums):
    zero_count = nums.count(0)
    
    # More than one zero: all products are 0
    if zero_count > 1:
        return [0] * len(nums)
    
    # Exactly one zero
    if zero_count == 1:
        # Product of all non-zero elements
        product = 1
        for num in nums:
            if num != 0:
                product *= num
        
        # Only the zero position gets the product
        return [0 if num != 0 else product for num in nums]
    
    # No zeros: use division
    total = 1
    for num in nums:
        total *= num
    
    return [total // num for num in nums]

Why This Works

Case 1: Multiple zeros → Every product includes at least one zero → all zeros

Case 2: One zero → Only the zero position excludes the zero → only it's non-zero

Case 3: No zeros → Safe to use division

Drawbacks

  • More complex logic
  • Uses division (some problems forbid it)
  • Doesn't teach the prefix/suffix pattern

Recommendation: Use prefix/suffix approach—it's cleaner and more general.

Common Mistakes

Mistake 1: Using Division

❌ Wrong:

python
total = 1
for num in nums:
    total *= num
return [total // num for num in nums]  # Breaks with zeros

✅ Correct:

python
# Use prefix/suffix (no division)

Mistake 2: Not Handling Multiple Zeros

❌ Wrong:

python
# Only checking for one zero
if 0 in nums:
    # Special case

✅ Correct:

python
# Prefix/suffix handles all cases automatically

Mistake 3: Forgetting to Initialize Result

❌ Wrong:

python
result = []  # Empty list
result[i] = prefix  # IndexError!

✅ Correct:

python
result = [1] * n  # Pre-allocate

The Correct Template

python
def productExceptSelf(nums: List[int]) -> List[int]:
    n = len(nums)
    result = [1] * n
    
    # Prefix pass: product of elements to the left
    prefix = 1
    for i in range(n):
        result[i] = prefix
        prefix *= nums[i]
    
    # Suffix pass: product of elements to the right
    suffix = 1
    for i in range(n - 1, -1, -1):
        result[i] *= suffix
        suffix *= nums[i]
    
    return result

Practice Problems

  1. Product of Array Except Self (#238) - Classic problem
  2. Maximum Product Subarray (#152) - DP variant
  3. Sign of the Product of an Array (#1822) - Count negatives and zeros

FAQ

Q: Why not just count zeros and handle them specially?

A: You can, but prefix/suffix is cleaner, more general, and doesn't require division.

Q: What if the problem allows division?

A: Still use prefix/suffix—it's more robust and handles all edge cases uniformly.

Q: Does this work with negative numbers?

A: Yes, perfectly. Negatives don't require special handling.

Q: What's the space complexity?

A: O(1) excluding the output array (we use the output array to store prefix products).

Q: Can I do this in one pass?

A: No, you need two passes (prefix and suffix). But it's still O(n) time.

Conclusion

Prefix product requires careful handling of zeros, but the prefix/suffix approach solves all edge cases elegantly.

Key takeaways:

  • Don't use division (breaks with zeros)
  • Use prefix/suffix products
  • No special cases needed for zeros or negatives
  • O(n) time, O(1) space

The template:

python
# Prefix pass
for i in range(n):
    result[i] = prefix
    prefix *= nums[i]

# Suffix pass
for i in range(n-1, -1, -1):
    result[i] *= suffix
    suffix *= nums[i]

Master this pattern, and you'll handle product problems with confidence. For more, see Prefix XOR and Product and the complete guide.

Next time you see "product except self," remember: prefix/suffix, no division.

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