CodeWithYash

Intermediate · Module 8 · Lesson 5

Prefix/suffix products

Combine directional precomputation.

15–25 min readLesson 46 of 123

Lesson 46 of 123

1d streak

Theory lesson · Step-by-step

Understand Prefix/suffix products

Combine directional precomputation.

Worked approach

Trace the reasoning step by step

  1. 1Define the cumulative or best-ending state required by Prefix/suffix products.
  2. 2Initialize the identity or first-element base case explicitly.
  3. 3Update the current state from the previous state and current value.
  4. 4Combine directional states or subtract boundaries using matching index conventions.
  5. 5Test all-negative values, empty prefixes, single cells, and boundary queries.

Concrete example

Compute product except self for [1,2,3,4].

  1. 1Left products are [1,1,2,6].
  2. 2Multiply by right products [24,12,4,1].

Result is [24,12,8,6] without division.

What must remain true (Loop Invariant)

Each stored prefix, suffix, or ending-state value in Prefix/suffix products summarizes exactly its declared contiguous range.

Cost Model

Complexity & Trade-offs

Two directional passes
O(n)
O(n) output
The bound follows from the number of states visited and the work performed per transition.

When to use it

  • Range queries, range updates, directional products, and best contiguous scores.
  • Replacing repeated range scans with cumulative state.
  • Combining left and right summaries.

Common mistakes

  • Mixing inclusive and exclusive prefix conventions.
  • Using zero as Kadane’s base when a non-empty all-negative answer is required.
  • Subtracting the wrong boundary or forgetting a neutral prefix.

Practice prompts

  • Reproduce the worked Prefix/suffix products example without looking at the trace.
  • Change one input condition and explain whether the Prefix/suffix products invariant still holds.
  • Implement the approach, derive its cost, and compare it with a straightforward baseline.

Mastery check

Can you derive it instead of memorizing it?

State the invariant before looking at code.
Explain why every transition is safe.
Construct a worst-case input.
Name one alternative and its trade-off.