Smooth n-Window Averages Fast

This question is about computing moving averages over an array with a fixed window size, in a way that scales efficiently as the array grows. The candidate must reason about how to transform a straightforward but slow method that repeatedly sums over overlapping ranges into something much faster. The setup is a classic one in time-series processing and streaming analytics: for each position in the input, you want a smoothed value that reflects either all data so far or just the most recent segment of fixed length. It asks you to turn this idea into a concrete algorithm that is both time- and space-efficient, and to recognize what happens at the beginning of the sequence before the window is "full."

The problem leans on prefix sums, sliding windows, and incremental updates to aggregates. A strong answer usually identifies the naive complexity, then finds a way to reuse previous work as the window moves. Interviewers watch for the ability to manage indices correctly, handle boundary cases cleanly, and argue about time complexity. They also look for clarity of thought in turning a mathematical definition of a moving average into tight, low-level iteration logic without off-by-one errors.

What it tests

The core structure behind moving averages and similar window-based computations is that overlapping intervals share most of their content, so their aggregate properties (like sums or means) can be updated incrementally. Instead of recalculating the sum for each window from scratch, you recognize that only one element leaves and one enters as the window slides, so the sum changes by subtracting the outgoing value and adding the incoming one. This principle is an instance of exploiting redundancy in sequential data: when a process involves repeated, slightly shifted subsets, their computations can be chained together efficiently. The reason this works is because the operation (addition/subtraction) is associative and commutative, so the order and grouping do not affect the result. This incremental approach transforms an apparently quadratic process into a linear one by minimizing repeated work.