Prefix Difference

Prefix difference means looking at gaps between neighbors: a[i] - a[i-1]. It is the companion picture to prefix sums—differences highlight jumps; sums rebuild the original curve.

Why this shows up in the real world

Daily stock returns are differences; their cumulative sum (from a starting price) reconstructs the level series. Energy meters log deltas; utilities integrate to bill totals. Sports lap splits show per-lap change while season totals integrate performance.

Core idea (explained for students)

For i > 0, define d[i] = a[i] - a[i-1]. Often d[0] = a[0] if you want accumulate(d) to equal a. Anytime you see piecewise-constant signals with rare jumps, store deltas and integrate when you need the full waveform.

Try this in Python

a = [2, 5, 5, 8, 1]
d = [a[0]] + [a[i] - a[i - 1] for i in range(1, len(a))]
print("deltas", d)

from itertools import accumulate

print("recover", list(accumulate(d)))

Common mistakes

• Inconsistent definition of d[0]—pick one convention and stick to it.
• Applying pairwise diff to unsorted timestamps without aligning to a grid first.
• Integer subtraction surprises in narrow fixed-width types (Python ints are arbitrary precision).

Key takeaways

• Differences = local story; prefix sums = global reconstruction.
• Practice translating a problem statement into “what jumps?” vs “what totals?”
• Keep indexing notes beside your scratch work.