Algorithms

An algorithm is a concrete method with an argument for its correctness and a bound on its cost — the difference between a program that works on your laptop and one that holds up at scale.

Correctness comes first. Before you ask whether an algorithm is fast, you ask whether it is right — an invariant that holds at every step, a base case that grounds the recursion, an argument that the loop must end.

The core skill is then reasoning about cost before you write a line of code. Asymptotic analysis lets you compare strategies on paper and predict which scales to a million inputs and which becomes impractical at a thousand.

Big-O strips away the constants and the hardware to leave the one thing that matters at scale: how the work grows as the input does. A linear pass and a quadratic one are indistinguishable on ten items and orders of magnitude apart on ten million.

Most hard problems yield to a handful of ideas — divide & conquer, greedy choices, dynamic programming, and well-chosen data structures. Learn the ideas once and you start to recognize them everywhere.