Counting steps: how algorithms grow

An algorithm is a recipe: a finite list of steps that turns an input into an answer. To compare two recipes you do not time them on one laptop, because a faster machine makes everything faster and tells you nothing. You count steps instead, and you count them as a function of the input size n. That count is what Big-O notation describes.

# Looking for one name in a list of n names. # Linear scan: check one by one. # n = 10 -> up to 10 steps # n = 1000 -> up to 1000 steps # n = 1 000 000 -> up to 1 000 000 steps -> O(n) # Halving: keep the half that can still contain the name. # n = 10 -> up to 4 steps # n = 1000 -> up to 10 steps # n = 1 000 000 -> up to 20 steps -> O(log n) # Doubling the list adds ONE step, not double the work.

O(n) means linear: twice the input, twice the work — checking every name one by one. O(log n) means the work barely grows: doubling the input adds a single step, which is what halving does. O(n²) means a loop inside a loop: ten times the input, a hundred times the work, and that is the one that quietly kills a program at scale.