Polynomial vs. Exponential

Broadly speaking, algorithms can be classified into two categories:

• "Polynomial time"
• "Exponential time"

Technically O(n!) is "factorial" time, but let's lump them together for simplicity

An algorithm runs in "Polynomial time" if its runtime does not grow faster than n^k, where k is any constant (e.g. n^2, n^3, etc) and n is the size of the input. Polynomial-time algorithms can be useful if they're not too slow.

In comparison, exponential-time algorithms are almost always too slow to be practical. (However, sometimes you're trying to force someone to be slow, like in the case of cryptography and security). Even when n is as low as 20, 2^n is already over a million!