# Speed of algorithms (with cats)

Let's see how programmers evaluate fast and slow algorithms. Since the topic is pretty boring, we'll use silly cat examples.

## Constant time: O(1)

This is your best option. The algorithm speed does not depend on the number of cats.

**🐾 Example**

You are the lucky owner of `N`

cats. Every kitten knows their name. If you call "Felix!", only one will come running, and the rest of the `N-1`

fluffs don't care.

## Logarithmic time: O(log n)

On `N`

cats the algorithm completes in `log(N)`

steps. It's fast! 1,000,000 kittens → 20 operations total.

**🐾 Example**

Cat's bowls are arranged alphabetically. When you adopt a new cat, the place for its bowl can be found in `log(N)`

steps.

## Linear time: O(n)

On `N`

cats the algorithm completes in `N`

steps. It means that every time you have to traverse all cats. Not great, not terrible.

**🐾 Example**

Your kittens rebelled and stopped responding to nicknames. Now you have to look through `N`

fluffs to find the right one.

## Log-linear time: O(n log n)

On `N`

cats the algorithm completes in `N`

× `log(N)`

steps. This is slower than in linear time, but not by much (the logarithm of `N`

is much smaller than `N`

, remember?).

**🐾 Example**

Waiting for guests, you decided to seat kittens in the order of their size. The quick sort algorithm will handle this in `N`

× `log(N)`

steps.

Next in line, we have lazy polynomial cats and snail-speed superpolynomial ones.

## Quadratic time: O(n²)

On `N`

cats the algorithm completes in `N²`

steps. So slow.

**🐾 Example**

Your competitor claims that his `N`

kittens are smoother and happier than yours. A special commission will compare the cats in pairs and make a fair verdict. You will need ~ `N²`

comparisons.

## Polynomial time: O(nᵏ)

On `N`

cats the algorithm completes in `N³`

steps, `N⁴`

steps, `N⁵`

steps, or even longer. Ugh. Don't be like that.

**🐾 Example**

Photoshoot! Each of the `N`

kittens will be photographed in pairs with others, and the photographer takes `N`

pictures for each pair. `N`

× `N`

× `N`

≃ `N³`

steps.

Polynomial algorithms are not famous for their speed. But compared to superpolynomial ones, they are as fast as a Flash. Sadly, the only "super" part of superpolynomial algorithms is a name. Let me show you.

## Exponential time: O(2ⁿ)

On `N`

cats the algorithm completes in `2ⁿ`

steps. It's a long time, you're probably not gonna wait.

**🐾 Example**

Kittens are going to the cat show. Everyone will be weighed and rated in stars. But the cat transport can handle a maximum of X kilograms (or pounds). How to choose the most stellar cast? The answer will require `2ⁿ`

steps.

## Factorial time: O(n!)

On `N`

cats the algorithm completes in `N`

×`(N-1)`

×`(N-2)`

×... × `1`

steps. This is crazy! With 20 cats it's already a couple of quintillion operations.

**🐾 Example**

Kittens spread out across the apartment. You want to pet everyone, but you're lazy and don't like walking. What is the shortest route to visit all fluffs? This is ~ `N!`

comparisons.

## Summary

Here are the algorithms we've covered:

- Constant
`O(1)`

- Logarithmic
`O(log n)`

- Linear
`O(n)`

- Log-Linear
`O(n log n)`

- Quadratic
`O(n²)`

- Polynomial
`O(nᵏ)`

- Exponential
`O(2ⁿ)`

- Factorial
`O(n!)`

A constant algorithm is always the best option, and a logarithmic one is almost always. Linear and polynomial ones are more complex — it all depends on the task with them. Sometimes it's a shame to choose `O(n)`

, and sometimes `O(n²)`

is a huge success.

`O(2ⁿ)`

and `O(n!)`

are insanely slow. So in practice, people often use suboptimal but fast algorithms.

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