# Asymptotic Analysis Basics

Content Note

This concept is a big reason why a strong math background is helpful for computer science, even when it’s not obvious that there are connections! Make sure you’re comfortable with Calculus concepts up to power series.

## An Abstract Introduction to Asymptotic Analysis #

The term **asymptotics,** or **asymptotic analysis,** refers to the idea of **analyzing functions when their inputs get really big.** This is like the **asymptotes** you might remember learning in math classes, where functions approach a value when they get very large inputs.

Here, we can see that $y= \dfrac{x^3}{x^2 + 1}$ looks basically identical to $y = x$ when x gets really big. Asymptotics is all about reducing functions to their eventual behaviors exactly like this!

## That’s cool, but how is it useful? #

Graphs and functions are great and all, but at this point it’s still a mystery as to how we can use these concepts for more practical uses. Now, we’ll see how we can **represent programs as mathematical functions** so that we can do cool things like:

**Figure out how much time or space a program will use****Objectively tell how one program is better than another program****Choose the optimal data structures for a specific purpose**

As you can see, this concept is **absolutely fundamental** to ensuring that you write **efficient algorithms** and choose the **correct data structures.** With the power of asymptotics, you can figure out if a program will take 100 seconds or 100 years to run without actually running it!

## How to Measure Programs #

In order to convert your `public static void Main(String[] args)`

or whatever into `y = log(x)`

, we need to figure out what `x`

and `y`

even represent!

**TLDR:** It depends, but the three most common measurements are **time, space,** and **complexity**.

**Time** is almost always useful to minimize because it could mean the difference between a program being able to run on a smartphone and needing a supercomputer. Time usually increases with the **number of operations** being run. Loops and recursion will increase this metric substantially. On Linux, the `time`

command can be used for measuring this.

**Space** is also often nice to reduce, but has become a smaller concern now that we can get terabytes (or even petabytes) of storage pretty easily! Usually, the things that take up lots of space are **big lists** and **a very large number of individual objects.** Reducing the size of lists to hold only what you need will be very helpful for this metric!

There is another common metric, which is known as **complexity** or **computational cost.** This is a less concrete concept compared to time or space, and cannot be measured easily; however, it is highly generalized and usually easier to think about. For complexity, we can simply assign basic operations (like println, adding, absolute value) a complexity of **1** and add up how many basic operation calls there are in a program.

## Simplifying Functions #

Since we **only care about the general shape of the function,** we can keep things as simple as possible! Here are the main rules:

**Only keep the****fastest growing term.**For example, $log(n) + n$ can be simplified to just $n$since $n$ grows faster out of the two terms.**Remove all constants.**For example, $5log(3n)$ can just be simplified to $log(n)$since constants don’t change the overall shape of a function.**Remove all other variables.**If a function is really $log(n + m)$ but we only care about n, then we can simply it into $log(n)$.

There are two cases where we can’t remove other variables and constants though, and they are:

- A polynomial term $n^c$(because $n^2$grows slower than $n^3$, for example), and
- An exponential term $c^n$(because $2^n$grows slower than $3^n$, for example).

## The Big Bounds #

There are **three** important types of runtime bounds that can be used to describe functions. These bounds put restrictions on how slow or fast we can expect that function to grow!

**Big O** is an **upper bound** for a function growth rate. That means that **the function grows slower or the same rate as the Big O function.** For example, a valid Big O bound for $log(n) + n$ is $O(n^2)$ since $n^2$ grows at a faster rate.

**Big Omega** is a **lower bound** for a function growth rate. That means that **the function grows faster or the same rate as the Big Omega function.** For example, a valid Big Omega bound for $log(n) + n$ is $\Omega(1)$ since $1$ (a constant) grows at a slower rate.

**Big Theta** is a **middle ground** that describes the function that grows at the **same rate** as the actual function. **Big Theta only exists if there is a valid Big O that is equal to a valid Big Omega.** For example, a valid Big Theta bound for $log(n) + n$ is $\Theta(n)$ since $n$ grows at the same rate (log n is much slower so it adds an insignificant amount).

## Orders of Growth #

There are some **common functions** that many runtimes will simply into. Here they are, from fastest to slowest:

Function | Name | Examples |
---|---|---|

$\Theta(1)$ | Constant | System.out.println, +, array accessing |

$\Theta(\log(n))$ | Log | Binary search |

$\Theta(n)$ | Linear | Iterating through each element of a list |

$\Theta(n\log(n))$ | nlogn 😅 | Quicksort, merge sort |

$\Theta(n^2)$ | Quadratic | Bubble sort, nested for loops |

$\Theta(2^n)$ | Exponential | Finding all possible subsets of a list, tree recursion |

$\Theta(n!)$ | Factorial | Bogo sort, getting all permutations of a list |

$\Theta(n^n)$ | n^n 😅😅 | Tetration |

Don’t worry about the examples you aren’t familiar with- I will go into much more detail on their respective pages.

## Asymptotic Analysis: Step by Step #

- Identify the function that needs to be analyzed.
- Identify the parameter to use as $n$.
- Identify the measurement that needs to be taken. (Time, space, etc.)
- Generate a function that represents the complexity. If you need help with this step, try some problems!
- Simplify the function (remove constants, smaller terms, and other variables).
- Select the correct bounds (O, Omega, Theta) for particular cases (best, worst, overall).