## Big O and friends

All software developers know about Big O. For instance, contemporary sorting algorithms usually have $\mathcal{O}(n\log{}n)$ complexity. Something like $\mathcal{O}(n\log{}n)$ looks like a function of a function, but it’s really a classification system.

We start by defining the complexity of an algorithm to be an estimate of the number of steps required to solve a problem. This count is going to vary by the size of the problem, and so the complexity must be a function of the problem size.

Sometimes this can result in a nasty looking functions like $f(x)=3x^2+2x+7$ or $f(x)=5x^2-999$. We say that both of these functions are in the category $\mathcal{O}(x^2)$ because they have similar growth properties. This is called Bachmann-Landau notation.

In mathematics, this notation describes the asymptotic behavior of a function. if $f(n) = \mathcal{O}(g(n))$, then we have a guarantee that after a certain point, the graph of $f(n)$ will fall below the graph of $g(n)$, if you’re willing to rotate $g(n)$ a bit. In computer science, we use this to understand how the performance of an algorithm will change as the size of the problem or input grows. There are actually several Bachmann-Landau notations, each of which gives us some kind of indication on the limits of the growth of a function.

Name Notation Analogy
Big O $f(n)$ is $\mathcal{O}(g(n))$ $f(n) \le g(n)$
Little o $f(n)$ is $o(g(n))$ $f(n)$ < $g(n)$
Big Omega $f(n)$ is $\Omega(g(n))$ $f(n) \ge g(n)$
Little omega $f(n)$ is $\omega(g(n))$ $f(n) > g(n)$
Theta $f(n)$ is $\Theta(g(n))$ $f(n) = g(n)$

Credit goes to MIT for the excellent table and analogies. The definitions of each can be found in the referenced material.

Additionally, some of these definitions can be expressed using calculus. We start with the definition of $o$ in the material says that $f(n)$ is $o(g(n))$ if : $\displaystyle \forall C > 0 , \exists k \,|\, ( x > k \rightarrow |f(n)| \le C|g(n)| )$

That’s a symbolic mouthful. It means, “For all positive values of C, there exists some k such that, if x>k, the absolute value of f(n) is less than or equal to the absolute value of g(n) multiplied by C.

Separately, any calculus text will give this definition for a limit at infinity. I like Paul’s Online Calculus Notes. It says that $\lim_{x\to\infty} f(x) = L$ if $\displaystyle \forall \epsilon > 0, \exists N \forall x>N : |f(x)-L|$ < $\epsilon$

Again in plain English, “For any positive value of $\epsilon$, there must exist some value for N such that for any value of x that is greater than N, this inequality holds.”

Let’s make some substitutions of variable names in the above equation. If we change out f(x) for f(x)/g(x), 0 for L, C for $\epsilon$, n for x, and k for N : $\displaystyle \forall C > 0, \exists k \forall n>k : |f(n)/g(n)|$ < $C$

or $\displaystyle \forall C > 0, \exists k \forall n>k : |f(n)|$ < $C |g(n)|$

This boils down to an alternative definition of $o$. $f(n)$ is $o(g(n))$ if : $\displaystyle \lim_{x\to\infty} f(x)/g(x) = 0$

This allows us to use other mathematical tricks, like L’Hopitals Rule, to figure out if two given functions follow this relationship.

## Mustache

Ah, it’s nice to be free from exams and assignments. Grading is over!

A lot of small but indispensable tools belong in the software developer’s toolkit. Take Markdown, for example. I am actually writing this post using WordPress Jetpack’s new Markdown composition features. So far, so good! I use Markdown extensively for creating documents. There is only a little bit to learn, but you can create nice documents very quickly. I use Typora for MD docs on my Mac, and I highly recommend it.

Mustache is a template system embedded in several front-end web frameworks. It is very simple to use, and the whole thing can be described in one unix-style man page.

I find it is most instructive to play with frameworks interactively using playgrounds or other REPL environments, so one npm install -g mustache later I am on the way to learning it.

Mustache works by embedding tags into any type of text data, such as plain text, HTML, or Markdown. Tags are encapsulated using mustache symbols (the curly brace) like this : {{tag}}. Take this example :

These tags will be replaced by variable values found in another file to follow. Some variables mark sections that can be repeated or left out all together based on their values. There values are found in a separate file.

In Mustache lingo, the above data comprises a template file. It will be formatted using variables contained in the following JSON file, called a view or hash :

The result is the following

You can see how this might be more useful if the JSON object was produced by loops in JavaScript and the template data was marked-up HTML instead of simple plain text. But that’s the general idea.