"Algorithms for Programmers" (A work in progress) - by Jörg Arndt

This is something of a "book review" for Jörg Arndt's "Algorithms for Programmers" which I have not read straight through but continually dig through and find interesting gems.

There are clearly two types of algorithms already showing up in this collection: those with a systems / optimization bent and those with a mathematical bent. Since I appreciate both optimization and mathematics for the sake of mathematics, I find no problem with this but very often books on Algorithms emphasize the absolutely most boring end of the mathematics of algorithms: complexity analysis. One almost never finds interesting or cool problems solved with equally interesting data-structures or algorithms. Instead Quicksort is presented and we get a chance to show that something is of O(n-logn) complexity. Lame / Boring / *Snore*. To Quicksort's credit it is one of the more mathematically beautiful sorting algorithms but once you've seen it, you've seen it, and can move one.

Jörg's collection of algorithms "for programmers" has a very "systems" or "bit twiddling" feel to it and provides algorithms for many day-to-day real programming problems that come up. Little (if any) time is spent on complexity analysis, in particular proving that an algorithm lives in a given class. Instead good pseudo-code is provided, often several solutions to the same problem, with a bit of explanation as to why one would use a certain variant over the others presented. Even cooler is there are three full chapters on mathematical algorithms including Orthogonal Transforms, Fast Arithmetic, and Algorithms for Finite Fields!

Check it out and be sure to thank Jörg.

Cool Ruby One-Liner - Sierpinski Gasket

First found here referencing a twitter-updated Ruby golf entry, a nice snippet which generates the Sierpinski Gasket (or triangle) I was completely intrigued in the console. Like Giles from the first link I figured I should understand the mathematics used in generating the image but just didn't.

I "improved" the code to my liking since I like parameters for investigative purposes...

ruby -le 's=32; s.times{|y|print" "*(s-y-1),(0..y).map{|x|~y&x>0?" .":" A"}}'

I figured that the reason this code worked was based on a nice mathematical description of the underlying points. The example which hinted at this is to me is the Cantor Set. It has the beautiful definition as the set of points in the closed interval [0,1] such that the trinary expansion at any point contains no 1's (only 0's and 2's). The punchline is the Sierpinski Gasket has a similar description (documented here) and that's why the code works.

Math Nerd Editor's Note: The proof that the cantor-set is uncountable is based on the trivial mapping of the trinary expansions containing only 0's and 2's to the binary expansions containing 0's and 1's which describe all of the reals in [0,1] which itself is uncountable. How cool is that?

Cool Data-Structure: Interval Tree

Geometric Algorithms are pretty darned cool. I've found them interesting for a long time; in fact my new-found love affair with cool algorithms came from my interest in them. The book Data Structures for Computer Graphics, while lacking in certain descriptions, contains a load of fun structures. The first one I just had to implement in Ruby was the Interval Tree, which as a mathematician allowed me to play with the structure theorem for open intervals in the reals in a cool way. I may even post my implementation here later, but for now read the Wiki and see if this is cool for you too!

For a snippet of the book check out this file: Geometric Data Structures for Computer Graphics [PDF]. This is by the authors of the above book and contains much of the same content and even some pretty pictures.

Coming Soon: my implementation from early in my Ruby explorations.

Cool Data-Structure: Radix Tree

My interest in Radix trees came from reading Cisco's Inside Cisco IOS Software Architecture (a surprisingly interesting book) after getting a job as a network engineer. My interest was once again piqued by Thomas Ptacek's post on the Matasano Chargen Blog (which is always a good read). Thomas thinks that the Wikipedia entry on them "sucks" (and I kind of agree but certainly haven't stepped up to improve it) but it is also worth reading I think. So check out the Wiki and Thomas' post and note how cool Radix Trees (Tries) are.

Cool Algorithm: Boyer-Moore string searching algorithm

This algorithm is super cool in that instead of searching the text for a key it searches the key for the text and very very quickly. How sweet is that? More at the almighty Wiki.

Cool Data-Structure: Bloom Filter

Bloom Filters share a relationship with hash tables but are optimized for space over time. They don't store values but will tell you if things have not been hashed (and usually if they have... but not always). More at the almighty-Wiki as well as my original inspiration in looking into them in the first place.

Coming soon... my implementation (again for those paying attention) once I get my stylesheets working. I am programmer not a web-designer.

Cool Algorithms and Data-Structures in general

Like many CS students (before I dropped out and studied real mathematics) I was continually frustrated by the standard algorithm courses we are forced to sit through. First you do weeks of recurrence relations and basic algorithmic analysis followed by quicksort and some trees. Along the way there are tons of implementations of various data-structures and algorithms with stupid patronizing applications.

After having left CS, and even mathematics for the most part, I find myself discovering tons of cool algorithms and data-structures that are never mentioned in classes that I've seen. So I am starting a list of cool algorithms everyone should see once and appreciate, even if they never ever use them.

I will tag them all as "cool algorithms" even if they are data-structures that imply certain algorithms.