"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?