Fixed-Points

Preparing lecture notes (or videos) sometimes brings you to revisit things you’ve known for a while, but haven’t really taken time to formalize properly. One of those things is fixed-point arithmetic.

Evaluating polynomials

Evaluating polynomials is not a thing I do very often. When I do, it’s for interpolation and splines; and traditionally those are done with relatively low degree polynomials—cubic at most. There are a few rather simple tricks you can use to evaluate them efficiently, and we’ll have a look at them.

Factorial Approximations

(and its logarithm) keep showing up in the analysis of algorithm. Unfortunately, it’s very often unwieldy, and we use approximations of (or ) to simplify things. Let’s examine a few!

YoU CanT MaKE BuBBleSorT FaSTER With ASseMbLY

In one of the classes I teach, we end up writing assembly language programs. And while I explain the (sometimes very relative) benefits of writing assembly language, I use bubble sort as an example where even carefully crafted assembly language doesn’t mean much: it’s a bad algorithm to start with.

YoU CanT MaKE BuBBleSorT FaSTER With ASseMbLY

Except that it’s not quite true.

Fast Exponentiation, revisited

Quite a while ago, I presented a fast exponentiation algorithm that uses the binary decomposition of the exponent to perform products to compute .

While discussing this algorithm in class, a student asked a very interesting question: what’s special about base 2? Couldn’t we use another base? Well, yes, yes we can.

(Sub)bit-fields (Coding with fractions of bits, Part II)

Last week, we used the 6×7×6 palette as an example of very simple fraction-of-a-bit coding¹. However, we can generalize still a bit more to allow single field extraction and modification

The 6×7×6 palette (Coding with fractions of bits, Part I)

Remember ye olde dayes when we had to be mindful of the so-called “web safe palette“? Once upon a time, screens could display 24-bits colors, but only 256 at a time in some “hi-res” modes. But that’s not what I’m going to tell you about: I’d rather tell you about the encoding of the palette, and about a somewhat better palette. And also about using fractions of bits for more efficient encodings.

Discrete Inversion (Generating Random Sequences XII)

While this sounds something like a shameful family secret, discrete inversion is only the finite-valued variation on the method of inversion for the generation of random numbers with a given distribution (as I’ve discussed quite a while ago here). The case we’ll consider here is a random variable with few possible outcomes, each with different odds

Mœud deux

Pairing functions are fun. Last week, we had a look at the Cantor/Hopcroft and Ullman function, and this week, we’ll have a look at the Rosenberg-Strong function—and we’ll modify it a bit.

Mœud

Pairing functions are used to reversibly map a pair of number onto a single number—think of a number-theoretical version of std::pair. Cantor was the first (or so I think) to propose one such function. His goal wasn’t data compression but to show that there are as many rationals as natural numbers.

Cantor’s function associates pairs (i,j) with a single number:

…but that’s not the only way of doing this. A much more fun—and spatially coherent—is the boustrophedonic pairing function.