Essential Math Weblog Thoughts on math for games

10/21/2013

New Thoughts on Random Numbers, Part Two

Filed under: Erratical,Mathematical — Jim @ 10:32 pm

Sorry about the long delay on this. I originally intended to post this right away, but then realized I should do some further testing… and for whatever reason that never happened until last week. In any case, as I mentioned in my previous post, I recently added a new pseudo-random number generator to the Skia test suite. The last post was about verifying that a new PRNG was needed, this one is about what I used to replace it.

My initial thoughts about how to replace the Linear Congruential Generator was to jump right to the Mersenne Twister. However, the Mersenne Twister requires a buffer of 624 integers. While this is not terrible, it’s probably overkill for what we need, and the large buffer is not great for low-memory situations. In addition, part of my research showed that it’s possible to get large random periods with other, simpler methods.

To cut the story short, because we do a lot of testing on mobile platforms I wanted something low-memory and super simple. I ended up taking one of Marsaglia’s generators and modifying it slightly. This is the “mother-of-all” generator that I mention in the random number generator chapter:

k=30345*(k&65535)+(k>>16);
j=18000*(j&65535)+(j>>16);
return((k << 16)+j);

where k and j are unsigned 32-bit integers, initialized to non-zero values.

When I ran this through the Tuftests suite, it passed the GCD and Birthday Spacings tests, but failed the Gorilla test on half of the bits. So I figured, what the hey, I’ll just combine the high bits of k into the low bits of the result as well, or

k=30345*(k&65535)+(k>>16);
j=18000*(j&65535)+(j>>16);
return(((k << 16) | (k >> 16)) + j);

Surprisingly, this passed all three tests. In addition, I ran a test to try to see what its period is. It ran over a weekend and still didn’t hit 2^64 values. While a good part of that time was running the test on each new value, it does show that in a game or your average application, we probably don’t need these large periods because practically we’ll never generate that many values (the exception might be a game server, which is up for days). For Skia’s particular bench and unit tests, we might generate a few thousand random values — as long as our period is more than that we should be okay.

I’ve since run the TestU01 suite on this, and it passes SmallCrush, fails one of the Birthday Spacing tests (#11) on Crush, and two other Birthday Spacing tests (#13 and #15) on BigCrush. Other than that it passes. So while it’s not as random as other generators, at this point it doesn’t seem worth it to change it and rebaseline all of our results again, just for those failures.

If that doesn’t work for you, an alternative worth trying might be Marsaglia’s KISS generator, which takes four different random number generators and mixes their bits together. He claims (or claimed: Dr. Marsaglia unfortunately passed on in 2011) that the 2011 version has a period of greater than 10^40000000. Regardless of quality, the 2011 version requires a large buffer which is probably not practical for low-memory mobile platforms. The 1993 version, on the other hand, requires only six values, and has a period of 2^123. It does still fail one Crush and one BigCrush test, but is still probably good enough for games.

One important note: KISS is not cryptologically safe, as proven here, and I strongly doubt the generator I describe above is either. And if you do need a cryptologically secure generator, it’s best to find an approved implementation and use that, as even slight changes in speed due to a naive implementation can be enough to give an attacker enough information to break your code. But of course, if the NSA was involved, you might want to be wary…

In any case, to get results suitable for game logic, you really don’t need Mersenne Twister. Using one of these simple generators will probably do, and are leaps and bounds better than any Linear Congruential Generator.

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