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Quasirandom Ramblings

Random numbers have a peculiar power, even when they are only pseudo- or quasirandom

Brian Hayes

Random Variations

The concept of randomness in a set of numbers has at least three components. First, randomly chosen numbers are unpredictable: There is no fixed rule governing their selection. Second, the numbers are independent, or uncorrelated: Knowing one number will not help you guess another. Finally, random numbers are unbiased, or uniformly distributed: No matter how you slice up the space of possible values, each region can expect to get its fair share.

These concepts provide a useful key for distinguishing between truly random, pseudorandom, quasirandom and orderly sets. True random numbers have all three characteristics: They are unpredictable, uncorrelated and unbiased. Pseudorandom numbers abandon unpredictability; they are generated by a definite arithmetic rule, and if you know the rule, you can reproduce the entire sequence. But pseudorandom numbers are still uncorrelated and unbiased (at least to a good approximation).

Quasirandom numbers are both predictable and highly correlated. There’s a definite rule for generating them, and the patterns they form, although not as rigid as a crystal lattice, nonetheless have a lot of regularity. The one element of randomness that quasirandom numbers preserve is the uniform or equitable distribution. They are spread out as fairly and evenly as possible.

A highly ordered set, such as a cubic lattice, preserves none of the properties of randomness. It’s obvious that these points fail the tests of unpredictability and independence, but perhaps it’s not so clear that they lack uniform distribution. After all, it’s possible to carve up an N-point lattice into N small cubes that each contain exactly one point. But that’s not enough to qualify the lattice as fairly and evenly distributed. The ideal of equal distribution demands an arrangement of N points that yields one point per region when the space is divided into any set of N identical regions. The rectilinear lattice fails this test when the regions are slices parallel to the axes.

These three aspects of randomness are important in different contexts. In the case of the other Monte Carlo—the casino in Monaco—unpredictability is everything. Cryptographic applications of randomness are similar. In both these cases, an adversary is actively trying to detect and exploit any hint of pattern.

Some kinds of computer simulations are very sensitive to correlations between successive random numbers, so independence is important. For the volume estimations under discussion here, however, uniformity of distribution is what matters most. And the quasirandom numbers, by giving up all pretense to unpredictability or lack of correlation, are able to achieve higher uniformity.

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