The Britney Spears Problem
Tracking who's hot and who's not presents an algorithmic challenge
Gently Down the Stream
Are there better algorithms for stream problems waiting to be discovered? Some of the limits of what's possible were outlined in 1996 by Noga Alon, Yossi Matias and Mario Szegedy, who were all then at AT&T Bell Laboratories. They framed the issue in terms of an infinite series of frequency moments , analogous to the more familiar statistical moments (mean, variance, and so on). The zero-order frequency moment F 0 is the number of distinct elements in the stream; the first-order moment F 1 is the total number of elements; F 2 is a quantity called the repeat rate; still higher moments describe the "skew" of the stream, giving greater emphasis to the most-frequent elements. All of the frequency moments are defined as sums of powers of m i , where m i is the number of times that item i appears in a stream, and the sum is taken over all possible values of i .
It's interesting that calculating F 1 is so easy. We can determine the exact length of a stream with a simple, deterministic algorithm that runs in constant space. Alon, Matias and Szegedy proved that no such algorithm exists for any other frequency moment. We can get an exact value of F 0 (the number of distinct elements) only by supplying much more memory. Even approximating F 0 in constant space is harder: It requires a nondeterministic algorithm, one that makes essential use of randomness. The same is true of F 2 . For the higher moments, F 6 and above, there are no constant-space methods at all.
All this mathematics and algorithmic engineering seems like a lot of work just for following the exploits of a famous "pop tart." But I like to think the effort might be justified. Years from now, someone will type "Britney Spears" into a search engine and will stumble upon this article listed among the results. Perhaps then a curious reader will be led into new lines of inquiry.
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