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The Britney Spears Problem

Tracking who's hot and who's not presents an algorithmic challenge

Brian Hayes

Majority Rules

Stream algorithms that require more than a constant amount of storage space are seldom of practical use in large-scale applications. Unfortunately, for tasks such as counting distinct elements and finding most-frequent elements, there is really no hope of creating a constant-space algorithm that's guaranteed always to give the correct answer. But before we grow too despondent about this bleak outlook, I should point out that there are also a few pleasant surprises in the world of stream algorithms.

An%20algorithm%20that%20uses%20just%20two%20memory%20registersClick to Enlarge Image Although identifying the most frequent item in a stream is hard in general, there is an ingenious way of doing it in one special case—namely, when the most common item is so popular that it accounts for a majority of the stream entries (more than half the elements). The algorithm that accomplishes this task requires just two registers, and it runs in a constant amount of time per stream element. (Before reading on you might want to try constructing such an algorithm for yourself.)

The majority-finding algorithm uses one of its registers for temporary storage of a single item from the stream; this item is the current candidate for majority element. The second register is a counter initialized to 0. For each element of the stream, we ask the algorithm to perform the following routine. If the counter reads 0, install the current stream element as the new majority candidate (displacing any other element that might already be in the register). Then, if the current element matches the majority candidate, increment the counter; otherwise, decrement the counter. At this point in the cycle, if the part of the stream seen so far has a majority element, that element is in the candidate register, and the counter holds a value greater than 0. What if there is no majority ele­ment? Without making a second pass through the data—which isn't possible in a stream environment—the algorithm cannot always give an unambiguous answer in this circumstance. It merely promises to correctly identify the majority element if there is one.

The majority algorithm was invented in 1982 by Michael E. Fischer and Steven L. Salzberg of Yale University. The version I have described here comes from a 2002 article by Erik D. Demaine of MIT and Alejandro López-Ortiz and J. Ian Munro of the University of Waterloo in Canada. Demaine and his colleagues have extended the algorithm to cover a more-general problem: Given a stream of length n, identify a set of size m that includes all the elements occurring with a frequency greater than n /( m +1). (In the case of m =1, this reduces to the majority problem.) The extended algorithm requires m registers for the candidate elements as well as m counters. The basic scheme of operation is analogous to that of the majority algorithm. When a stream ele­ment matches one of the candidates, the corresponding counter is incremented; when there is no match to any candidate, all of the counters are decremented; if a counter is at 0, the associated candidate is replaced by a new element from the stream.

Again the results carry only a weak guarantee: If any elements of the stream exceed the threshold frequency, those elements will appear among the candidates, but not all the candidates are necessarily above the threshold. Even with this drawback, the algorithm performs impressive feats, such as scanning a stream of Web search queries for all terms that make up at least 1 percent of the traffic.

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