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Unformatted text preview: 1 More StreamMining Counting Distinct Elements Computing Moments Frequent Itemsets Elephants and Troops Exponentially Decaying Windows 2 Counting Distinct Elements Problem : a data stream consists of elements chosen from a set of size n . Maintain a count of the number of distinct elements seen so far. Obvious approach : maintain the set of elements seen. 3 Applications How many different words are found among the Web pages being crawled at a site? Unusually low or high numbers could indicate artificial pages (spam?). How many different Web pages does each customer request in a week? 4 Using Small Storage Real Problem : what if we do not have space to store the complete set? Estimate the count in an unbiased way. Accept that the count may be in error, but limit the probability that the error is large. 5 FlajoletMartin* Approach Pick a hash function h that maps each of the n elements to at least log 2 n bits. For each stream element a , let r ( a ) be the number of trailing 0s in h ( a ). Record R = the maximum r ( a ) seen. Estimate = 2 R . * Really based on a variant due to AMS (Alon, Matias, and Szegedy) 6 Why It Works The probability that a given h ( a ) ends in at least r 0s is 2r . If there are m different elements, the probability that R r is 1 (1  2r ) m . Prob. a given h(a) ends in fewer than r 0s. Prob. all h(a)s end in fewer than r 0s. 7 Why It Works (2) Since 2r is small, 1  (12r ) m 1  e m2 . If 2 r >> m , 1  (1  2r ) m 1  (1  m2r ) m /2 r 0. If 2 r << m , 1  (1  2r ) m 1  e m2 1. Thus, 2 R will almost always be around m .rr First 2 terms of the Taylor expansion of e 8 Why It Doesnt Work E(2 R ) is actually infinite. Probability halves when R > R +1, but value doubles. Workaround involves using many hash functions and getting many samples. How are samples combined?...
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This document was uploaded on 03/04/2012.
 Fall '09

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