1. Ph.D. SeminarUniversity of Alberta1 Approximation Algorithms for Frequency Related Query Processing on Streaming Data Presented by Fan Deng Supervisor: Dr. Davood Rafiei May 24, 2007

2. Ph.D. SeminarUniversity of Alberta2 A sequence of data records Examples –Document/URL streams from a Web crawler –IP packet streams –Web advertisement click streams –Sensor reading streams –... Data stream

3. Ph.D. SeminarUniversity of Alberta3 One pass processing –Online stream (one scan required) –Massive offline stream (one scan preferred) Challenges –Huge data volume –Fast processing requirement –Relatively small fast storage space Processing in one pass

4. Ph.D. SeminarUniversity of Alberta4 Approximation algorithms Exact query answers –can be slow to obtain –may need large storage space –sometimes are not necessary Approximate query answers –can take much less time –may need less space –with acceptable errors

5. Ph.D. SeminarUniversity of Alberta5 Frequency related queries Frequency –# of occurrences Continuous membership query Point query Similarity self-join size estimation

6. Ph.D. SeminarUniversity of Alberta6 Outline Introduction Continuous membership query –Motivating application –Problem statement –Existing solutions and our solution –Theoretical and experimental results Point query Similarity self-join size estimation Conclusions and future work

7. Ph.D. SeminarUniversity of Alberta7 A Motivating Application Duplicate URL detection in Web crawling Search engines [Broder et al. WWW03] –Fetch web pages continuously –Extract URLs within each downloaded page –Check each URL (duplicate detection) If never seen before Then fetch it Else skip it

8. Ph.D. SeminarUniversity of Alberta8 A Motivating Application (cont.) Problems –Huge number of distinct URLs –Memory is usually not large enough –Disks are slow Errors are usually acceptable –A false positive (false alarms) A distinct URL is wrongly reported as a duplicate Consequence: this URL will not be crawled –A false negative (misses) A duplicate URL is wrongly reported as distinct Consequence: this URL will be crawled redundantly or searched on disks

9. Ph.D. SeminarUniversity of Alberta9 Problem statement A sequence of elements with order Storage space M –Not large enough to store all distinct elements Continuous membership query Appeared before? Yes or No …d g a f b e a d c b a Our goal –Minimize the # of errors –Fast M

10. Ph.D. SeminarUniversity of Alberta10 An existing solution (caching) Store as many distinct elements as possible in a buffer Duplicate detection process –Upon element arrival, search the buffer –if found then report “duplicate” else “distinct” Update the buffer using some replacement policies –LRU, FIFO, Random, …

11. Ph.D. SeminarUniversity of Alberta11 Another solution (Bloom filters) A bitmap, originally all “0” Duplicate detection process –Hash each incoming element into some bits –If any bit is “0” then report “distinct” else “duplicate” Update process - sets corresponding bits to “1” x h1(x) h2(x) 1 2 3 4 5 6 a 1 2 b 1 3 c 2 4 a 1 2 000011 000111 001111 001111

12. Ph.D. SeminarUniversity of Alberta12 Another solution (Bloom filters, cont.) False positives (false alarms) Bloom Filters will be “full” - All distinct URLs will be reported as duplicates, and thus skipped! 111111

13. Ph.D. SeminarUniversity of Alberta13 Our solution (Stable Bloom Filters) Kick “elements” out of the Bloom filters Change bits to “cells” (“cellmap”) 110101 031203

14. Ph.D. SeminarUniversity of Alberta14 Stable Bloom Filters (SBF, cont.) A “cellmap”, originally all “0” Duplicate detection –Hash each element into some cells, check those cells –If any cell is “0”, report “distinct” else “duplicate” Kick “elements” –Randomly choose some cells and deduct them by 1 Update the “cellmap” –Set cells into a predefined value, Max > 0 –Use the same hash functions as in the detection stage

15. Ph.D. SeminarUniversity of Alberta15 SBF theoretical results SBF will be stable –The expected # of “0”s will become a constant after a number of updates –Converge at an exponential rate –Monotonic False positive rates become constant An upper bound of false positive rates –(a function of 4 parameters: SBF size, # of hash functions, max cell values, and kick-out rates) Setting the optimal parameters (partially empirical)

16. Ph.D. SeminarUniversity of Alberta16 SBF experimental results Experimental comparison between SBF, and Caching/Buffering method (LRU) –URL fingerprint data set, originally obtained from Internet Archive (~ 700M URLs) To fairly compare, we introduce FPBuffering –Let Caching generate some false positives FPBuffering –If an element is not found in the buffer, report “duplicate” with certain probabilities

17. Ph.D. SeminarUniversity of Alberta17 SBF experimental results (cont.) SBF generates 3-13% less false negatives than FPBuffering, while having exactly the same # of false positives (<10%)

18. Ph.D. SeminarUniversity of Alberta18 SBF experimental results (cont.)

19. Ph.D. SeminarUniversity of Alberta19 SBF experimental results (cont.) MIN, [Broder et al. WWW03], theoretically optimal –assumes “the entire sequence of requests is known in advance” –beats LRU caching by <5% in most cases More false positives allowed, SBF gains more

20. Ph.D. SeminarUniversity of Alberta20 Outline Introduction Continuous membership query Point query –Motivating application –Problem statement –Existing solutions and our solution –Theoretical and experimental results Similarity self-join size estimation Conclusions and future work

21. Ph.D. SeminarUniversity of Alberta21 Motivating application Internet traffic monitoring –Query the # of IP packets sent by a particular IP address in the past one hour Phone call record analysis –Query the # of calls to a given phone # yesterday

22. Ph.D. SeminarUniversity of Alberta22 Problem statement Point query –Summarize a stream of elements –Estimate the frequency of a given element Goal: minimize the space cost and answer the query fast

23. Ph.D. SeminarUniversity of Alberta23 Existing solutions Fast-AGMS sketch [AMS97, Charikar et al. 2002] Count-min sketch (counting Bloom filters) –e.g. an element is hashed to 4 counters –Take the min counter value as the estimate 121113 110214 132111 001152

24. Ph.D. SeminarUniversity of Alberta24 Our solution Count-median-mean (CMM) –Count-min based –Take the value of the counter the element is hashed to –Deduct the median/mean value of all other counters –Remainder from deducting the mean is an unbiased estimate (in the case of deducting mean) –Basic idea: all counters are expected to have the same value Example: –counter value = 3 –mean value of all other counters = 2 (median = 2, more robust) –remainder = 1, so frequency estimate = 3-2 = 1 122413

25. Ph.D. SeminarUniversity of Alberta25 Theoretical results Unbiased estimate (deduct mean) Estimate variance is the same as that of Fast- AGMS (in the case deducting mean) For less skewed data set – the estimation accuracies of CMM and Fast- AGMS are exactly the same

26. Ph.D. SeminarUniversity of Alberta26 Experimental results and analysis For skewed data sets – Accuracy (given the same space): CMM-median = Fast-AGMS > CMM-mean Time cost analysis –CMM-mean = Fast-AGMS < CMM-median –but the difference is small Advantage of CMM –More flexible (with estimate upper bound) –More powerful (Count-min can be more accurate for the very skewed data set)

27. Ph.D. SeminarUniversity of Alberta27 Outline Introduction Continuous membership query Point query Similarity self-join size estimation –Motivating application –Problem statement –Existing solutions and our solution –Theoretical and experimental results Conclusions and future work

28. Ph.D. SeminarUniversity of Alberta28 Motivating application Near-duplicate document detection for search engines [Broder 99, Henzinger 06] –Very slow (30M pages, 10 days in 1997; 2006?) –Good to predict the time –How? Estimate the number of similar pairs Data cleaning in general (similarity self-join) –To find a better query plan (query optimization) –Estimates of similarity self-join size is needed

29. Ph.D. SeminarUniversity of Alberta29 Problem statement Similarity self-join size –Given a set of records with d attributes, estimate the # of record pairs that at least s-similar An s-similar pair –A pair of records with s attributes in common –E.g. & are 3-similar

30. Ph.D. SeminarUniversity of Alberta30 Existing solutions A straightforward solution –Compare each record with all other records –Count the number of pairs at least s-similar –Time cost O(n 2 ) for n records Random sampling –Take a sample of size m uniformly at random –Count the number of pairs at least s-similar –Scale it by a factor of c = n(n-1)/m(m-1)

31. Ph.D. SeminarUniversity of Alberta31 Our solution Offline SimParCount (Step 1- data processing) –Linearly scan all records once –For each record for each k=s…d Randomly pick k different attribute values, and concatenate them into one k-super-value Repeat this process l_k times –Look at all k-super-values as a stream –Store the (d-s+1) super-value streams on disks

32. Ph.D. SeminarUniversity of Alberta32 Our solution (cont.) Offline SimParCount (Step 2 - Result generating) –Obtain the self-join size of those 1-dimensional super- value streams –Based on the d-s+1 self-join sizes, estimate the similarity self-join size Online SimParCount –Use small sketches to estimate stream self-join sizes rather than expensive external sorting

33. Ph.D. SeminarUniversity of Alberta33 Our solution (cont.) Key idea –Convert similarity self-join size estimation to stream self-join size estimation –A similar record pair will have certain chance to have a match in the super-value stream records --- 2-super-values --- …

34. Ph.D. SeminarUniversity of Alberta34 Theoretical results Unbiased estimate Standard deviation bound of the estimate Time and space cost (For both offline and online SimParCount)

35. Ph.D. SeminarUniversity of Alberta35 Experimental results Online SimParCount v.s. Random sampling –Given the same amount of space –Error = (estimate – trueValue) / trueValue –Dataset: DBLP paper titles Each converted into a record with 6 attributes Using min-wise independent hashing

36. Ph.D. SeminarUniversity of Alberta36 Similarity self-join size estimation – Experimental results (cont.)

37. Ph.D. SeminarUniversity of Alberta37 Conclusions and future work Streaming algorithms –found real applications (important) –can lead to theoretical results (fun) –More work to be done Current direction: multi-dimensional streaming algorithms E.g Estimating the # of outliers in one pass

38. Ph.D. SeminarUniversity of Alberta38 Questions/Comments? Thanks!