Aggregation computation over distributed data streams
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This ppt is my another learning report addressed in the lab which was composed by myself. I hope it is of help and use to you friends.
![Flajolet-Martin Sketch(Cont.) Preliminary what do we need? the Multi-set M, containing all items/records, and |M| = n the upper bound on the number of distinct items/records U, which is more than n the bitmap B, consisting of L elements, and 2 L = U the hash function h(x: item/record), transforming each items into a binary string distributed uniformly over the range of [1…2 L ], just like b 1 b 2 …b L , in which b 1 is the lowest digit, and b L is the highest the p(x), attaining the left most position of ‘1’ 12/15/11 Middleware, CCNT, ZJU counting not computing 1 1 … 0 B 0 PPT VS Whiteboard ? x record h(x) 1 L](https://image.slidesharecdn.com/aggregationcomputationoverdistributeddatastreams-111215074354-phpapp01/85/Aggregation-computation-over-distributed-data-streams-9-320.jpg)
![Flajolet-Martin Sketch(Cont.) The algorithm itself the core task: Remarking the position of which the leftmost ‘1’ of the hash value recorded by p(x) in bitmap B 12/15/11 Middleware, CCNT, ZJU for i:=1 to L do bitmap[i] :=0 for all x in M do begin index := p(hash(x)); if bitmap[index] = the bitmap[index] :=1; end Why?](https://image.slidesharecdn.com/aggregationcomputationoverdistributeddatastreams-111215074354-phpapp01/85/Aggregation-computation-over-distributed-data-streams-10-320.jpg)
![Flajolet-Martin Sketch(Cont.) The explanation The fact: bitmap[k] equals to 1 iff after execution a pattern of the form 0 k-1 1 has appeared amongst hashed values of records in M The probability: the occurrence probability of the pattern 0 k-1 1 is 1/2 k Occurrence times: so if |M| = n, then bitmap[1] is accessed approximately n/2 times, bitmap[2] approximately n/4 times Extension: bitmap[k] will almost certainly be zero if k >> log 2 (n) and one if k << log 2 (n) wit a fringe of 0 and 1 for k ≈ log 2 (n) Selection: the leftmost 0, the rightmost 1 or something else 12/15/11 Middleware, CCNT, ZJU U The most practical part is over, and the left is very complicated taken for proving and error analysis, namely all about mathematic for i:=1 to L do bitmap[i] :=0 for all x in M do begin index := p(hash(x)); if bitmap[index] = the bitmap[index] :=1; end](https://image.slidesharecdn.com/aggregationcomputationoverdistributeddatastreams-111215074354-phpapp01/85/Aggregation-computation-over-distributed-data-streams-11-320.jpg)
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Aggregation computation over distributed data streams
- 1. A ggregation C omputation O ver D istributed D ata S treams (partial content) Yueshen Xu Middleware, CCNT Zhejiang University Middleware, CCNT, ZJU 12/15/11
- 2. Paper reference What's Different: Distributed, Continuous Monitoring of Duplicate- Resilient Aggregates on Data Streams Published in ICDE, 2006 Cited by 61 times By Graham Cormode, S. Muthukrishnan etc. 12/15/11 Middleware, CCNT, ZJU I think it’s a good reading suitable for freshmen on distributed data streams Bell Lab Expert/27 Rutgers Expert/45 !
- 3. Background Distributed Data Streams Where and why? Large scale monitoring applications Many sensors distributed over a wide area 12/15/11 Middleware, CCNT, ZJU Just one example Distributed Streaming Model Query paradigm Centralized Decentralized VS
- 4. Constraints and Features Constraints Space Embedded equipments don’t have enough memory Processing power The same reason Communication capability Unreliable, spotty and sporadic 12/15/11 Middleware, CCNT, ZJU All resources are restricted Features Different from ad hoc queries in DBMS, but continuous What’s different?
- 5. Trouble Duplication Why? Wide scale monitoring invariably encounters the same events at different points 12/15/11 Middleware, CCNT, ZJU Instances The same flow will be observed in different routers The same individual will be observed by several mobile sensors Requirement Duplicate-resilient aggregate Two vital questions What is the amount of duplication in the network? What are the versions of classical aggregates in the presence of duplicates? root of all evil
- 6. Topic What kind of topics are researchers interested in ? Aggregation computation Routing algorithms … What is the aggregation? Summarization, namely a statistic variable describing the original data sets Examples min, max, quantile, heavy hitter distinct counts, average, sum … 12/15/11 Middleware, CCNT, ZJU Not strange contacting with data streams Why aggregation? transaction
- 7. Problems and Concerns Distinct count To obtain the number of distinct data (item, record, etc) in multi-sets, namely the cardinality Distinct sample Important, but I’m sorry that I haven’t finished this part 12/15/11 Middleware, CCNT, ZJU What does this paper concern about? Priority: correctness, communication cost Computational cost, space cost ! Features attached to those algorithms applied to distributed environments
- 8. Distinct Counting: Flajolet-Martin Sketch Flajolet-Martin Sketch P. Flajolet, G. Martin . Probabilistic Counting Algorithms for Data Base Applications . Journal of Computer and System Sciences, 1985(Cited by 628) Goal: To estimate the cardinalities of multi-sets of data using relative small space by one pass scan The sketch is a kind of data structure, which is the way to obtain the aggregation results. (skyline) I think this method can be regarded as the classical application of probability without complexity. 12/15/11 Middleware, CCNT, ZJU Give a question: How about you dealing with this problem? The computing paradigm of sketching Be appropriate for using in data streams inherently
- 9. Flajolet-Martin Sketch(Cont.) Preliminary what do we need? the Multi-set M, containing all items/records, and |M| = n the upper bound on the number of distinct items/records U, which is more than n the bitmap B, consisting of L elements, and 2 L = U the hash function h(x: item/record), transforming each items into a binary string distributed uniformly over the range of [1…2 L ], just like b 1 b 2 …b L , in which b 1 is the lowest digit, and b L is the highest the p(x), attaining the left most position of ‘1’ 12/15/11 Middleware, CCNT, ZJU counting not computing 1 1 … 0 B 0 PPT VS Whiteboard ? x record h(x) 1 L
- 10. Flajolet-Martin Sketch(Cont.) The algorithm itself the core task: Remarking the position of which the leftmost ‘1’ of the hash value recorded by p(x) in bitmap B 12/15/11 Middleware, CCNT, ZJU for i:=1 to L do bitmap[i] :=0 for all x in M do begin index := p(hash(x)); if bitmap[index] = the bitmap[index] :=1; end Why?
- 11. Flajolet-Martin Sketch(Cont.) The explanation The fact: bitmap[k] equals to 1 iff after execution a pattern of the form 0 k-1 1 has appeared amongst hashed values of records in M The probability: the occurrence probability of the pattern 0 k-1 1 is 1/2 k Occurrence times: so if |M| = n, then bitmap[1] is accessed approximately n/2 times, bitmap[2] approximately n/4 times Extension: bitmap[k] will almost certainly be zero if k >> log 2 (n) and one if k << log 2 (n) wit a fringe of 0 and 1 for k ≈ log 2 (n) Selection: the leftmost 0, the rightmost 1 or something else 12/15/11 Middleware, CCNT, ZJU U The most practical part is over, and the left is very complicated taken for proving and error analysis, namely all about mathematic for i:=1 to L do bitmap[i] :=0 for all x in M do begin index := p(hash(x)); if bitmap[index] = the bitmap[index] :=1; end
- 12. Flajolet-Martin Sketch(Cont.) Conclusion Analysis Bit-based, reducing the space complexity by constant level Space complexity O(log(n)) O(log(log(n))) Duplicate-insensitive duplicate-resilient and flexible Order-insensitive stable and robust Additivity The ability to merge two FM sketches together, and the merger is simply the bitwise-or of each pair of corresponding bitmaps Questions How to make the value of U? What’s the relationship of U and n? How to make the analysis to the error? … 12/15/11 Middleware, CCNT, ZJU nice qualities for distributed aggregation
- 13. Question What’s the relationship between sketch and skyline? Are they the same? Or… Does the aggregation computation belong to the research fields of data mining ? No, I think 12/15/11 Middleware, CCNT, ZJU
- 14. Q&A 12/15/11 Middleware, CCNT, ZJU