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Approximation and Load Shedding for QoS in DSMS*

Approximation and Load Shedding for QoS in DSMS*. CS240B Notes By Carlo Zaniolo CSD--UCLA. ________________________________________ * Notes based on a VLDB’02 tutorial by Minos Garofalakis, Johannes Gehrke, and Rajeev Rastogi. Synopses and Approximation.

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Approximation and Load Shedding for QoS in DSMS*

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  1. Approximation and Load Sheddingfor QoS in DSMS* CS240B Notes By Carlo Zaniolo CSD--UCLA ________________________________________ * Notes based on a VLDB’02 tutorial by Minos Garofalakis, Johannes Gehrke, and Rajeev Rastogi

  2. Synopses and Approximation • Synopsis:bounded-memory history-approximation • Succinct summary of old stream tuples • Like indexes/materialized-views, but base data is unavailable • Examples • Sliding Windows • Samples • Histograms • Wavelet representation • Sketching techniques • Approximate Algorithms: e.g., median, quantiles,… • Fast and light Data Mining algorithms

  3. Overview of Stream Synopses • Windows: logical, physical (covered) • Samples: Answering queries using samples • Histograms: Equi-depth histograms, On-line quantile computation • Wavelets: Haar-wavelet histogram construction & maintenance

  4. Sampling: Basics • Idea: A small random sample S of the data often well-represents all the data • For a fast approx answer, apply “modified” query to S • Example:select agg from R where odd(R.e) (n=12) • If agg is avg, return average of odd elements in S • If agg is count, return average over all elements e in S of • 1 if e is odd • 0 if e is even Data stream: 9 3 5 2 7 1 6 5 8 4 9 1 Sample S: 9 5 1 8 answer: 5 answer: 12*3/4 =9 • Unbiased: For expressions involving count, sum, avg: the estimator • is unbiased, i.e., the expected value of the answer is the actual answer

  5. Probabilistic Guarantees • Example: Actual answer is within 5 ± 1 with prob  0.9 • Use Tail Inequalities to give probabilistic bounds on returned answer • Markov Inequality • Chebyshev’s Inequality • Hoeffding’s Inequality • Chernoff Bound

  6. Sampling—some background • Reservoir Sampling [Vit85]:Maintains a sample S having a pre-assigned size M on a stream of arbitrary size • Add each new element to S with probability M/n, where n is the current number of stream elements • If add an element, evict a random element from S • Instead of flipping a coin for each element, determine the number of elements to skip before the next to be added to S • Concise sampling [GM98]: Duplicates in sample S stored as <value, count> pairs (thus, potentially boosting actual sample size) • Counting Samples [GM98]: for answering hot list queries (k most frequent values) • Window Sampling [BDM02,BOZ08]. Maintains a sample S having a pre-assigned size M on a window on a stream—reservoir sampling with expiring tuples.

  7. Load Shedding Using Samples • Given a complex Query graph how to use/manage the sampling process [BDM04] • More about this later [LawZ02]

  8. Overview • Windows: logical, physical (covered) • Samples: Answering queries using samples • Histograms: Equi-depth histograms, On-line quantile computation • Wavelets: Haar-wavelet histogram construction & maintenance • Sketches

  9. Histograms • Histograms approximate the frequency distribution of element values in a stream • A histogram (typically) consists of • A partitioning of element domain values into buckets • A count per bucket B (of the number of elements in B) • Widely used in DBMS query optimization Many Types of Proposed: • Equi-Depth Histograms: select buckets such that counts per bucket are equal • V-Optimal Histograms: select buckets to minimize frequency variance within buckets • Wavelet-based Histograms

  10. Types of Histograms • Equi-Depth Histograms • Idea: Select buckets such that counts per bucket are equal • V-Optimal Histograms [IP95] [JKM98] • Idea: Select buckets to minimize frequency variance within buckets Count for bucket 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Domain values Count for bucket 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Domain values

  11. Equi-Depth Histogram Construction • For histogram with b buckets, compute elements with rank n/b, 2n/b, ..., (b-1)n/b • Example: (n=12, b=4) Data stream: 9 3 5 2 7 1 6 5 8 4 9 1 After sort: 1 1 2 3 4 5 5 6 7 8 9 9 rank = 9 (.75-quantile) rank = 3 (.25-quantile) rank = 6 (.5-quantile)

  12. Answering Queries Histograms [IP99] answer: 3.5 * • (Implicitly) map the histogram back to an approximate relation, & apply the query to the approximate relation • Example: select count(*) from R where 4 <= R.e <= 15 • For equi-depth histograms, maximum error: Count spread evenly among bucket values 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 4  R.e  15

  13. Approximate Algorithms • Quantiles Using Samples • Quantiles from Synopses • One pass algorithms for approximate samples … • Much work in this area … omitted

  14. Overview • Windows: logical, physical (covered) • Samples: Answering queries using samples • Histograms: Equi-depth histograms, On-line quantile computation • Wavelets: Haar-wavelet histogram construction & maintenance • Sketches

  15. One-Dimensional Haar Wavelets [1.5, 4] [0.5, 0] [2.75] [-1.25] Haar wavelet decomposition: [2.75, -1.25, 0.5, 0, 0, -1, -1, 0] • Wavelets: Mathematical tool for hierarchical decomposition of functions/signals • Haar wavelets: Simplest wavelet basis, easy to understand and implement • Recursive pairwise averaging and differencing at different resolutions Resolution Averages Detail Coefficients 3 [2, 2, 0, 2, 3, 5, 4, 4] ---- 2 [2, 1, 4, 4] [0, -1, -1, 0] 1 0

  16. Haar Wavelet Coefficients 2.75 - 2.75 + -1.25 -1.25 - 0 -1 0.5 -1 0 0 + - 0.5 + 0 - + - + 0 + - -1 + - -1 + 0 • Hierarchical decomposition structure (a.k.a. “error tree”) Coefficient “Supports” + + - + - + - + - + - + - + - 2 2 0 2 3 5 4 4 Original frequency distribution

  17. Compressed Wavelet Representations Key idea: Use a compact subset of Haar/linear wavelet coefficients for approximating frequency distribution Steps • Compute cumulative frequency distribution C • Compute linear wavelet transform of C • Greedy heuristic methods • Retain coefficients leading to large error reduction • Throw away coefficients that give small increase in error

  18. Overview • Windows: logical, physical (covered) • Samples: Answering queries using samples • Histograms: Equi-depth histograms, On-line quantile computation • Wavelets: Haar-wavelet histogram construction & maintenance • Sketches

  19. Sketches • Conventional data summaries fall short: • Quantiles and 1-d histograms: Cannot capture attribute correlations • Samples (e.g., using Reservoir Sampling) perform poorly for joins • Multi-d histograms/wavelets: Construction requires multiple passes over the data • Different approach: Randomized sketch synopses • Only logarithmic space • Probabilistic guarantees on the quality of the approximate answer • Can handle extreme cases.

  20. Overview • Windows: logical, physical (covered) • Samples: Answering queries using samples • Histograms: Equi-depth histograms, On-line quantile computation • Wavelets: Haar-wavelet histogram construction & maintenance • Sketches • QoS by load shedding.

  21. QoS and Load Schedding • When input stream rate exceeds system capacity a stream manager can shed load (tuples) • Load shedding affects queries and their answers: drop the tasks and the tuples that will cause least loss • Introducing load shedding in a data stream manager is a challenging problem • Random load shedding or semantic load shedding

  22. Load Shedding in Aurora • QoS for each application as a function relating output to its utility – Delay based, drop based, value based • Techniques for introducing load shedding operators in a plan such that QoS isdisrupted the least – Determining when, where and how much load to shed

  23. Load Shedding in STREAM • Formulate load shedding as an optimization problem for multiple sliding window aggregate queries – Minimize inaccuracy in answers subject to output rate matching or exceeding arrival rate • Consider placement of load shedding operators in query plan – Each operator sheds load uniformly with probability pi

  24. References [BDM02] B. Babcock, M. Datar, R. Motwani, ”Sampling from a moving window over streaming data”,Proceedingsof the thirteenth annual ACM-SIAM Symposium on Discrete Algorithms, p.633–634, 2002. [BOZ 08]Vladimir Braverman, Rafail Ostrovsky, Carlo Zaniolo Succinct Sampling on Streams, submitted for publication. [Vit85] J. S. Vitter. “Random Sampling with a Reservoir”. ACM TOMS, 1985. [GM98] P. B. Gibbons and Y. Matias. “New Sampling-Based Summary Statistics for Improving Approximate Query Answers”. ACM SIGMOD 1998. [BDM04] Brian Babcock, Mayur Datar, Rajeev Motwani: Load Shedding for Aggregation Queries over Data Streams. ICDE 2004: 350-361. [lawZ08] Yan-Nei Law and Carlo Zaniolo: Improving the Accuracy of Continuous Aggregates and Mining Queries on Data Streams under Load Shedding. International Journal of Business Intelligence and Data Mining, 2008.

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