1. New Sampling-Based Summary Statistics for Improving Approximate Query Answers Yinghui Wang 02-07-2006

2. 2 Outline Introduction Concise samples Counting samples Application to hot list queries Conclusion Reference

3. 3 Introduction In large data recording and warehousing environments, it is often advantageous to provide fast, approximate answers to queries, whenever possible. Effectiveness of a synopsis is evaluated as a function of its footprint, i.e., the number of memory words to store the synopsis. Data Warehouse New Data Queries Response Figure 1:A traditional data warehouse Data Warehouse New Data Queries Response Figure 2: Data warehouse set-up for providing approximate query answers. Approx. Answer Engine

4. 4 Definition Concise samples a uniform random sample of the data set such that values appearing more than once in the sample are represented as a value and a count, ex:. Counting samples a variation on concise samples in which the counts are used to keep track of all occurrences of a value inserted into the relation since the value was selected for the sample. Hot list queries request an ordered set of pairs for the k most frequently occurring data values, for some k. ex: the top selling items in a database of sales transactions.

5. 5 Outline Introduction Concise samples Counting samples Application to hot list queries Conclusion Reference

6. 6 Concise samples Consider a relation R with n tuples and an attribute A. The goal is to obtain a uniform random sample of R.A, i.e., the values of A for a random subset of the tuples in R. Definition: Let S = {,…,, v j+1,..., v l } be a consice sample. Then sample-size(S) = l-j+∑ j i = 1 c i, and footprint(S) = l+j Lemma 1 For any footprint m ≥ 2, there exists data sets for which the sample-size of a consice sample is n/m times lager than its footprint, where n is the size of the data set.

7. 7 Concise samples – offline/static Offline/static computation Repeat m times: select a random tuple from the relation and extract its value for attribute A. Semi-sort the set of values, and replace every value occurring multiple times with a pair. Continue to sample until either adding the sample point would increase the concise sample footprint to m+1 or n samples have been taken. For each new value sampled, look-up to see if it is already in the concise sample and then either add a new singleton value, convert a singleton to a pair, or increment the count for a pair.

8. 8 Concise samples – online With concise samples, the sample-size depends on the data distribution to date, and any changes in the data distribution must be reflected in the sampling frequency. Maintenance algorithm: Let S be the current concise sample and consider a new tuple t. Set up an entry thresholdβ(initially 1) for new tuples to be selected for the sample. I. Add t.A to S with probability 1/ β. II. Do a look-up on t.A in S. a) if it is represented by a pair, its count is incremented. b) if t.A is a singleton in S, a pair is created, c) if it is not in S, a singleton is created. III. Increase footprint by 1 in cases b) and c) IV. Raise the threshold to some β’. Subject each sample point in S to this higher threshold. Subsequent inserts are selected for the sample with probability 1/ β’

9. 9 Concise samples (cont.) Theorem 2 Consider the family of exponential distributions: for I = 1,2,…,Pr(v=i) = α -i (α-1), for α>1. For any footprint m≥2, the expected sample-size of a concise sample with footprint m is at least α m/2 Theorem 3 For any data set, when using a concise sample S with sample-size m, the expected gain is E[m-number of distinct values in S] =

10. 10 Concise samples (cont.) Update time overheads The coin flips that must be performed to decide which inserts are added to the concise sample and to evict values from the concise sample when the threshold is raised The lookups into the current concise sample to see if a value is already present in the sample

11. 11 Concise Samples – experimental evaluation Figure 3: Comparing sample-sizes of concise and traditional samples as a function of skew, for varying footprints and D/m ratios. In (a) and (b), authors compare footprint 100 and footprint 1000, respectively, for the same data sets. In (c) and (d), authors compare D/m = 50 and D/m = 5, respectively, for the same footprint 1000. D: potential number of distinct values m: footprint size

12. 12 Outline Introduction Concise samples Counting samples Application to hot list queries Conclusion Reference

13. 13 Counting samples Counting samples – a variation on concise samples in which the counts are used to keep track of all occurrences of a value inserted into the relation since the value was selected for the sample. Definition: A counting sample for R.A with thresholdβis any subset of R.A obtained as follows: 1. For each value v occurring c times in R, we flip a coin with probability 1/βof heads until the first heads, up to at most c coin tosses in all; if the ith coin toss is heads, then v occurs c-i+1 times in the subset, else v is not in the subset. 2. Each value v occurring c>1 times in the subset is represented as a pair, and each value v occurring exactly once is represented as a singleton v.

14. 14 Counting samples (cont.) An algorithm for incremental maintenance is introduced. Theorem 4 Let R be an arbitrary relation, and let β be the currentthreshold for a counting sample S. (i) Any value v that occurs at least βtimes in R is expected to be in S. (ii) Any value v that occurs f v times in R will be in S with probability 1-(1-1/β) fv. (iii) For all α>1, if f v ≥ αβ, then with probability ≥ 1 - e -α, the value will be in S and its count will be at least fv - αβ

15. 15 Outline Introduction Concise samples Counting samples Application to hot list queries Conclusion Reference

16. 16 Application to hot list queries Hot list queries request an ordered set of pairs for the k most frequently occurring data values, for some k. Algorithms Using traditional samples Using concise samples Using counting samples Using histogram on disk – maintains a full histogram on disk, i.e., pairs for all distinct values in R, with a copy of the top m/2 pairs stored as a synopsis within the approximate answer engine. -- is considered only as a baseline for accuracy comparisons

17. 17 Application to hot list queries (cont.) x-axis: rank of a value y-axis: count for the values

18. 18 Application to hot list queries (cont.)

19. 19 Application to hot list queries (cont.)

20. 20 Application to hot list queries – overheads

21. 21 Conclusion Using concise samples may offer the best choice when considering both accuracy and overheads. In this paper, a batch-like processing of data warehouse inserts, in which inserts and queries do not intermix, is assumed. To address the more general case, issues of concurrency bottlenecks need to be addressed. Future work is to explore the effectiveness of using concise samples and counting samples for other concrete approximate answer scenarios.

22. 22 Reference P. B. Gibbons and Y. Matias. New Sampling-Based Summary Statistics for Improving Approximate Query Answers. ACM SIGMOD 1998.