1. Hashed Samples Selectivity Estimators for Set Similarity Selection Queries

2. Set Similarity: An Application Find similar strings Decompose strings into 3-grams. Represent strings as sets of 3-grams. Compare strings by comparing their respective 3- gram sets. “Nick Koudas”: { ‘Nic’, ‘ick’, …, ‘das’ } “Nick Arkoudas”: {‘Nic’, …, ‘das’ } We can use TF/IDF similarity (or other metrics) to evaluate set similarity.

3. Indexes For Set similarity Evaluation Current approaches use inverted lists Compute IDF of set elements (e.g., 3-grams). Create one inverted list per set element consisting of one entry per database set containing the respective elements (e.g., one entry per string containing the respective 3- gram). Use various algorithms and sorting/compression schemes for fast merging of inverted lists.

4. Motivation Set similarity queries are very important String matching. Data cleaning. Set-valued attributes in ORDBMS. A variety of set similarity operators have been proposed (for join, selection queries) Selectivity estimation is important for query optimization

5. The Problem Let I be a predefined set similarity measure. Let D a collection of sets. Given query set q and threshold τ, a set similarity selection query returns the answer set A: { s  D, s.t. I(q, s) > τ }. A set similarity selectivity estimation query estimates the size of A.

6. Naïve Solutions

7. Random Sampling Maintain a sample S, of sets s  D. Size of answer: |A| = |A s ||D| / |S|, where |A s | = {s  S: I(q, s) > τ}. Drawbacks: Query independent. Large variance. Needs to store complete sets in the sample. Cannot handle updates.

8. One Sample Per List Use the existing inverted index Compute one sample per inverted list. Compute independent estimates per list corresponding to the query set elements only (query specific) Report median, max, average… Drawbacks Ignores correlations between lists. Needs to store complete sets in inverted lists. Will not be better than simple random sampling. Cannot handle updates.

9. Sample Union Compute the sample union of the samples corresponding to the query set elements. Drawbacks Results in a biased sample. –There are duplicate elements in those lists. –Even if we eliminate duplicates we still need to compute the distinct set size of the sample union (for scaling up). –This is more expensive than answering the set similarity query exactly to begin with. Needs to store complete sets in inverted lists. Cannot handle updates.

10. Dynamically Computed Samples Given the query Use reservoir sampling to compute a sample union from the inverted lists on the fly. Drawbacks: Produces a biased sample –Skips part of the input. –Duplicates. Need to store complete sets in the inverted lists.

11. Hashed Samples

12. Hashed Sample An a priori computed sample that Builds uniform samples from arbitrary combinations of inverted lists. Does not need to store complete sets in the sample (only set ids). Leverages partial weight information contained in the lists. Eliminates the need to store distinct value estimation synopses. Provides unbiased estimates. Handles updates gracefully.

13. Construction We cannot draw independent samples per list Draw samples deterministically –In order to leverage partial weights contained in lists for computing I(q, s) efficiently. –Guarantee that if a set id is sampled from one list, it will be always sampled in all other lists. Guarantee that union of list samples is a uniform random sample. We impose a random permutation on the domain of set ids –Use hashing and sample a consistent subset.

14. Construction 2 Randomly choose a hash function h from a family of universal hash functions. Assume that h hashes in [1, 100] Values h(s 1 ), h(s 2 ), … appear as i.i.d. (empirically) Choose a value x and sample from every list sets s: h(s) < x.

15. Hashed Sample Properties We get an x% sample per list on average. We get an overall x% sample. The union of samples of any set of inverted lists is an x% sample of the respective lists. Let q = {q 1, q 2, …, q n }: |A| = |A s |  |q 1  …  q n | d / |q s1  …  q sn | d. Computing |A s | is simple Run any exact evaluation algorithm on the sampled lists! Performance improvement with respect to exact evaluation is directly proportional to the size of the sample. We still need the distinct number of set ids in q to scale up the results.

16. The K-Minimum Values Synopsis Estimating the distinct size of arbitrary list unions: The sampled lists themselves can be used as a KMV synopsis, by contsruction. The r-th smallest hash value h r of a set of elements gives an unbiased estimator of the distinct number of elements in the set: –|S| d  |P| (r – 1) / h r Given that sample lists contain all elements s.t. h(s)  x, we can deduce the rank of h m = x.

17. Experimental Evaluation

18. Setup IMDB, DBLP, YellowPages Decompose strings into 3-grams and build inverted index for TF/IDF similarity. Build list samples: 1%, 5%, 10%. Draw queries from the data: 100 queries per workload. Each set contains queries of preset selectivity. Evaluate estimation accuracy and runtime.

19. Storing Sets VS. Storing Set Ids

20. Reservoir Sampling: Accuracy

21. Reservoir Sampling: Cost

22. Hashed Sampling: Accuracy

23. Hashed Sampling: Cost

24. Hashed Sampling: Threshold

25. Hashed Sampling: Answer Size

26. Hashed Sampling: KMV Accuracy