Hashed Samples

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Hashed Samples

• Selectivity Estimators for
• Set Similarity Selection Queries

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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.

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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.

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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

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The Problem

• Let I be a predefined set similarity measure.
• Let D a collection of sets.
• Given query set q and threshold t, a set
  similarity selection query returns the answer set
  A s ? D, s.t. I(q, s) gt t .
• A set similarity selectivity estimation query
  estimates the size of A.

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Naïve Solutions
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Random Sampling

• Maintain a sample S, of sets s ? D.
• Size of answer
• A As??D / S,
• where As s ? S I(q, s) gt t.
• Drawbacks
• Query independent.
• Large variance.
• Needs to store complete sets in the sample.
• Cannot handle updates.

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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.

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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.

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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.

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Hashed Samples
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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.

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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.

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Construction 2

• Randomly choose a hash function h from a family
  of universal hash functions.
• Assume that h hashes in 1, 100
• Values h(s1), h(s2), appear as i.i.d.
  (empirically)
• Choose a value x and sample from every list sets
  s h(s) lt x.

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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 q1, q2, , qn
• A As ? q1 ? ? qnd / qs1 ?? ? qsnd.
• Computing As 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.

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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 hr of a set of
  elements gives an unbiased estimator of the
  distinct number of elements in the set
• Sd ?? P (r 1) / hr
• Given that sample lists contain all elements s.t.
  h(s) ?? x, we can deduce the rank of hm x.

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Experimental Evaluation
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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.

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Storing Sets VS. Storing Set Ids
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Reservoir Sampling Accuracy
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Reservoir Sampling Cost
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Hashed Sampling Accuracy
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Hashed Sampling Cost
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Hashed Sampling Threshold
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Hashed Sampling Answer Size
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Hashed Sampling KMV Accuracy
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