CONGRESSIONAL SAMPLES FOR APPROXIMATE ANSWERING OF

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CONGRESSIONAL SAMPLES FOR APPROXIMATE
ANSWERING OF GROUP-BY QUERIES Swarup Acharya
Phillip Gibbons Viswanath Poosala (Information
Sciences Research Center, Bell Labs, New
Jersey) Divya Rao
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Outline

• Introduction
• Background
• Aqua System
• Problem Formulation
• Solutions
• Query Rewriting strategies
• Experiment
• Conclusion

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Introduction

• Group-by queries- most important class of
  queries in decision support systems.
• Congressional Samples- A hybrid union of uniform
  and biased samples
• Seek to propose techniques for obtaining fast,
  highly-accurate answers for Group-by queries

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Background

• Uniform random sampling is not effective for
  group-by queries.
• Ex A group by query on the US Census database to
  determine the per-capita income of every state.
• Huge discrepancies in the sizes of different
  groups like California is 70 times more populated
  than Wyoming.
• This leads to poor accuracy of answers of those
  groups which have fewer number of tuples than the
  larger ones as accuracy is highly dependent on
  the number of sample tuples that belong to that
  group.

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Background

• Uniform Random Sampling are more appropriate only
  when the utility of data to the user mirrors the
  data distribution
• Multi-table query When different data have equal
  representation but their utility to the user is
  skewed
• Ex Data warehouses where the usefulness of data
  degrades with time
• This means the approximate sample has to collect
  more samples from the recent data which cannot be
  achieved through uniform random sample over the
  entire warehouse.

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Biased Sampling

• Use precomputed samples to address the problem of
  unbiased query
• Advantages of using precomputed biased samples
• Queries can be answered without accessing the
  original data at query run time
• Storing queries in disk blocks avoids the
  overhead of random scanning
• Disadvantage Biased samples must commit to the
  sample before seeing the query. Hence not
  suitable for user controlled progressive
  refinement.

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Aqua System

• Aqua is an efficient decision support system
  providing approximate answers to queries

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Aqua System(Contd.)

• Aqua is a Middleware tool that can sit atop any
  DBMS managing a data warehouse
• Aqua maintains statistical summaries of data in
  Synopses and uses them to answer queries
• The aqua system provides probabilistic error/
  confidence bounds on the answer

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Aqua System(Contd.)
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Aqua System(Contd.)
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Problem Formulation

• Main aim is to provide accurate answers to
  group-by queries in an approx. query answering
  system
• If ci and ci' be the exact and apprx. aggregate
  values in the group gi. Then error is the
  percentage relative error e in the estimation of
  ci is
• e ( ci ci' )/ci 100

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Solutions

• Theorem Divide the sample space X equally among
  the groups and take uniform random sample within
  each group.
• Map this theorem to various classes of group-by
  queries with arbitrary mixes of groupings.
• Ex US Congress

HOUSE
SENATE
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House

• The House has representatives from each state
  proportional to the state's population
• Applying theorem T to the House we have,
• For the aggregate operation, the quality of
  approx. answers increases with the query
  selectivity
• Answers to the queries with the same aggregate
  and equal selectivities will typically have
  similar quality guarantees.

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Senate

• Senate has equal number of representatives from
  each state
• Applying the theorem to the Senate we have,
• Each group in the sample will have atleast as
  many sample points as any other group in the
  entire sample

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Problems with House and Senate

• Using Samples from House would result in very few
  sample points for smaller groups
• Senate allocates fewer tuples to the larger
  groups compared to the House.
• Hence we have another technique called the
  Congress-collect both the House and the Senate
  samples

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Basic Congress

• Apply the theorem to the aggregate queries
  containing group-by queries on a set of
  attributes and queries with no group-bys at all.
• Collect both the House and Senate samples
• Reduce this factor by 2

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Congress

• For the sample space X, the final sample size
  allocated to each group is given by,
• Where the expected sample space allocated to g is

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Query rewriting

• Scaling up the aggregate expressions
• Deriving error bounds on the estimate
• Generating unbiased answers using tuples in the
  biased sample
• Scale factor is the inverse of sampling rate

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Rewriting Strategies

• The key step in scaling is to efficiently
  associate each tuple with its corresponding scale
  factor
• a) Store the scale factor with each tuple
• i) Integrated Rewriting
• ii) Nested-integrated Rewriting
• b) Use a separate table to store the scale factor
• iii) Normalized Rewriting
• iv) Key-normalized Rewriting

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Experiments

• Experimental Testbed Aqua system with Oracle v7
  as the back-end DBMS

Parameter Range of values Default value
Table size(T) 100k-6M tuples 1M
Sample Percentage(SP) 1-75 7
Num.groups 10-200k 1000
Group-size skew(z) 0-1.5 0.86
Experimental Parameters
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Experiment(Contd.)

• Study to identify a scheme that can provide
  consistently good performance

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Performance of various allocation strategies

• Performance of Different query sets
• Queries with no group-bys House performs well
  Congress technique performs consistently the best
  for queries of all types
• Queries with three group-bys Senate has low
  errors
• Queries with two group-bysBoth senate and House
  perform poorly in this case
• Congress performs close to best for queries of
  all types. Other techniques perform well only in
  a limited part of the spectrum

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• Performance of different sample sizes
• The errors in Congress drop as the sample space
  increases

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• Performance of group count
• Integrated and Nested-integrated perform better
  than Normalized and Key-normalized due to the
  absence of a join operation
• Nested-integrated performs better than Integrated
  due to significantly fewer multiplications.

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Conclusions

• Demonstrated that uniform samples are not enough
  to accurately answer all group-by queries
• Proposed new techniques based on biased sampling
• Congressional sampling concept was introduced and
  the sampling strategies were validated
  experimentally to produce accurate estimates to
  group-by queries and in their execution
  efficiency
• All the techniques have been incorporated into
  the Aqua System.

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Questions??