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

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

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

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.

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.

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.

Aqua System Aqua is an efficient decision support system

providing approximate answers to queries

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

Aqua System(Contd.)

Aqua System(Contd.)

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 ε in the estimation of ci is

ε = ( ci – ci' )/ci * 100

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

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.

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

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

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

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

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

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

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 0.86Experimental Parameters

Experiment(Contd.)

Study to identify a scheme that can provide consistently good performance

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-bys:Both 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

Performance of different sample sizes:

The errors in Congress drop as the sample space increases

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.

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.

Questions??