Approximate Query Processing using Wavelets

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Approximate Query Processing using Wavelets

• Kaushik Chakrabarti(Univ Of Illinois)
• Minos Garofalakis(Bell Labs)
• Rajeev Rastogi(Bell Labs)
• Kyuseok Shim(KAIST and AITrc)
• Presented at 26th VLDB Conference, Cairo, Egypt
• Presented By
• Supriya Sudheendra

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Outline
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Introduction

• Approximate Query Processing is a viable solution
  for
• Huge amounts of data
• High query complexities
• Stringent response-time requirements
• Decision Support Systems
• Support business and organizational
  decision-making activities
• Helps decision makers compile useful information
  from raw data, solve problems and make decisions

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Introduction

• DSS users pose very complex queries to the DBMS
• Requires complex operations over GB or TBs of
  disk-resident data
• Very long time to execute and produce exact
  answers
• Number of scenarios where users prefer a fast,
  approximate answers

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Prior Work

• Previous Approximate query processing techniques
• Focused on specific forms of aggregate queries
• Data reduction mechanism how to obtain the
  synopses of data
• Sampling-based Techniques
• A join-operator on 2 uniform random samples
  results in a non-uniform sample having very few
  tuples
• For non-aggregate queries, it produces a small
  subset of the exact answer which might be empty
  when joins are involved.

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Prior Work

• Histogram Based Techniques
• Problematic for high-dimensional data
• Storage overhead
• High construction cost
• Wavelet Based Techniques
• Mathematical tool for hierarchical decomposition
  of functions
• Apply wavelet decomposition to input data
  collection gt data synopsis
• Avoids high construction costs and storage
  overhead

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Contribution of the Paper

• Viability and effectiveness of wavelets as a
  generic tool for high-dimensional DSS
• New, I/O-efficient wavelet decomposition
  algorithm for relational tables
• Novel Query processing algebra for
  Wavelet-Co-Efficient Data Synopses
• Extensive Experiments

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Background

• Mathematical tool to hierarchically decompose
  functions
• Coarse overall approximation together with detail
  coefficients that influence function at various
  scales
• Haar wavelets are conceptually simple, fast to
  compute
• Variety of applications like image editing and
  querying

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One-Dimensional Haar Wavelets

• How to compute, given a data array
• Average the values together pairwise to get a
  lower-resolution representation of data
• Detailed coefficients-gt differences of the
  averaged value from the computed pairwise average
• Reconstruction of the data array possible
• Why Detail Coefficients

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One-dimensional Haar Wavelets

• Wavelet Transform Overall average followed by
  detail coefficients in increasing order of
  resolution. Each entry-gtwavelet coefficient
• WA 4, -2, 0, -1
• For vectors containing similar values,
• most detail coefficients have small values that
  can be eliminated
• Introduces only small errors

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One-dimensional Haar Wavelets

• Overall average more important than any detail
  coefficient
• To normalize the final entries of WA, each
  wavelet coefficient is divided by ?2l
• l level of resolution
• WA 4, -2, 0, -1/?2

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Multi-dimensional Haar Wavelets

• Haar wavelets can be extended to
  multi-dimensional array
• Standard Decomposition
• Fix an ordering for the data dimensions(1,2,d)
• Apply complete 1-D wavelet transform for each 1-d
  row of array cells along dimension k

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• Nonstandard Decomposition
• Alternates between dimensions during successive
  steps of pairwise averaging and differencing for
  each 1-D row of array cells along dimension k
• Repeated recursively on quadrant containing all
  averages across all dimensions

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Non-standard Decomposition

• Pairwise averaging and differencing for one
  positioning of 2x2 box with root 2i1, 2i2
• Distribution of the results in the wavelet
  transform array
• Process is recursed on lower-left quadrant of WA

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Example Decomposition of a 4 X 4 Array
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Multi-dimensional Haar coefficients Semantics
and Representation

• D-dimensional Haar basis function corresponding
  to Wavelet w is defined by
• D-dimensional rectangular support region
• Quadrant sign information

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Support Regions for 16 Nonstandard 2-D Haar Basis
Function

• Blank areas regions of A whose reconstruction
  is independent of the coefficient
• WA0,0 overall average
• WA3,3 contributes only to upper right
  quadrant

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Haar CoEfficients Semantics and Representation

• W ltR, S, vgt
• W.R d-dimensional support hyper-rectangle of W
  encloses all cells in A to which W contributes
• Hyper-rectangle represented by low and high
  boundaries across each dimension j, 1lt j ltd
• W.R.boundaryj.lo and W.R.boundaryj.hi
• W contributes to each data cell Ai1,id where
• W.R.boundaryj.lo lt ij lt W.R.boundaryj.hi
  for all j

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• W.S sign information for all d-dimensional
  quadrants of W.R
• Denoted by W.S.signj.lo and W.S.signj.hi
  corresponding to lower and upper half of W.Rs
  extent along j
• Computed as the product of d sign-vector entries
  that map to that quadrant
• W.v scalar magnitude of W
• Quantity that W contributes to all data array
  cells enclosed in W.R

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Building Wavelet Coefficient Synopses

• Relation R with d attributes X1, X2, Xd
• Can represent R as a d-dimensional array AR
• Jth dimension is indexed by the values of
  attribute Xj
• Cells contain the count of tuples in R having the
  corresponding combination of attribute values
• AR joint frequency distribution of all
  attributes of R

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• Chunk-based organization of relational tables
• Joint frequency array AR split into
  d-dimensional chunks
• Tuples of R of same chunk are stored contiguously
  on disk
• If R is not chunked, one extra pre-processing
  step to reorganize R on disk

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ComputeWavelet Algorithm

• When a chunk is loaded for the first time,
  ComputeWavelet can perform entire computation for
  decomposing
• Pairwise averaging and differencing is performed
  as soon as 2d averages are accumulated
• Memory efficient- no more than one active
  sub-array at a time for each level of resolution

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Processing Relational Queries in Wavelet
Coefficient Domain
Wavelet-Coefficient Synopses WT1, WT2,WTk
Wavelet-Coefficient Synopses WT1, WT2,WTk
Op(WT1,.WTk)
Render(WT1WTk)
RS of Wavelet Coefficients WS
Approximate Relations T1, T2,.Tk
Op(T1, T2. Tk)
Render(WS)
Approx. Result Relation S
Approx. Result Relation S
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Selection Operator

• Our selection operator has the general form
  selectpred(WT ), where pred represents a generic
  conjunctive predicate on a subset of the d
  attributes in T that is,
  
• pred (li1 Xi1 hi1 ) ? . . . ? (lik Xik
  hik ), where lij and hij denote the low and
  high boundaries of the selected range along each
  selection dimension Dij , j 1, 2, , k, k
  d.

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Selection - Relational Domain
Relation
Joint Data Distribution Array
3
3
2
1
Dim. D1
2
3
1
7
6
3
4
8
6
Dim. D2
Query Range

• In relational domain, interested in only those
  cells inside query range
• In wavelet domain, interested in only the
  coefficients that contribute to those cells

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Projection Operator
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Projection- Wavelet Domain
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Join Operator
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Join Operator- Wavelet Domain
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Experimental Study

• Improved answer quality
• Low synopsis construction costs
• Fast query execution

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Query Execution Times
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SELECT-JOIN-SUM
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SELECT Query errors on real-life data
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Conclusion

• Multidimensional wavelets as an effective tool
  for general purpose approximate query processing
  in modern, high dimensional applications
• The query processing algorithms operate directly
  on the wavelet-coefficient synopses of relational
  data, thus allowing for very fast processing of
  arbitrarily complex queries entirely in the
  wavelet-coefficient domain
• Extensive experimental study with synthetic as
  well as real-life data sets that verifies the
  effectiveness of the wavelet-based approach
  compared to both sampling and histograms

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Thank you?