Kaushik Chakrabarti(Univ Of Illinois) Minos Garofalakis(Bell
Labs) Rajeev Rastogi(Bell Labs) Kyuseok Shim(KAIST and AITrc)
Presented at 26 th VLDB Conference, Cairo, Egypt Presented By
Supriya Sudheendra Slide 2 Outline Slide 3 Introduction o
Approximate Query Processing is a viable solution for: Huge amounts
of data High query complexities Stringent response-time
requirements o Decision Support Systems Support business and
organizational decision-making activities Helps decision makers
compile useful information from raw data, solve problems and make
decisions Slide 4 Introduction o 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 Slide 5 Prior Work o Previous Approximate query processing
techniques Focused on specific forms of aggregate queries Data
reduction mechanism how to obtain the synopses of data o
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. Slide 6 Prior
Work o Histogram Based Techniques Problematic for high-dimensional
data Storage overhead High construction cost o Wavelet Based
Techniques Mathematical tool for hierarchical decomposition of
functions Apply wavelet decomposition to input data collection >
data synopsis Avoids high construction costs and storage overhead
Slide 7 Contribution of the Paper o Viability and effectiveness of
wavelets as a generic tool for high-dimensional DSS o New,
I/O-efficient wavelet decomposition algorithm for relational tables
o Novel Query processing algebra for Wavelet-Co- Efficient Data
Synopses o Extensive Experiments Slide 8 Background o Mathematical
tool to hierarchically decompose functions o Coarse overall
approximation together with detail coefficients that influence
function at various scales o Haar wavelets are conceptually simple,
fast to compute o Variety of applications like image editing and
querying Slide 9 One-Dimensional Haar Wavelets o How to compute,
given a data array: Average the values together pairwise to get a
lower- resolution representation of data Detailed coefficients->
differences of the averages from the computed pairwise average
Reconstruction of the data array possible Why Detail Coefficients
Slide 10 One-dimensional Haar Wavelets o Wavelet Transform: Overall
average followed by detail coefficients in increasing order of
resolution. Each entry->wavelet coefficient o W A = [4, -2, 0,
-1] o For vectors containing similar values, most detail
coefficients have small values that can be eliminated Introduces
only small errors Slide 11 One-dimensional Haar Wavelets o Overall
average more important than any detail coefficient o To normalize
the final entries of W A, each wavelet coefficient is divided by 2
l l: level of resolution W A = [4, -2, 0, -1/ 2] Slide 12
Multi-dimensional Haar Wavelets o 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 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 Slide 13 Non-standard
Decomposition Pairwise averaging and differencing for one
positioning of 2x2 box with root [2i 1, 2i 2 ] Distribution of the
results in the wavelet transform array Process is recursed on
lower-left quadrant of W A Slide 14 Example Decomposition of a 4 X
4 Array Slide 15 Multi-dimensional Haar coefficients: Semantics and
Representation o D-dimensional Haar basis function corresponding to
w is defined by: D-dimensional rectangular support region Quadrant
sign information Slide 16 Support Regions for 16 Nonstandard 2-D
Haar Basis Function Blank areas regions of A whose reconstruction
is independent of the coefficient WA[0,0] overall average WA[3,3]
contributes only to upper right quadrant Slide 17 Haar
CoEfficients: Semantics and Representation o W = 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, 1
