Wavelet Synopses with Predefined Error Bounds: Windfalls of Duality Panagiotis Karras DB seminar, 23 March, 2006

Algorithms for Maximum-Error Wavelet Synopses Restricted Space-Bounded Direct GK04,05 Guha05 Unrestricted GH05,06 Error-Bounded Muthukrishnan05 Indirect ?

Compact Data Synopses useful in: Approximate Query Processing (exact answers not always required) Learning, Classification, Event Detection Data Mining, Selectivity Estimation Situations where massive data arrives in a stream

Haar Wavelets 18 :Wavelet transform: orthogonal transform for the hierarchical representation of functions and signals :Haar wavelets: simplest wavelet system, easy to understand and implement Haar tree: structure for the visualization of decomposition and value reconstructions Synopsis: Wavelet representation with few non-zero terms.

Maximum-Error Metrics Maximum-Error Metrics Error Metrics providing tight error guarantees for all reconstructed values: –Maximum Absolute Error –Maximum Relative Error with Sanity Bound (to avoid domination by small data values) Aim at minimization of these metrics

Restricted Synopses Compute Haar wavelet decomposition of D Preserve best coefficient subset that satisfies bound Space-Bounded ProblemSpace-Bounded Problem: [GK04,05,Guha05] Bound B on number of non-zero coefficients Error-Bounded ProblemError-Bounded Problem : [Muthukrishnan05] Bound ε on maximum error Faster Indirect solution to Space-Bounded Problem

How does it work? Space-Bounded Problem GK04,05: Global Tabulation Guha05: Local Tabulation - Tabulate four one-dimensional arrays: - Extract from these four, delete them - At most arrays concurrently stored - Derive solution at the top, solve the problem again below time, space iLiL iRiR S = subset of selected ancestors root i + -

How does it work? Error-Bounded Problem Muthukrishnan05 iLiL iRiR + root S = subset of selected ancestors i root - - At levels from bottom stop recursion, enter local search - time, space No need to tabulate The solution to this problem is more economic Dual Space-Bounded solved Indirectly via binary search

Unrestricted Synopses Unrestricted Synopses [GH05,06] Forget about actual coefficient values Choose a best set of non-zero wavelet terms of any values In practice: Examined values are multiples of resolution step δ

Unrestricted Synopses Unrestricted Synopses [GH05,06] Approximation quality better than restricted Time asymptotically linear to n But: - Examined values bounded by M [GH05] - Multiple Guesses of error result [GH06] - Space-Bounded Problem: Two-Dimensional Tabulation E(b,v) on each tree node → High Running Time and Space demands

Our Approach: Wavelet Synopses with Predefined Error Bounds Error-Bounded ProblemError-Bounded Problem DP algorithm: - Demarcates examined values using error bound ε - Tabulates only S(v), one dimension per node Space-Bounded ProblemSpace-Bounded Problem Enhanced Solution: - Calculate upper bound for error, use it to bound values Indirect Solution: - Use binary search on Error-Bounded problem

How does it work? One-dimensional tabulation on values only Examined incoming values v bounded by error bound Examined assigned values z also bounded Strong version of problem: minimize error within space

Complexity Error-Bounded Problem: Time or Space or Space-Bounded Problem: Time vs. Space vs.

Experiments: Error-Bounded Problem

Experiments: Space-Bounded Problem

Related Work M. Garofalakis and A. Kumar. Deterministic wavelet thresholding for maximum-error metrics. PODS 2004 S. Guha. Space efficiency in synopsis construction algorithms. VLDB 2005 S. Guha and B. Harb. Wavelet Synopses for Data Streams: Minimizing Non-Euclidean Error. KDD 2005 S. Muthukrishnan. Subquadratic algorithms for workload-aware haar wavelet synopses. FSTTCS 2005 S. Guha and B. Harb. Approxmation algorithms for wavelet transform coding of data streams. SODA 2006

Thank you! Questions?