1. One-Pass Wavelet Synopses for Maximum-Error Metrics Panagiotis Karras Trondheim, August 31st, 2005 Research at HKU with Nikos Mamoulis

2. Outline Preliminaries & Motivation –Usefulness of Synopses –Haar wavelet decomposition, conventional wavelet synopses –The maximum error guarantee problem Earlier Approach: Wavelet Synopses with Optimal Error Guarantees –Impracticability of this approach Solution: Practicable Wavelet Synopses for Maximum Error Metrics –Low-Complexity Algorithms that provide near-optimal error results Extension to Data Streams –One-Pass adaptations of the proposed algorithms Conclusions & Future Directions

3. 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

4. 34 16 2 20 20 0 36 16 0 18 7 -8 9 -9 10 25 11 10 26 Haar Wavelet Decomposition 18 Wavelet decomposition:Wavelet decomposition: orthogonal transform for the hierarchical representation of functions and signals Haar wavelets:Haar wavelets: simplest wavelet system, easy to understand and implement Extensible to many dimensions Error treeError tree: structure for the visualization of decomposition and value reconstructions Reconstructions require logarithmically many terms, along appropriate error tree paths

5. Wavelet Synopses Compute Haar wavelet decomposition of D Coefficient thresholding : retain B coefficients, B<<|D| Approximate query engine can operate over such compact synopses –[MVW, SIGMOD’98]; [VW, SIGMOD’99]; [CGRS, VLDB’00 ] Conventional approach: Retain B largest coefficients in absolute normalized value –Normalized Haar basis: divide coefficients at resolution j by –Minimizes the Total Squared (L 2 ) Error However…

6. The Problem with Conventional Synopses 34 16 2 20 20 0 36 16 0 18 7 -8 9 -9 10 + - + + + ++ + + - - ---- Example data vector and synopsis (|D|=8, B=4) Original Data Reconstruction 18 18 18 18 20 0 36 16 Large variation in answer quality Root cause –Aggregate error measure may be optimal, but error distributed unevenly among individual values

7. Solution: Thresholding for 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

8. Former Approach: Optimal Thresholding for Maximum-Error Metrics [GK, PODS’04] Based on Dynamic-Programming Formulation Relies on recursive function that computes minimum maximum error for a coefficient’s sub-tree given an allocated storage space Optimally distributes allocated space b between a node’s two child sub-trees and decides whether to retain the coefficient on this node Approximation schemes for multiple dimensions, also applicable in one dimension

9. Challenge:Challenge: –Design efficient, low-complexity thresholding schemes that achieve competitive results in comparison to the optimal solution and are extensible to streaming data However: Complexity: time (reducible to ) space (reducible to ) 1-D Approximation Schemes Impractical for the purpose it is meant for All Inapplicable in Streaming Environments

10. Solution: Greedy Thresholding for Maximum-Error Metrics Key Idea: Greedy solution that makes the best choice of next coefficient to discard at each step Each error-tree node stores the Maximum Potential Error that will be affected when the coefficient on it is discarded: –For Absolute Error: –For Relative Error: Global Heap structure returns node of Least Maximum Potential Error For Absolute Error:For Absolute Error: –Max and Min values of Accumulated Error below maintained on nodes For Relative Error:For Relative Error: –Accumulated Error on data level stored on leaf nodes –Heaps returning leaf of Maximum Potential Error augmented on nodes

11. Changes in Accumulated Error values propagated up and down the tree On each affected node: For Absolute Error:For Absolute Error: –Max, Min Accumulated Error updated Absolute –New Maximum Potential Absolute Error calculated as: For Relative Error:For Relative Error: –Descendants’ Heap updated Relative –New Maximum Potential Relative Error returned from Heap Update node’s position in Global Heap After each discarding operation: Solution: Greedy Thresholding for Maximum-Error Metrics

12. An Example (absolute error) First drop coefficient -1 Error accumulates on leaf nodes Next drop coefficient 2 of maximum potential error 3 And so on… 11 -1 -6 8 -2 6 6 10 4 2 -3 6 -7 -2 -4 + - + + + ++ + + - - ---- 1 1 1 1 -1 -1 -1 -1 -1 -1 -3 -31 -3

13. Complexity Analysis AbsoluteAbsolute Error Algorithm: Time: O(Nlog 2 N) Space: O(N) RelativeRelative Error Algorithm: Time: O(Nlog 3 N) Space: O(NlogN)

14. Extension to Data Streams Major application area Existing methods inapplicable Assumption: O(B ) available memory budget Further Problem:Further Problem: –Extend proposed methods to streams –One-pass overall process –Construct and truncate error-tree on-the-fly

15. Solution for Absolute Error After first B data, pair of coefficients discarded for every arriving data pair Scope limited to error-tree constructed so far Higher tree level for higher power of 2 #data Frontline structure storing: –Hanging coefficient nodes –Temporary average of data in hanging subtree –Error information from deleted orphan nodes Error propagation similar to static case, with some elaboration in upward propagation due to tree sparseness

16. 9 3 9 -5 5 13 13 17 14 -2 9 7 7 3... -4 2 -3 3 7 -2 -4 - + ++ -- 5 7 Error Tree Frontline 8 1 8 - + Data Stream 2 Example: Classic Error-Tree

17. 9 3 9 -5 5 13 13 17 14 -2 9 7 7 3... -4 2 -3 3 7 -2 -4 5 7 Error Tree Frontline 8 1 8 Data Stream 2 Example: Sibling Error-Tree

18. 9 3 9 -5 5 13... 2 3 7 -4 9 4 Error Tree Frontline Data Stream Example: B = 6, after 6 values

19. 9 3 9 -5 5 13 13 17... -4 -3 3 7 -4 9 4 Error Tree Frontline 8 Data Stream Example: B = 6, after 8 values -2 2

20. 9 3 9 -5 5 13 13 17 14 -2... -4 7 6 - Error Tree Frontline 8 8 Data Stream -3 Example: B = 6, after 10 values

21. 9 3 9 -5 5 13 13 17 14 -2 9 7... -4 7 - 7 Error Tree Frontline 8 8 Data Stream -3 Example: B = 6, after 12 values

22. 9 3 9 -5 5 13 13 17 14 -2 9 7 7 3 -4 7 5 7 Error Tree Frontline 8 8 Data Stream Example: B = 6, after 14 values

23. 4 4 11 -3 12 12 12 12 15 -1 7 7 5 5 -4 7 8 Reconstruction 1 1 7 Error Tree Example: B = 6, after padding

24. Solution for Relative Error Analogous Extension not feasible Solution: Heuristic Techniques Estimate of MR k calculated based on: –4 quantities as in Absolute Error (with denominators) –Minimum Absolute values in each subtree (with errors) –A sample value (with error) for each subtree, initialized as Minimum Absolute value beneath, changed by error propagation process when a sample below involves larger relative error Heuristic Estimate set as Maximum Relative Error among these 8 positions

25. Experimental Setting Experiments with Real DataExperiments with Real Data: –Frequency counts in US Forest Service Database –Photon counts by Voyager 2 stellar occultation experiments –Temperature measures from equatorial Pacific Comparison of both Static and Stream Algorithms with the Optimal Solution and the Conventional MethodComparison of both Static and Stream Algorithms with the Optimal Solution and the Conventional Method Streaming Algorithm can produce window-based synopses by discarding those retained coefficients whose scope falls entirely outside the window of interestStreaming Algorithm can produce window-based synopses by discarding those retained coefficients whose scope falls entirely outside the window of interest We present results for fixed data sets arriving in stream in order to preserve comparability with those of the non- streaming algorithmsWe present results for fixed data sets arriving in stream in order to preserve comparability with those of the non- streaming algorithms We present results for the relative error heuristic in the static case as wellWe present results for the relative error heuristic in the static case as well

26. Experimental Results Run-time, B = N / 16, Relative ErrorRun-time, B = N / 16, Relative Error

27. Experimental Results Quality, Absolute Error, Real Data (frequency counts), N = 360Quality, Absolute Error, Real Data (frequency counts), N = 360

28. Experimental Results Quality, Relative Error, Real Data (frequency counts), N = 360Quality, Relative Error, Real Data (frequency counts), N = 360

29. Experimental Results Scalability, Absolute Error, Real Data (photon counts), N = 16KScalability, Absolute Error, Real Data (photon counts), N = 16K

30. Experimental Results Scalability, Relative Error, Real Data (photon counts), N = 16KScalability, Relative Error, Real Data (photon counts), N = 16K

31. Experimental Results Scalability, Absolute Error, Real Data (temperature measures), B = N / 16Scalability, Absolute Error, Real Data (temperature measures), B = N / 16

32. Experimental Results Scalability, Relative Error, Real Data (temperature measures), B = N / 16Scalability, Relative Error, Real Data (temperature measures), B = N / 16

33. Conclusions & Future Directions Feasibility of Wavelet Synopses with near- optimal Error Guarantees at near-linear cost for both Static and Streaming DataFeasibility of Wavelet Synopses with near- optimal Error Guarantees at near-linear cost for both Static and Streaming Data Extension to Multidimensional Wavelets?Extension to Multidimensional Wavelets? Alternative Relative Error Heuristics?Alternative Relative Error Heuristics? Variable Coefficients?Variable Coefficients? Theoretical Worst-case Guarantee?Theoretical Worst-case Guarantee?

34. Related Work Y. Matias, J. S. Vitter, and M. Wang. Wavelet-based histograms for selectivity estimation. SIGMOD 1998 J. S. Vitter and M. Wang. Approximate computation of multidimensional aggregates of sparse data using wavelets. SIGMOD 1999 K. Chakrabarti, M. Garofalakis, R. Rastogi, and K. Shim. Approximate query processing using wavelets. VLDB Journal 2001 A. Gilbert, Y. Kotidis, S. Muthukrishnan and Martin Strauss. Surfing Wavelets on Streams: One-Pass Summaries for Approximate Aggregate Queries. VLDB 2001 M. Garofalakis and A. Kumar. Deterministic wavelet thresholding for maximum-error metrics. PODS 2004 S. Guha and B. Harb. Wavelet Synopses for Data Streams: Minimizing Non-Euclidean Error. KDD 2005

35. Thank you! Questions?