Quotient Cube: How to Summarize the Semantics of a Data Cube

Transcription

1 Quotient Cube: How to Summarize the Semantics of a Data Cube Laks V.S. Lakshmanan (Univ. of British Columbia) * Jian Pei (State Univ. of New York at Buffalo) * Jiawei Han (Univ. of Illinois at Urbana-Champaign) + * The work is partially supported by NSERC and NCE/IRIS + The work is partially supported by NSF, UI, and Microsoft Research

2 Outline Introduction and motivation Cube lattice partitions Semantics preserving partitions Algorithms Experimental results Discussion and summary Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 2

3 Data Cube Base table Dimensions Measure Store S1 S1 S2 Product P1 P2 P1 Season Spring Spring Fall Sales Store S1 S1 Dimensions Product Season P1 Spring P2 Spring Measure AVG(Sales) 6 12 S2 P1 Fall 9 S1 * Spring 9 Aggregation * * * 9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 3

4 Previous Work: Efficient Cube Computation Compute a cube from a base table: e.g. (Agarwal et al. 98), (Zhao et al. 97) View materialization with space constraint: e.g. Harinarayann et al. 96 Handling scarcity (Ross & Srivastava 97) Cube compression: e.g. (Sismanis et al. 02), (Shanmugasundaram et al. 99), (Want et al. 02) Approximation: e.g. (Barbara & Sullivan 97), (Barbara & Xu 00), (Vitter et al. 98) Constrained cube construction: e.g. (Beyer & Ramakrishnan 99) Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 4

5 Previous Work: Extracting Semantics From Cubes General contexts of patterns (Sathe & Sarawagi 01) Generalize association rules (Imielinski et al. 00) Cube gradient analysis (Dong et al. 01) Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 5

6 Cube (Cell) Lattice Many cells have same aggregate values Can we summarize the semantics of the cube by grouping cells by aggregate values? (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*):9(*,P1,f):9 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 6

7 A Naïve Attempt Put all cells having same aggregate value in a class (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 C1 C2 C3 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*):9(*,P1,f):9 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 C4 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 7

8 Problems w/ the Naïve Attempt The result is not a lattice anymore! rollup rollup Anomaly C3 C4 C3 The rollup/drilldown semantics is lost (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 C1 C2 C3 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*):9(*,P1,f):9 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 8 C4

9 A Better Partitioning Quotient cube: partitioning reserving the rollup/drilldown semantics (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 C1 C2 C3 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*) (*,P1,f):9 C4 C5 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 9

10 Problem Statement Given a cube, characterize a good way (quotient cube) of partitioning its cells into classes such that The partition generates a reduced lattice preserving the rollup/drilldown semantics The partition is optimal: # classes as small as possible Compute quotient cubes efficiently Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 10

11 Why A Quotient Cube Useful? Semantic compression Semantic OLAP browsing (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 C3 (S1,*,s):9(S1,P1,*):6(*,P1,s):6(S1,P2,*):12(*,P2,s):12(S2,*,f):9 (S2,P1,*)(*,P1,f):9 C1 C2 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 C4 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 11 C5

12 Why A Quotient Cube Useful? (S2,P1,f):9 Semantic compression Semantic OLAP browsing (S2,*,f):9 (S2,P1,*) (*,P1,f):9 (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 (*,*,f):9 (S2,*,*):9 (S1,*,s):9(S1,P1,*):6(*,P1,s):6(S1,P2,*):12(*,P2,s):12(S2,*,f):9 (S2,P1,*)(*,P1,f):9 C1 C2 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 C4 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 12 C5

13 Outline Introduction and motivation Cube lattice partitions Semantics preserving partitions Algorithms Experimental results Discussion and summary Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 13

14 Convex Partitions A convex partition retains semantics c rollup rollup c2 c3, c1 c3 CLS c2 1, CLS (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 C1 C2 C3 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*) (*,P1,f):9 C4 C5 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 14

15 A Non-convex Partition Anomaly rollup rollup C3 C4 C3 The rollup/drilldown semantics is lost (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 C1 C2 C3 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*):9(*,P1,f):9 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 15 C4

16 Connected Partitions Cells c1 and c2 are connected if a series of rollup/drilldown operation starting from c1 can touch c2 Intuitively, (each class of) a partition should be connected Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 16

17 Cover Partition For a cell c, a tuple t in base table is in c s cover if t can be rolled up to c E.g., Cov(S1,*,spring)={(S1,P1,spring), (S1,P2,spring)} Store S1 S1 S2 Dimensions Product Season P1 Spring P2 Spring P1 Fall Measure Sales Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 17

18 Cover Partitions Are Convex All cells having the same cover are in a class (S1,P2,s) and (*,P2,*) cover same tuples in the base table (S1,P2,*) and (*,P2,s) are in the same class. (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*) (*,P1,f):9 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 18

19 Cover Partitions Are Connected Cells c1 and c2 have the same cover there must be some common ancestor c3 of c1 and c2 st c3 has the same cover Cells c1 and c2 are in the same class and connected (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*) (*,P1,f):9 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 19

20 Cover Partitions & Aggregates All cells in a cover partition carry the same aggregate value w.r.t. any aggregate function But cells in a class of MIN() may have different covers For COUNT() and SUM() (positive), cover equivalence coincides with aggregate equivalence Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 20

21 Outline Introduction and motivation Cube lattice partitions Semantics preserving partitions Algorithms Experimental results Discussion and summary Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 21

22 Weak Congruence Weak congruence preserves semantics c Class 1 c c c rollup rollup imply rollup Class 1 = Class 2 rollup d Class 2 d d d Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 22

23 Weak Congruence = Convex Convex no hole in the class weak congruence They preserve the rollup/drilldown semantics Quotient cube lattice is the lattice of convex classes How to derive the coarsest quotient cube? Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 23

24 Monotone Aggregate Functions Monotone functions S T f(s) f(t) S T f(s) f(t) MIN(), MAX(), COUNT(), PSUM(), The aggregate function f is monotone f is the unique coarsest partition MIN(): put all cells having the same MIN() value into a class Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 24

25 Non-monotone Functions Bad news: f may or may not be a convex/weak congruence. Good news: cover partition is convex (I.e., weak congruence) and always yields a quotient cube w.r.t. any aggregate function! Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 25

26 Outline Introduction and motivation Cube lattice partitions Semantics preserving partitions Algorithms Experimental results Discussion and summary Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 26

27 How to Compute A QC Aggregate functions Monotone functions Non-monotone functions Settings The cube is available Only the base table is available Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 27

28 Monotone Functions The cube is available grab all cells with the same aggregate value and put them into a class Only the base table is available bottom-up, depth-first search For a cell, compute its cover, find the upper bound having the same aggregate value Group lower bounds by upper bounds Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 28

29 Example: Cover QC (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*) (*,P1,f):9 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 29

30 Non-monotone Functions Class merging Find cover partition classes Merge classes as long as convexity is retained Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 30

31 Example: AVG QC (S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9 (S1,*,s):9 (S1,P1,*):6 (*,P1,s):6 (S1,P2,*):12 (*,P2,s):12 (S2,*,f):9 (S2,P1,*) (*,P1,f):9 (S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9 (*,*,*):9 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 31

32 Outline Introduction and motivation Cube lattice partitions Semantics preserving partitions Algorithms Experimental results Discussion and summary Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 32

33 Reduction Ratio vs. Dimensionality Reduction ratio (%) Dimensionality MinCube QC_Cov QC_MIN # base tuples = 200k Zipf factor = 2.0 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 33

34 Reduction Ratio vs. Zipf Factor MinCube QC_Cov QC_MIN Reduction ratio (%) Zipf factor # base tuples = 200k # dimensions = 6 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 34

35 Reduction Ratio vs. Base Table Size Reduction ratio (%) MinCube QC_Cov QC_MIN Number of tuples (k) Zipf factor = 2.0 # dimensions = 6 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 35

36 Runtime Runtime (seconds) MinCube QC_Cov QC_MIN BUC Number of tuples (k) Zipf factor = 2.0 # dimensions = 6 Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 36

37 Compression Ratio on Weather Data Set MinCube QC_Cov Reduction ratio (%) Reduction ratio (%) Number of dimensions Number of dimensions Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube QC_Cov QC_AVG

38 Outline Introduction and motivation Cube lattice partitions Semantics preserving partitions Algorithms Experimental results Discussion and summary Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 38

39 Semantic Cube Exploration Theoretical foundation for semantic summarization in data cube concept and properties of quotient cubes Efficient algorithms for quotient cube construction Quotient cubes can be computed directly from base tables Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 39

40 Ongoing Research Efficient implementation of quotient cube-based OLAP system Data warehouse built using quotient cubes Hierarchies and constraints Incremental maintenance Semantics based OLAP and mining Efficient query answering Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 40

41 References (1) R. Agrawal and R. Srikant. Fast Algorithms for Mining Association Rules in Large Databases. VLDB 1994 S. Agarwal, R. Agrawal, P.M. Deshpande, A. Gupta, J.F. Naughton, R. Ramakrishnan, and S. Sarawagi. On the computation of multidimensional aggregates. VLDB, D. Barbara and M. Sullivan. Quasi-cubes: Exploiting approximation in multidimensional databases. SIGMOD Record, 26:12--17, D. Barbara and X. Wu. Using loglinear models to compress datacube. In WAIM'2000}, pages , K. Beyer and R. Ramakrishnan. Bottom-up computation of sparse and iceberg cubes. In SIGMOD'99. Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 41

42 Reference (2) G. Birkhoff, Lattice Theory, 2 nd edition, New York, American Mathematical Society (Colloquium Publications, vol. 25), S. Geffner, D. Agrawal, A. El Abbadi, and T. R. Smith. Relative prefix sums: An efficient approach for querying dynamic OLAP data cubes. In ICDE'99. Jim Gray, Adam Bosworth, Andrew Layman, Hamid Pirahesh. Data Cube: A Relational Aggregation Operator Generalizing Group-By, Cross-Tab, and Sub-Total. ICDE'96. C.-T. Ho, J. Bruck, and R. Agrawal. Partial-sum queries in data cubes using covering codes. In PODS'97. J. Han, J. Pei, G. Dong, and K. Wang. Efficient Computation of Iceberg Cubes with Complex Measures. In SIGMOD'01. Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 42

43 Reference (3) V. Harinarayan, A. Rajaraman, and J. D. Ullman. Implementing data cubes efficiently. In SIGMOD'96. T. Imielinski, L. Khachiyan, and A. Abdulghani. Cubegrades: Generalizing Association Rules. Technical Report, Rutgers University, August H. V. Jagadish, J. Madar, R.T. Ng. Semantic Compression and Pattern Extraction with Fascicles. VLDB'99. K. Ross and D. Srivastava. Fast computation of sparse datacubes. In VLDB'97. G. Sathe and S. Sarawagi. Intelligent Rollups in Multidimensional OLAP Data. VLDB'01. Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 43

44 Reference (4) J. Shanmugasundaram, U.M. Fayyad, and P. S. Bradley. Compressed Data Cubes for OLAP Aggregate Query Approximation on Continuous Dimensions. SIGKDD 99. J. S. Vitter, M. Wang, and B. R. Iyer. Data cube approximation and historgrams via wavelets. In CIKM'98. W. Wang, H. Lu, J. Feng, and J. X. Yu. Condensed cube: An effective approach to reducing data cube size. In ICDE'02. Y. Zhao, P. M. Deshpande, and J. F. Naughton. An array-based algorithm for simultaneous multidimensional aggregates. In SIGMOD'97. G.K. Zipf. Human Behavior and The Principle of Least Effort Addison-Wesley, Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 44