quotient cube: how to summarize the semantics of a data cube laks v.s. lakshmanan (univ. of british...

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

Lakshmanan, Pei & Han. Quotient Cube: How to Summarize the Semantics of a Data Cube 2

Outline

• Introduction and motivation

• Cube lattice partitions

• Semantics preserving partitions

• Algorithms

• Experimental results

• Discussion and summary


Data CubeBase table

Dimensions Measure

Store Product Season AVG(Sales)

S1 P1 Spring 6

S1 P2 Spring 12

S2 P1 Fall 9

S1 * Spring 9

… … … …

* * * 9

Dimensions Measure

Store Product Season Sales

S1 P1 Spring 6

S1 P2 Spring 12

S2 P1 Fall 9

Aggregation


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)


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)


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


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


(S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9

(*,*,*):9

C1 C2 C3

C4


Problems w/ the Naïve Attempt

• The result is not a lattice anymore!– Anomaly– The rollup/drilldown semantics is lost

C1 C2 C3

C4

343 CCC rolluprollup

(S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9


(S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9

(*,*,*):9


A Better Partitioning

• Quotient cube: partitioning reserving the rollup/drilldown semantics

(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

C1 C3

C5

C4

C2


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


Why A Quotient Cube Useful?

• Semantic compression

• Semantic OLAP browsing(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

C1 C2

C5

C4

C3


Why A Quotient Cube Useful?

• Semantic compression

• Semantic OLAP browsing(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

C1 C2

C5

C4

(S2,P1,f):9

(S2,*,f):9 (S2,P1,*) (*,P1,f):9

(*,*,f):9 (S2,*,*):9


Outline




• Algorithms




Convex Partitions

• A convex partition retains semantics

CLScCLSccccc rolluprollup 231321 ,,

(S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9


(S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9

(*,*,*):9

C1 C3

C5

C4

C2


A Non-convex Partition

• Anomaly

• The rollup/drilldown semantics is lost

C1 C2 C3

C4

343 CCC rolluprollup

(S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9


(S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9

(*,*,*):9


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


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)}

Dimensions Measure

Store Product Season Sales

S1 P1 Spring 6

S1 P2 Spring 12

S2 P1 Fall 9


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,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9

(*,*,*):9


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,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9

(*,*,*):9


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


Outline




• Algorithms




Class 1 = Class 2

Class 1

Weak Congruence

• Weak congruence preserves semantics

Class 2

c c’

d d’

roll

up

roll

up

c c’

d d’ro

llup

roll

upimply


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?


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


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!


Outline




• Algorithms




How to Compute A QC

• Aggregate functions– Monotone functions– Non-monotone functions

• Settings– The cube is available– Only the base table is available


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


Example: Cover QC(S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9


(S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9

(*,*,*):9


Non-monotone Functions

• Class merging

• Find cover partition classes

• Merge classes as long as convexity is retained


Example: AVG QC(S1,P1,s):6 (S1,P2,s):12 (S2,P1,f):9


(S1,*,*):9 (*,*,s):9 (*,P1,*):7.5 (*,P2,*):12 (*,*,f):9 (S2,*,*):9

(*,*,*):9


Outline




• Algorithms




Reduction Ratio vs. Dimensionality

0

10

20

30

40

50

60

70

80

90

100

2 3 4 5 6 7 8 9 10

Re

duction

ratio

(%

)

Dimensionality

MinCubeQC_CovQC_MIN

# base tuples = 200k Zipf factor = 2.0


Reduction Ratio vs. Zipf Factor

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3

Reduction r

atio (

%)

Zipf factor

MinCubeQC_CovQC_MIN

# base tuples = 200k # dimensions = 6


Reduction Ratio vs. Base Table Size

0

10

20

30

40

50

60

70

80

0 200 400 600 800 1000 1200 1400

Red

uct

ion r

atio

(%

)

Number of tuples (k)

MinCubeQC_CovQC_MIN

Zipf factor = 2.0 # dimensions = 6


Runtime

0

500

1000

1500

2000

2500

3000

0 200 400 600 800 1000 1200 1400

Runtim

e (

seconds)

Number of tuples (k)

MinCubeQC_CovQC_MIN

BUC

Zipf factor = 2.0 # dimensions = 6


Compression Ratio on Weather Data Set

0

10

20

30

40

50

60

70

80

90

100

2 3 4 5 6 7

Red

uctio

n ra

tio (

%)

Number of dimensions

QC_CovQC_AVG

0

10

20

30

40

50

60

2 3 4 5 6 7 8 9

Red

uctio

n ra

tio (

%)

Number of dimensions

MinCubeQC_Cov


Outline




• Algorithms




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


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


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, 1996.

• D. Barbara and M. Sullivan. Quasi-cubes: Exploiting approximation in multidimensional databases. SIGMOD Record, 26:12--17, 1997.

• D. Barbara and X. Wu. Using loglinear models to compress datacube. In WAIM'2000}, pages 311--322, 2000.

• K. Beyer and R. Ramakrishnan. Bottom-up computation of sparse and iceberg cubes. In SIGMOD'99.


Reference (2)• G. Birkhoff, Lattice Theory, 2nd edition, New York, American

Mathematical Society (Colloquium Publications, vol. 25), 1948.• 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.


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 2000.

• 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.


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, 1949.

quotient cube: how to summarize the semantics of a data cube laks v.s. lakshmanan (univ. of british...

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