Query Processing on Cubes Mapped from Ontologies to Dimension Hierarchies

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1 Query Processing on Cubes Mapped from Ontologies to Dimension Hierarchies Carlos Garcia-Alvarado Greenplum EMC USA Carlos Ordonez University of Houston USA

2 Scenario Dimension Measurements Explore Digital Library Document 2

3 Dimension Hierarchies 3

4 Problem Efficient summarization of text corpora mapped from ontologies to dimension hierarchies. OLAP Cube is an excellent candidate to represent concept hierarchies and perform efficient aggregations on multiple combinations. Previous work required star schema or cubes on demand with all concepts [1][2][3]. 4

5 Definitions Let a collection C, or corpus, of n documents. Each document has {D 1, D 2,,D k } dimensions. Each document has a measurements {A 1, A 2,,A e } Fact table is in vertical format: F(i,D j,a 1,...,A e ) An ontology O is mapped to a dimension hierarchy as a tree-like structure. A query Q is a subset of dimensions from F which builds an OLAP cube. H (1,1) H (2,1). H (h 2,1) D 1 H (1,2). H 0... H 1,k1 H 2,k2. H (h 2,kh 2 ) D k Figure 4: Hierarchies. 5

6 CUBO: CUBed Ontologies Take advantage of sparse frequency matrix. Perform single pass through the data. Store the result in a Hash- table. Load the Ontology in main memory for summarizaeon by level. 6

7 Fact Table Computation Algorithm 1: CUBO Input: O,F,Q,T,{A 1,...} Output: R /* Init CUBO struct */ 1 R ; /* Load ontology in main memory. */ 2 O LoadOntologyFromOWL(); /* Filter F to consider only those D j in Q */ 3 ˆF {t i t i F D j s.t.d j t i D j Q} ; /* Single data set scan. */ 4 t ; 5 while row in ˆF do 6 if document changed then /* Add document to CUBO */ 7 R R BuildCube(t,O,T,R,{A 1,...}); 8 end /* D j row. */ 9 t t {D j }; 10 end /* Add last document to CUBO */ 11 BuildCube(t,O,T,R,{A 1,...}); 7

8 Build Cube per Document Algorithm 2: BuildCube Input: t,o,t,r,{a 1,...} Output: R s h Combos(); /* Aggregate all the existing combos of the h-1 level. */ foreach combo do R R {1,combo,{A 1,...}}; end /* Recursive function to extract all unique concepts by level h-2 to 0. */ s 0,...,h 2 CombosForOntologyLevel(s,O,T); /* Increments found combos by level. */ foreach lins 0,...,h 2 do foreach combo in s l do R R {l, combo, {A 1,...}}; end end return R; 8

9 Example Q={Parallel processors, Array and vector processors, Distributed databases} Fact Table 3, Array and Vector Processors, 50 3, Distributed databases, 50 3, Parallel Processors, 50 A 1 = 50 Cube per Document H h- 1 {Array and Vector Processors, Distributed databases} {Distributed databases, Parallel Processors} {Array and Vector Processors, Parallel Processors} H h- 2 {Parallel Processors, Distributed Systems} H 0 {Computer Science} Hash Table H h- 1 Level 2 Hash Table H h- 2 Level 1 Hash Table H 0 Level 0 9

10 Time Complexity Traditional data cube computation O(nh2 k ) The average number of k and h is small. Our algorithm has a worst time complexity of O(n2 kh ), but on average performs less computations. 10

11 Experiments in a DBMS CUBO is a User- Defined FuncEon in C#. Our experiments were run on: Intel Xeon Dual GHz 1 TB Hard drive 4 GB RAM SQL SERVER

12 Data Sets Table 1: TPCH Corpora. n Max k j Min k j Total k 1K K K M M Table 2: dbpedia Corpora. n Max k j Min k j Total k 1K K K M M

13 Experiments Table 3: Performance of Traditional Cube and CUBO (* unable to compute) d Traditional Single Level CUBO * * * 96 13

14 Experiments Figure 6: Varying Corpus Size. Figure 7: Varying Number of Dimensions. Figure 8: CUBO Size when Varying Number of Dimensions. Table 5: Varying Ontology Levels in TPCH (time in seconds). n ALL MAX 2 MAX 1 1K K K M M

15 Conclusions CUBO is an efficient and single pass algorithm for summarizing hierarchical data. CUBO is faster than using a traditional OLAP algorithm. CUBO performs faster than the theoretical upper bound. CUBO not sensitive to the branching factor. 15

16 Future Work Support ontologies that do not fit in main memory. Improve scalability on h (more than 5 levels deep). Support unbalanced trees (ontologies) and ontologies with multiple parents. Support incremental computation of new dimensions. CUBO needs to be explored in MPP databases. 16

17 References [1] J. Lee, D. Grossman, O. Frieder, and M.C. McCabe. IntegraEng structured data and text: A muledimensional approach. In Proc. of IEEE InternaEonal Conference on InformaEon Technology: Coding and CompuEng, pages , [2] C.X. Lin, B. Ding, J. Han, F. Zhu, and B. Zhao. Text cube: CompuEng IR measures for muledimensional text database analysis. In Proc. of IEEE ICDM, pages , [3] D. Zhang, C. Zhai, and J. Han. Topic cube: Topic modeling for OLAP on muledimensional text databases. In Proc. of SIAM SDM Conference,

Application Tool for Experiments on SQL Server 2005 Transactions

Proceedings of the 5th WSEAS Int. Conf. on DATA NETWORKS, COMMUNICATIONS & COMPUTERS, Bucharest, Romania, October 16-17, 2006 30 Application Tool for Experiments on SQL Server 2005 Transactions ŞERBAN