Christian Böhm & Hans-Peter Kriegel, Ludwig Maximilians Universität München A Cost Model and Index Architecture fo

1. Christian Böhm & Hans-Peter Kriegel,Ludwig Maximilians Universität MünchenA Cost Model and Index Architecture for the Similarity Join

2. Feature Based Similarity

3. Simple Similarity Queries • Specify query object and • Find similar objects – range query • Find the k most similar objects – nearest neighbor q.

4. R S Join Applications: Catalogue Matching • Catalogue matching • E.g. Astronomic catalogues

5. Join Applications: Clustering • Clustering (e.g. DBSCAN) • Similarity self-join

6. e R-tree Spatial Join (RSJ) procedure r_tree_sim_join (R, S, e)if IsDirpg (R) Ù IsDirpg (S) thenforeachrÎR.children do foreachsÎS.children doif mindist (r,s) £ethen CacheLoad(r); CacheLoad(s);r_tree_sim_join (r,s,e) ;else (* assume R,S both DataPg *)foreachpÎR.points do foreachqÎS.points do if |p - q| £ ethen report (p,q); R S

7. Cost Modeling • Single similarity queries: Access prob. of pages modeled using the concept of Minkowski Sum

8. Cost Modeling • Binomial formula:

9. Cost Modeling • Mating probability of index pages: • Probability that distance between two pages £ e • Two-fold application of Minkowski sum

10. Page Capacity Optimization • Cost model can determine index selectivity which depends on various parameters • Page capacity (number of stored points) is an important parameter • Known from similarity search: Page capacity optimization yields considerable improvement

11. Analysis of the Index Overhead • Assuming 100% selectivity (index doesnt work)How much more expensive is index usage ? • CPU: • Distance betw. boxes more expensive to compute than distance betw. points: a » 5 • Smaller capacity  more box distance computations

12. Analysis of the Index Overhead • Disk I/O: • High constant cost per page access (move disk head) • Page access is by factor b » 10000 / d more expensive than continuous reading of a point • Smaller capacity  more disk head movement

13. Analysis of the Index Overhead • What selectivity is needed that index pays off ?

14. Optimization • I/O cost function:is optimized by • CPU cost function:is optimized by:

15. Optimization • I/O cost: • Large capacity optimum (several 10,000 points, typically) • CPU cost: • Small capacity optimum (< 100 points, typically) • No compromise achievable

16. Multipage Index (MuX) ® CPU-performance like CPU optimized index ® I/O- performance like I/O optimized index separate optimization

17. Experimental Evaluation Uniform 4D Uniform 8D

18. Experimental Evaluation CAD Data 16D Color Images 64D

19. Conclusions • Summary • High potential for performance gains of the similarity join by page capacity optimization • Necessary to separately optimize I/O and CPU • Future research potential • Similarity join for metric index structures • Approximate similarity join • Parallel similarity join algorithms

20. Consequences • Assume for I/O optimization selectivity » 100% • Page accesses in a nested block loop like style: fill cache with pages of R (1 page free) ; foreachS-page sdo ifs joins some of the cached R-pg then load (s) ; foreach joining R-page r in cache do if mindist(r,s) < ethen join (r,s) ;

21. e R-tree Spatial Join (RSJ) procedure r_tree_sim_join (R, S, e)if IsDirpg (R) Ù IsDirpg (S) thenforeachrÎR.children do foreachsÎS.children doif mindist (r,s) £ethen CacheLoad(r); CacheLoad(s);r_tree_sim_join (r,s,e) ;else (* assume R,S both DataPg *)foreachpÎR.points do foreachqÎS.points do if |p - q| £ ethen report (p,q); R S