1. Hierarchical Constraint Satisfaction in Spatial Database Dimitris Papadias, Panos Kalnis And Nikos Mamoulis

2. Purpose of the Paper Show how systematic and local search make use of hierarchical decomposition of space. To efficiently guide search. Show conditions when hierarchical constraint satisfaction outperforms traditional methods

3. Helps solves queries in spatial database and geographical information systems. For e.x. a user is searching for a residential area that covers a commercial center and the commercial center meets a park. Hierarchical Constraint Satisfaction

4. Introduction Content based queries can be modeled as CSP. All objects in query  variables. relation between variables  constraints. The domain of the variables consists of objects in the database. For e.x. find residential areas(v1) that cover commercial centers(v2) that meets a park(v3).

5. x1 x2 Disjoint(x1, x2) x1 x2 Meet(x1, x2) x1 x2 Overlap(x1, x2) x2 x1 Cover( x1, x2) Topological Relations x1 x2 Equal(x1, x2) x2 x1 Contain(x1, x2) x1 x2 Covered-by(x1, x2) x1 x2 Inside(x1, x2)

6. Minimum Bounding Rectangles(MBR) are actual area objects on the map the R-tree is built by grouping rectangles at the lower level. R-trees are used by CSP algorithms to accelerate search.

7. R-tree Are an extension of B + -trees to many dimensions. B + -trees is a balanced search tree which maintains an ordered set of data and in which the keys are stored in a the leaves Is a height Balanced Tree that consists of intermediate and leaf nodes corresponding to disk in secondary memory. If h is the height of the tree the root is at level h-1 and the leaf is at the level 0. The intermediate levels are built by grouping rectangles at lower level. There is a R-tree for each type of object.

8. R-tree

9. Used for window queries. Two Steps are involved. Filter Step – retrieve a set of candidates that includes all the results and some false hits. Refinement Step – each candidate is examined and false hits are eliminated. The method can be extended for topological relations.

10. R-tree

11. R-trees Join (RTJ) The most influential algorithm for processing intersection joins using R-trees. Based on enclosure property. Like window queries, in order to process arbitrary topological relations using RTJ we need to define conditions for intermediate nodes. The problem is viewed as a multi-way spatial join and processed by computing the result of one pair-wise join and joining the result with v3.

12. Hierarchical CSPs using R-trees A set of variables v1,v2,v3…vn. Domain di for variable vi is for level 0 : {x i,1,…… x i,ci } for level 1 to h-1:{X i,1,…… X i,ci } For each pair of variables the binary constraint is for level 0 : C ij is a disjunction of topological relations as specified by the query. for level 1 to h-1: C ij is derived by replacing each relation in C ij by the corresponding condition for intermediate nodes in Table 2.

13. Table 2

14. Two preprocessing heuristic –Space Restriction. –Path Consistency Space Restriction – scans the domains of all variables, removing the entries that cannot satisfy the query constraints given their positions w.r.t. to other nodes. Path Consistency – is a form of semantic query optimization to discard inconsistent queries. Hierarchical CSPs using R-trees

15. Using systematic and local search algorithms there are three cases : –Hierarchical systematic search. –Hierarchical local search. –Hierarchical local/systematic search. Hierarchical CSPs using R-trees

16. Experiments Problems were randomly generated by modifying the paramters n,m,p 1, p 2. n = number of variables. m = size of datasets p 1 = is the probability that a random pair of variables is constrained (network density). p 2 = is the probability that assignment for a constrained pair is inconsistent (tightness).

17. Typical values takes for the problem m = 10 4 |x| =.0045  d .2( typical value for real datasets) h=3 C=50-200 Using these values problems were randomly generated.

18. Hierarchical Systematic Search Using Forward Checking with fail first dynamic variable ordering heuristic. 50 randomly generated problems. P 1 = 1. n=5. m=10 4  D  0.2

19. Hierarchical Systematic Search Two searches were used –Forward Checking (FC) –Hierarchical FC (H-FC) Three types of problems were tested –Varying P 2 without disjoint –Varying P 2 with disjoint –Varying n with one solution

20. Results Varying P 2 without disjoint –H-FC outperforms FC by two orders of magnitude. Varying P 2 with disjoint –For dense graphs the H-FC outperforms FC by two orders of magnitude. –As tightness decreases the performance converges. Hierarchical Systematic Search

21. Results Varying n with one solution –The performances converges as the number of variables increases. –For n>25 FC outperforms H-FC. Hierarchical Systematic Search

22. Local search used is Hill Climbing with min- conflicts(MC) heuristic. Following Variations used –Flat MC –Hierarchical uninformed MC (HU-MC) –Hierarchical informed MC (HI-MC) –Hierarchical root MC (HR-MC) –Hierarchical root MC/FC (HR-MC/FC) Hierarchical Local Search

23. Problems created with the parameters –n=5 –m=10 3, 10 4, 10 5. All algorithms were executed 10 times for every setting. Their execution was terminated is solution could not be obtained after 10 9 checks. Hierarchical Local Search

24. Results HR-MC outperforms HU-MC by at least one order of magnitude. HI-MC’s performance is between HR-MC and HU-MC. When m= 10 3 MC is better than hierarchical local search When m= 10 5 HR-MC outperforms MC by one order of magnitude. Hierarchical Local Search

25. Results Due to large number of sultions at the upper level hierarchical local search succeeds fast but spends more time trying to find a soultion at the leaf level this motivated the replacement of MC at leaf level with FC For larger domains HR-MC/FC outperforms HR-MC by almost an order of magnitude. Hierarchical Local Search

26. Conclusion Provides a methodology for hierarchical constraint satisfaction in spatial database using R-trees. Systematic search is significantly faster in the case of hierarchical CSPs for m  10 4 and n  10 Hierarchical local search is better for very large domains. Provides hints to improve performance.