Hierarchical Constraint Satisfaction in Spatial Database

Dimitris Papadias, Panos Kalnis And Nikos Mamoulis

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

• 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

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).

x1 x2

Disjoint(x1, x2)

x1 x2

Meet(x1, x2)

x1 x2

Overlap(x1, x2)

x2x1

Cover( x1, x2)

Topological Relations

x1 x2

Equal(x1, x2)

x2x1

Contain(x1, x2)

x1x2

Covered-by(x1, x2)

x1x2

Inside(x1, x2)

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.

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.

R-tree

R-tree

• 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.

R-tree

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.

Hierarchical CSPs using R-trees• A set of variables v1,v2,v3…vn.• Domain di for variable vi is

for level 0 : {xi,1,…… xi,ci}

for level 1 to h-1:{Xi,1,…… Xi,ci}• For each pair of variables the binary constraint is for level 0 : Cij is a disjunction of topological

relations as specified by the query. for level 1 to h-1: Cij is derived by replacing each

relation in Cij by the corresponding condition for intermediate nodes in Table 2.

Table 2

• 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

• 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

Experiments

• Problems were randomly generated by modifying the paramters n,m,p1, p2.

• n = number of variables.

• m = size of datasets

• p1 = is the probability that a random pair of variables is constrained (network density).

• p2 = is the probability that assignment for a constrained pair is inconsistent (tightness).

• Typical values takes for the problem

• m = 104

• |x| = .0045 d .2( typical value for real datasets)

• h=3

• C=50-200

Using these values problems were randomly

generated.

Hierarchical Systematic Search

• Using Forward Checking with fail first dynamic variable ordering heuristic.

• 50 randomly generated problems.

• P1 = 1.

• n=5.

• m=104

D 0.2

Hierarchical Systematic Search

• Two searches were used – Forward Checking (FC)– Hierarchical FC (H-FC)

• Three types of problems were tested – Varying P2 without disjoint– Varying P2 with disjoint– Varying n with one solution

Results

• Varying P2 without disjoint

– H-FC outperforms FC by two orders of magnitude.

• Varying P2 with disjoint– For dense graphs the H-FC outperforms FC by two

orders of magnitude.– As tightness decreases the performance converges.

Hierarchical Systematic Search

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

• 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

• Problems created with the parameters– n=5

– m=103, 104, 105.

• All algorithms were executed 10 times for every setting.

• Their execution was terminated is solution could not be obtained after 109 checks.

Hierarchical Local Search

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= 103 MC is better than hierarchical local search

• When m= 105 HR-MC outperforms MC by one order of magnitude.

Hierarchical Local Search

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

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 104 and n10

• Hierarchical local search is better for very large domains.

• Provides hints to improve performance.