Topological Reasoning between Complex Regions in Databases with Frequent Updates Arif Khan & Markus Schneider Department of Computer and Information Science and Engineering University of Florida Presented by: Hechen Liu

Motivation Topological relationships are important in many applications, e.g., AI, cognitive science, and spatial databases It is impossible to find all topological facts It is impractical to keep all topological facts Simple regions are not enough to represent real life scenarios

Complex Objects Complex regions: Multiple Components: faces Each face may have single or multiple holes Interior: A Exterior: A - Boundary: A

9-Intersection Model

33 Relationships of Complex Regions [1] M. Schneider and T. Behr. Topological Relationships between Complex Spatial Objects. ACM Transactions on Database Systems, 31(1):39-81, 2006.

Inference Composition Rx(A,B), Ry(B,C) Rz(A, C) Rx o Ry Rz inside(A, B) o inside(B, C) inside(A,C) Determined by the inference rules

Overview of the Reasoning Process Local Inference Apply inference rules Interpret reasoning result and identify relationship(s) Global Inference Extend the inference to N complex regions Binary Spatial Constraint Network (BSCN)

Local Inference Interior can characterize a complex region 8 possible interior-interior set relations exist between two complex regions. A B: A B A - B B - A 8*8=64 combinations possible between A and C.

Inference Rules Consider, (A B ¬AB) (B C ¬BC) A B B C A C A C = 1 (interior-interior intersection) with the same input, A C = 0 (interior-exterior intersection) A B C C

Inference Rules Consider, A B and B C A o C o = unknown (interior-interior intersection)

Inference Rules

Relationship Identifying Process If all 9 predicates are deterministic, then inferred relationship is a single relationship. If there is any unknown value, then the inferred relationship is a disjunction. For example:

Decision Tree of the Relation Space Brute force method: 33*8=264 comparisons Recursively divide the relationship space based on a predicate value at each level, until we reach a single relationship e.g.,18 relationships have false in the interior-boundary (P2) value. 33 relationships form a tree of height 6 Deterministic values have 6 comparisons instead of 264: 97% improvement Indeterminate values have at most 32 comparisons: 88% improvement

Global Inference Extend the reasoning process to N objects. Binary Spatial Constraint Network (BSCN)

Reasoning in Dynamic Databases Find BSCN paths Each time a change occurs in the database, the algorithm should run Intermediate objects are thrown out when the query is committed

Most Specific Relationship The relationship which has the least number of disjunctions Shortest path does not guarantee most specific relationship A C B D E E A D B C

Most Specific Relationship The relationship which has the least number of disjunctions. Shortest path does not guarantee most specific relationship. overlap o overlap unknown A C B E A D B C

Most Specific Relationship The relationship which has the least number of disjunctions Shortest path does not guarantee most specific relationship inside o inside inside A C D E E A D B C

Most Specific Relationship The relationship which has the least number of disjunctions Shortest path does not guarantee most specific relationship inside o disjoint disjoint A C E E A D B C

Most Specific Relationship The relationship which has the least number of disjunctions Shortest path does not guarantee most specific relationship In fact, there is no relation between the length of the path and the most specific relationship

Most Specific Relationship Solution: consider all paths and take the intersection Problem: number of paths is O(n!) Interesting Facts: Worst case scenario when the graph is complete (then, we even do not need reasoning) Consider sparse graphs

K-Shortest Paths Let us not consider all the paths. Instead, we consider k-paths K-shortest path algorithm: O(m+nlogn+k) [2] Reasoning between complex regions: – Total complexity: O(n 2 log n) [2] D. Eppstein. Finding the k shortest paths. SIAM Journal on Computing, 28(2):652–673, 1999.

Simulation and Result Random graph Edges are Power Law distributed All edges have unit weight Number of paths considered: k = cn

Simulation and Results

Conclusions and Future Work Derived a complete set of inference rules Proposed BSCN and a dynamic reasoning approach Will introduce more robust heuristics Weighted BSCN. Will extend to other data types line-line line-region

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Thank you!