topological reasoning between complex regions in databases with frequent updates arif khan &...
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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 RulesConsider,
(A B⊂ ¬∂A∂B) (B∧ ∧ C⊂ ¬∂B∂C)∧A B⊂ B∧ C⊂
A 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
Ao ∩ Co = 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 CBD
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 CB
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 CD
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(n2 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
Thank you!