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: ABoundary: ∂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 ∧ ¬∂A∂B) ∧ (B ⊂ C∧ ¬∂B∂C)
A ⊂ B ∧ B ⊂ C
A ⊂ C
A
B
C
A ∩ C ≠ ∅
A ∩ C = 1 (interior-interior intersection)
with the same input,
A ∩ C− = 0 (interior-exterior intersection)
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
E
D
A
B
C
D
E
B
C
Most Specific Relationship
 The relationship which has the least
number of disjunctions.
−
Shortest path does not guarantee most
specific relationship.
A
A
B
C
D
overlap o overlap  unknown
E
B
C
Most Specific Relationship
 The relationship which has the least
number of disjunctions
−
Shortest path does not guarantee most
specific relationship
A
E
D
A
C
inside o inside  inside
D
E
B
C
Most Specific Relationship
 The relationship which has the least
number of disjunctions
−
Shortest path does not guarantee most
specific relationship
A
E
A
C
inside o disjoint  disjoint
D
E
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
Questions and Comments?
Please contact Mr. Arif Khan:
[email protected]
Thank you!