1. CS 599 – Spatial and Temporal Databases Realm based Spatial data types: The Rose Algebra Ralf Hartmut Guting Markus Schneider

2. Spatial data types or algebras for database systems should satisfy the following criteria: Generality: Geometric objects used as SDT values should be as general as possible. The domains of data types points, lines and region should be closed under union, intersection and difference. Rigorous definition: The semantics of SDT ’s, the possible values for the types and the functions associated with the operations, must be defined formally to avoid ambiguities. Finite resolution : The formal definitions must take into account the finite representations available inside computers Treatment of geometric consistency: The definition of SDT ’s must enforce geometric consistency General object model interface: SDT definition should be based on abstract interface to the DBMS data model

3. Realm concepts: A realm is a set of points and non intersecting lines segments over a discrete domain that is a grid. Values of spatial data types can be composed from the objects present in a realm.

4. Realm for points, lines and regions Regions : A, B Lines : C Points : D

5. Why use a realm as a basis for spatial data types It enforces geometric consistency of related spatial objects, For example, the common part of the borders of countries A and B is exactly the same for both objects. It guarantees nice closure properties for computation with spatial databases Shields geometric consistency in query processing from numeric correctness and robustness problems

6. Transforming to a realm Move the intersection point to the nearest point on the grid. Application data at the lowest level of abstraction can be viewed as a set of points and intersecting line segments.

7. A problem and a solution Error in resolving the point of intersection

8. A solution Define an envelope as a collection of grid points that are immediately above, below or on the line

9. A formal definition of realm based SDT’ s It is organized as a series of layers Layer 1: Robust geometric primitives Layer 2: Realms and realm based primitives Layer 3: Spatial algebra primitives Layer 4: Rose operations

10. Layer 1: Robust geometric primitives Consists of a finite discrete space N X N Points and line segments over this space Simple predicates and operations over them Defines an N point and an N segment Primitives such as ‘meet’, ’overlap’, ’intersect’, ’disjoint’, ’on’, ‘in’ and ‘intersection’ are defined.

11. Layer 1 Continued: We define a Realm over N It represent a finite, user-defined set of points and non-intersecting line segments over a discrete domain. We define R points and R segments for the realm We have an added restriction over N R segments don’t intersect and they don’t overlap

12. Realm based structures An R cycle is a cycle in the graph interpretation of a realm It is a set of R segments Relations between a N point p and R cycle c p on c, p in c, p out c c partitions the set of all N points into Pin(c ) Pon(c ) Pout(c )

13. Realm based structures Relations between two cycles c1 and c2 Possible relationships between two R cycles

14. Realm based structures C1 and C2 adjacent C1 and c2 meet c1 c2

15. Realm based structures Ways an R segment s can lie within an R cycle c Ways an R point p can lie within an R cycle c

16. Realm based structures An R face f, consists of a cycle c and a possibly empty set of R cycles h such that All cycles in h are edge-inside c All cycles in h are edge-disjoint with respect to each other Segments in f can either form a cycle c or one of the cycles in h and no other cycle This is necessary in order to achieve closure under operations

17. Realm based structures

20. An R-block b connected sub graph in the graph interpretation of a realm

21. Realm based Spatial data types A flat view: A structured view:

22. Examples of Spatial values

23. Second order signature ROSE algebra is a system of spatial data types together with operations between these types. Many of the operations are applicable to several types. Hence we need a framework and notations to describe polymorphic operations. A system of several sets and functions between these sets is called a many sorted algebra A many sorted signature describes the syntactic aspect of a many sorted algebra It consists of two sets of symbols called sorts and operators Second order signature is a system of two coupled many sorted signatures where the top level offers kinds as sorts and type constructors as operators.

24. Second order signature A second, bottom level signature uses the types defined by the top level signature as sorts.

25. The type of a partition The partition is a disjoint subdivision of the plane into regions with associated attributes. We would like to test conditions such as adjacency or overlay The kind regions area-disjoint contains all types whose carriers are sets of regions values such that any two distinct values of the type are area-disjoint.

26. The object model interface The SDT definition should be based on an abstract interface to the DBMS data model. There are basic concepts and operations in the object model that are needed to define the ROSE algebra There are constructs and notations needed to embed the ROSE algebra into the query language that is to use the ROSE algebra

27. Object model interface concepts Object types/ classes Collection of objects Functions for accessing values from objects Data types int, real, bool A pool of names Object aggregation function Object extension function

28. Concept for embedding ROSE algebra into DBMS query lang Denote a spatial data type value Denote a collection of objects together with an attribute Extend objects by derived values Allow naming of an SDT value or a new attribute Offer a grouping operation

29. The Rose algebra: Robust spatial extension

30. Spatial predicates These operations compare to spatial values with respect to their topological relationships and return a Boolean value

31. Operators returning Spatial data type values

32. Spatial operators returning numbers

33. Spatial operators on Sets of objects

34. Integrating with a DBMS query language

35. Example query List the names and the land use of districts which are neighbors with the same land use. Which states are enclosed by which other states?

36. Conclusion Implementation of the ROSE algebra. Invariance under redrawing. Objects and operations violating realm closure.