Spatial Data Types

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• Spatial Data Types
• and their
• Embedding in Query Languages (II)

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In-Class Presentation Part II
Padmavati Sridhar

• As Partial Course Requirement for
• Spatial Databases as a Foundation of
• Geographical Information Systems
• (CIS 6930,Fall 2004)
• Instructor Dr. Markus Schneider

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Objective

• To provide a detailed overview of the
• Modeling
• Design and
• Implementation
• approaches of Spatial data types and operations
• Emphasis Understanding the basic concepts

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Outline

• Introduction to Spatial Databases
• Spatial Modeling
• Data Models
• Modeling of Spatial Data Types (SDTs)
• Modeling of Spatial Operations
• Database Design and Query Examples
• Design
• Formal Definition Methods
• Implementation
• Data Structures for SDTs
• Algorithms for atomic SDT Operations

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Quick Recap

• Introduction to Spatial Databases
• Spatial Data and Querying
• User Classes
• SDBMS definition
• 3 tier Architecture
• Spatial Modeling
• Data Models Field and Object Based
• Modeling of Spatial Data Types (SDTs)
• Single Objects Fundamental Abstractions
  Point, Line, Region
• Spatially related Collections of Objects
  Partitions and Networks
• Modeling of Spatial Operations
• 3 Step Database Design
• Query Examples
• Design
• Formal Definition Methods
• Implementation
• Data Structures for SDTs
• Algorithms for atomic SDT Operations

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Introduction to Spatial Databases (Revised)

• What Is Spatial Data ?
• To extract information, data has to be processed
  with respect to a Spatial frame of Reference (Eg
  Latitude/Longitude )
• Data related to Space where Space can be
• Geographic Space
• Layout of a VLSI design
• Space representing the arrangement of chains of
  protein molecules
• Image Database Systems
• Include Analysis techniques to extract objects in
  space from Images and are prepared to store,
  manipulate and retrieve raster images as discrete
  entities
• Spatial Database System
• Is a Database system
• Offers SDTs in its model and Query Language
• Supports SDTs in its implementation, providing at
  least Spatial Indexing and efficient algorithms
  for Spatial Join

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Modeling Spatially Related Collections of Objects
Partition
Network

• Partition
• Set of Region Objects that are required to be
  disjoint
• Interest Adjacency (There exist pairs of Region
  Objects with a Common boundary)
• Use Thematic Maps
• Network
• Graph embedded into the Plane, consisting of
• Nodes A Set of Point Objects
• Edges A Set of Line Objects
• Use Map of Highways, Rivers

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Quick Recap

• Introduction to Spatial Databases
• Spatial Data and Querying
• User Classes
• SDBMS definition
• 3 tier Architecture
• Spatial Modeling
• Data Models Field and Object Based
• Modeling of Spatial Data Types (SDTs)
• Single Objects Fundamental Abstractions
  Point, Line, Region
• Spatially related Collections of Objects
  Partitions and Networks
• Modeling of Spatial Operations
• 3 Step Database Design
• Query Examples
• Design
• Formal Definition Methods
• Implementation
• Data Structures for SDTs
• Algorithms for atomic SDT Operations

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Conceptual Model Illustration

• ER Model
• 3 Entities Country, City, River
• 2 Relationships capital-of, originates-in

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Sample Data
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Spatial Query Types

• Simple SQL SELECT_FROM_WHERE examples
• Spatial analysis operations
• Unary operator Area
• Binary operator Distance
• Boolean Topological spatial operations - WHERE
  clause
• Touch
• Cross
• Using spatial analysis and topological operations
• Buffer, overlap
• Complex SQL examples
• Using spatial analysis and topological operations
• Aggreagate SQL queries
• Nested queries

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Spatial Query Examples (1)

• Using spatial operation in SELECT clause
• Query List the name, population, and area of
  each country listed in the Country table.
• SELECT C.Name,C.Pop, Area(C.Shape)AS "Area"
• FROM Country C
• Note This query uses spatial operation,
  Area().Note the use of spatial
• operation in place of a column in SELECT clause.

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Spatial Query Examples (2)

• Using Spatial Operation in WHERE clause
• Query Find the names of all countries which are
  neighbors of the United States (USA) in the
  Country table.
• SELECT C1.Name AS "Neighbors of USA"
• FROM Country C1,Country C2
• WHERE Touch(C1.Shape,C2.Shape)1
• AND C2.Name USA
• Note Spatial operator Touch() is used in WHERE
  clause to join Country table
• with itself. This query is an
  example of spatial self join operation

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Spatial Query Examples (3)

• Spatial Query with multiple tables
• Query For all the rivers listed in the River
  table, find the countries through which they
  pass.
• SELECT R.Name, C.Name
• FROM River R, Country C
• WHERE Cross(R.Shape,C.Shape)1
• Note Spatial operation Cross is used to join
  River and Country tables. This
• query represents a spatial join operation

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Spatial Query Examples (4)

• Using spatial operation in an aggregate query
• Query List all countries, ordered by number of
  neighboring countries.
• SELECT Co.Name, Count(Co1.Name)
• FROM Country Co, Country Co1
• WHERE Touch(Co.Shape,Co1.Shape)
• GROUP BY Co.Name
• ORDER BY Count(Co1.Name)

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Outline

• Introduction to Spatial Databases
• Spatial Modeling
• Data Models
• Modeling of Spatial Data Types (SDTs)
• Modeling of Spatial Operations
• Database Design and Query Examples
• Design
• Formal Definition Methods
• Implementation
• Data Structures for SDTs
• Algorithms for atomic SDT Operations

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Design Criteria for Spatial Data Modeling
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Closure Properties

• Closure
• The domains of SDTs (Point, Line, Regions) should
  be closed under Geometric Union, Intersection and
  difference
• Important The result of the geometric operation
  must be a well-defined object AND correspond to
  the definition of spatial data types

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Euclidean Geometry vs Discrete Geometric Bases

• Euclidean Space
• Number System is continuous and infinite
• Basis of Computational Geometry Algorithms
• Compute number systems are finite and discrete
• Leads to Numerical Robustness and Topological
  Correctness problems
• NEED Theory for modeling and design of spatial
  data that is compatible with the finiteness of
  computers
• Left to implementer - gt leads to errors in Query
  processing
• D moved to D due to rounding errors

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Numerical Robustness approaches

• Proposed Solution
• avoid computation of any new intersection points
  within geometric operations
• The Geometric basis is updated before treatment
  of numerical problems
• Approaches
• Simplicial Complexes
• Frank Kuhn 1986
• Egenhofer, Frank Jackson 1989
• Realms
• Güting Schneider 1993
• Schneider 1997

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Simplicial Complexes

• Based on Combinatorial Topology
• Topological relations are separately recorded and
  independent of metric positions
• For each dimension d, a d-simplex is a minimal
  object in that
• dimension
• - Rule Any d-simplex is composed of (d1)
  simplices of dimension d-1
• - Component of a simplex is called face (Edges
  and Vertices)

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Simplicial complex

• finite set of simplices such that the
  intersection of any two simplices is empty or a
  face

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Simplicial complex

• Two completeness principles
• - Completeness of Incidence the intersection of
  two k-simplices is either empty or a face of both
  simplices
• no line intersection at points which are not
  start or end points of the lines
• no two geometric objects may exist at the same
  location (geometry only recorded once)
• - Completeness of Inclusion every k-simplex is
  a face of a (k1)-simplex
• all point are end points of lines, all lines are
  boundaries of triangles, etc
• no isolated points, no lines which are not part
  of a boundary

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Simplicial complex

• Advantages
• maintenance of topological consistency
• approach fulfils closure properties
• Drawbacks
• - No spatial algebra has been defined on top of
  this approach
• - triangulation of space would lead to
  space-consuming representations of spatial
  objects
• - no treatment of numerical problems additional
  data structures needed to realize (at least
  imprecise) metric operations

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Realms

• description of the complete underlying geometry
  (all points and
• lines) of an application or application
  space
• A finite set of points and non-intersecting line
  segments defined over a grid such that
• each point and each end point of a segment is a
  grid point
• each end point of a segment is also a point of
  the realm
• no realm point lies within a segment
• any two distinct segments do neither properly
  intersect nor overlap

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Realms

• A realm is a spatially embedded planar graph
• All numerical problems are treated below the
  realm layer
• input application data that are sets of points
  and intersecting line segments
• output realmified data that have become
  acquainted with each other
• basic idea slightly distort/perturb both
  segments

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Realms

• Advantages
• definition of distinct SDTs on a common domain,
  guarantee of closure properties
• protection of geometric computation in query
  processing from problems of numerical
• robustness and topological correctness
• enforcement of geometric consistency of related
  spatial objects
• Disadvantages
• no SDT operations possible that create new
  geometries (leave the realm closure), e.g.,
  convex_hull, voronoi
• integration of realms into database systems
  somewhat difficult, propagation of realm updates
  to realm-based attribute values in database
  objects

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Data Model Independence- Use of ADTs

• Modeling aspects
• Separation of DBMS data model and
  application-specific data types/algebras
• Modularity, conceptual clarity
• Reusability of ADTs for different DBMS data
  models
• Extensibility of DBMS data models
• Implementation aspects
• Modularity, information hiding, exchange of
  implementations
• Employment of specialized methods (e.g.
  Computational Geometry for SDTs)
• Efficiency of data structures for data types and
  algorithms for operations

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Formal Definition Methods

• Need for Formal Definition Methods
• Users level
• Guaranteed clarity and consistency
• Designers level
• Better understanding of the complex semantics of
  spatial objects and operations
• Implementation Level
• formal specification of SDTs for a possible
  realization
• First step towards standardization of Spatial
  Data Types (SDTs)
• Formal definition of SDTs ? Formal definition of
  corresponding
• spatial operations
• consideration of the finiteness of computers and
  the problems of numerical robustness and
  topological correctness

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Formal Definition Methods

• Point Set Theory
• Point Set Topology
• Finite Set Theory
• Other Formal Approaches

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Point Set Theory

• Basic Assumption
• Space is composed of infinitely many points and
  contains a set of spatial objects
• Each spatial object can be regarded as the point
  set occupied by that object
• Point Set is a SET ? Set Operations can be
  applied to Point Sets

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Point Set Topology

• Same basic assumptions as point set theory
• Investigates properties that are independent of
  an underlying distance or coordinate measure
  (metric) and that are preserved under continuous
  topological transformations

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Outline

• Introduction to Spatial Databases
• Spatial Modeling
• Data Models
• Modeling of Spatial Data Types (SDTs)
• Modeling of Spatial Operations
• Database Design and Query Examples
• Design
• Formal Definition Methods
• Implementation
• Data Structures for SDTs
• Algorithms for atomic SDT Operations

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SDT Implementation

• Goal
• Implementation of a spatial type system (spatial
  algebra) so that it can be integrated into a DBMS
  (query processing, storage management, user
  interface, etc.), fulfilling the design criteria
• Tools
• Data Structures and Algorithms
• Data Structures Representation for the SDTs
• Algorithms Implementation of Atomic SDT
  Operations

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Representing SDT Values

• The representation of a value of a SDT (e.g
  Region) has to be simultaneously be compatible
  with 2 views
• Database System view
• Spatial Algebra view

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DBMS Perspective

• The SDT Representation
• Is the same as that of attribute values of other
  types with respect to generic Operations
• Can have varying and possibly very large size
• resides permanently on disk and is stored in one
  page or a set of pages
• can efficiently be loaded into main memory, where
  it is given as a value of some variable
  (typically, a pointer variable) to the procedures
  implementing operations of the spatial algebra
• offers a number of type-specific implementations
  of generic operations needed by the DBMS

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DBMS Perspective

• To fulfill DBMS Requirements SDT Representation
  must
• Be a Paged Data Structure
• Consist of a single contiguous byte block fitting
  into a page, or a large byte-block cut into
  page-sized pieces - For efficient loading and
  storing on disk
• Large value Split into small
• info part summary info DBMS object
  representation
• Exact Geometry part Long sequence of vertices
  logical pointer from info part to page sequence
• Generic Approximations are needed

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Spatial Algebra implementation Perspective

• The SDT Representation
• is a value of some programming language data
  type, e.g. region, is some arbitrary data
  structure which is possibly quite complex,
• supports efficient computational geometry
  algorithms for spatial algebra operations,
• is not geared only to one particular algorithm
  but is balanced to support many operations well
  enough.

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Spatial Algebra View

• representation contains approximations (e.g.,
  bounding box) in the info part
• representation contains stored values of unary
  functions (e.g., area, perimeter, length,
• number of components, etc.) in the info part
• representation contains plane sweep sequence in
  the geometric part

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Implementation of SDT Operations

• Should use efficient algorithms from
  Computational Geometry
• Precheck on Approximations
• Stored function look up
• Plane Sweep

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Conclusion

• Conceptual and Detailed Overview of
• Spatial Modeling
• Design
• Implementation
• Basis for further discussions and development of
  the concepts of Spatial Databases

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• Discussion

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• Thank You