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R-Trees
Index Structures for Spatial Data
Outlines
Spatial Applications
 Requirements for indexing spatial data
 Properties of R-trees
 Insertion
 Deletion
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Spatial Applications
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Maps
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Google Earth
Navigation Systems
Computer Aided Design
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Automobile design
Interior design
Computer Graphics
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Requirements for Indexing Spatial Data
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Support both point and region data.
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Support data in 2-, 3-, 4-dimensional spaces.
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Support spatial queries such as overlap, adjacent,
in, contain, to the north of, …
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Types of Spatial Data
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Point Data
Points in a multidimensional space
 E.g., Raster data such as satellite imagery, where
each pixel stores a measured value
 E.g., Feature vectors extracted from text
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Region Data
Objects have spatial extent with location and
Boundary
 DB typically uses geometric approximations
constructed using line segments, polygons, etc.,
called vector data.
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II. Spatial Databases
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Types of Spatial Queries
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Spatial Range Queries
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Nearest-Neighbor Queries
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Find the 10 cities nearest to Madison
Results must be ordered by proximity
Spatial Join Queries
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Find all cities within 50 miles of Madison
Query has associated region (location, boundary)
Answer includes ovelapping or contained data regions
Find all cities near a lake
Expensive, join condition involves regions and proximity
II. Spatial Databases
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Applications of Spatial Data
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Geographic Information Systems (GIS)
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Computer-Aided Design/Manufacturing
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Store spatial objects such as surface of airplane
fuselage
Range queries and spatial join queries are common
Multimedia Databases
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E.g., ESRI’s ArcInfo; OpenGIS Consortium
Geospatial information
All classes of spatial queries and data are common
Images, video, text, etc. stored and retrieved by content
First converted to feature vector form; high
dimensionality
Nearest-neighbor queries are the most common
II. Spatial Databases
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Single-Dimensional Indexes
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B+ trees are fundamentally single-dimensional indexes.
When we create a composite search key B+ tree, e.g., an index
on <age, sal>, we effectively linearize the 2-dimensional space
since we sort entries first by age and then by sal.
Consider entries:
<11, 80>, <12, 10>
<12, 20>, <13, 75>
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B+ tree
order
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Concepts of R-Trees
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The minimum bounding
rectangle (MBR) of an object is
the smallest box in the ndimensional space bounding
the object.
In an n-dimensional space, the
index of a set of spatial objects
is an n-dimensional box
bounding the set of objects.
Bounding boxes are used as
indices in R-trees.
Similar to B+ trees, but indices
are allowed to be overlapped.
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Properties of R-Trees
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An index record is composed of
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Each node contains at most M
index records.
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a bounding rectangle B, and
a pointer to a child node or a
pointer to a data record.
Non-root nodes contain between
M/2 and M index records.
The root node contains at least 2
index records.
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R-trees are height-balanced.
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Example
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Query I
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Query II
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Insertion
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Choose a leaf node L to insert
the new record E.
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IF there is a room in L:
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Different criteria for choosing
the node.
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THEN insert E in L.
ELSE split L into L and L’, with
records in L and E.
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Add the index of L’ in the
parent of L.
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Insertion
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Insert without Split
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Insert with Split
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Splitting a Node
Splitting can be propagated up the tree.
 If the root node is split, a new root node is created.
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Splitting
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Splitting
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Splitting
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Splitting
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Deletion
Find the leaf node L containing the record E to be
deleted.
 Remove E from L.
Set Q = {}.
 WHILE (E is not the root and has < m elements)
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Remove the pointer to E from its parent node.
 Adjust the bounding rectangle of the parent of E.
 Set Q = Q U {E}, E=parent of E.
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For each node N in Q
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Re-insert all elements in N in the correct level.
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How to split
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Variations of R-trees
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