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Jeremy Iverson & Zhang Yun
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Chapter 6 Key Concepts◦ Structures and access methods◦ R-Tree
R*-Tree
Mobile Object Indexing
Questions
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Indexes are used to efficiently locate data on hard disk
1D◦ Indexes that are based on one key value
B and B+-trees
2D◦ Indexes based on two key values
Ordered tilings
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Structures store data for efficient modification and querying
Types of data to store◦ Raster (Region quadtrees)◦ Point Object (2D trees)◦ Linear (PM quadtrees)◦ Collections of objects (R-trees)◦ Spherical (QTM region quadtrees)
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R-tree A balanced tree to index spatial objects Shape of objects is approximated by minimum bounding
rectangle Rectangles at any level may overlap
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R-tree review
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R-tree and its limitations
How to build R-tree Given a set of spatial objects, build a R-tree is based on heuristic R-tree is designed to minimize the area of containing rectanglesLimitations Heuristic of R-tree may cause much overlap Cause other problems like uneven distribution
Spatial objects set
One split method Another split method
R-tree prefer this
New index Better than R-tree performance Support Multi spatial object types (e.g. point, polygon)
Possible applications Support spatial query processing (e.g. online map service) Support imagine processing
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Motivations
More heuristics H1:The area covered by directory rectangles should be minimized H2:The overlap between directory rectangles should be minimized H3:Make bounding rectangles as square as possible H4:The storage utilization should be optimized—reduce height of tree
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R*-tree
Heuristics may conflictChoose best design from experiments
Spatial objects set
H1: area minimum
H2: overlap minimum
Insert new object Minimize the overlap Choose the entry in R*-tree whose rectangle needs least overlap
enlargement to include the new object
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R*-tree Operations
R-tree: minimize area enlargement
R*-tree: minimize overlap enlargement
Spatial objects set
R-tree
R*-tree (also R-tree) suffer from the sequences of insertions Reorganization of tree is necessary Compute the distance between the centers of their rectangles and the center
of the bounding rectangle, remove top k rectangle with maximum distance Invoke insert operation for removed rectangles
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R*-tree Reinsertion
Spatial Objects
Calculate distance
Remove object A
Reinsert object A
R*-tree highlights Use more heuristics, design validated from experiments Perform significantly better than R-tree
Limitations No concept for moving object Not designed for spatio-temporal objects
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R*-tree and its limitations
Naïve Approach◦ y(t)=vt+a◦ v: velocity◦ t: time◦ a: intercept
◦ Query is expressed as 2D interval [(y1q,y2q),(t1q,t2q)]
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Benefits◦ Intuitive representation
Drawbacks◦ Length of lines is infinite
A lot of redundancy High overhead for updates
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Time-Parameterized R-Tree◦ Actually extends the R*-tree
A moving object o is represented with◦ MBR◦ Velocity Bounding Rectangle (VBR) of the form
oV={oV1-,oV1+,oV2-,oV2+} ovi- represents the lower bound for velocity in
dimension i ovi+ represents the upper bound for velocity in
dimension i
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av={1,1,1,1} bv={-2,-2,-2,-2} cv={-2,0,0,-2} dv={-1,-1,1,1}
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*MBRs for non-leaf nodes are not required to always be minimum, only minimum at some time step.
TPR-Tree allows one to index and query moving objects
TPR-Tree creates index structures much worse than optimal [Tao et al.]◦ Thus, the TPR*-Tree is introduced, which
considers multiple paths when inserting an object into the index structure, creating an index much closer to optimal
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