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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