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R-TreesIndex Structures for Spatial Data

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2301474R-Trees: Index Structures for Spartial Data2

Outlines Spatial Applications Requirements for indexing spatial data

Properties of R-trees

Insertion Deletion

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Spatial Applications Maps

Google Earth

Navigation Systems

Computer Graphics

Computer Aided Design

Automobile design

Interior design

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Requirements for Indexing Spatial Data

Support both point and region data.

Support data in 2-, 3-, 4-dimensional spaces.

Support spatial queries such as overlap, adjacent,

in, contain, to the north of,

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Concepts of R-Trees The minimum bounding

rectangle (MBR) of an object isthe smallest box in the n-dimensional space boundingthe object.

In an n-dimensional space, theindex of a set of spatial objectsis an n-dimensional boxbounding the set of objects.

Bounding boxes are used asindices in R-trees.

Similar to B+ trees, but indicesare allowed to be overlapped.

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Properties of R-Trees

An index record is composed of a bounding rectangle B, and

a pointer to a child node or a

pointer to a data record.

Each node contains at most Mindex records.

Non-root nodes contain between

M/2 and M index records.

The root node contains at least 2

index records.

R-trees are height-balanced.

R

R

P

PQ

S

Q S

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ExampleA B

C D

R

F GE

M N P QI J K LH

A

BC

D

F G

EH

IJ

KL

M

NPQ

R

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Query IA B

C D

R

F GE

M N P QI J K LH

A

BC

D

F G

EH

IJ

KL

M

NPQ

R

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Query IIA B

C D

R

F GE

M N P QI J K LH

A

BC

D

F G

EH

IJ

KL

M

NPQ

R

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Insertion

Choose a leafnode

Lto insertthe new record E.

Different criteria for choosing

the node.

IF there is a room in L: THEN insert E in L.

ELSE split L into L and L, with

records in L and E.

Add the index ofL in theparent ofL.

4 5

51 2 3 4

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Insertion

4 551 2 3 4

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Insert without SplitA B

C D

R

F GE

M N P QI J K LH

A

BC

D

F G

EH

IJ

K LM

NPQ

R

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Insert with SplitA B

C D

R

F GE

M N P QI J K LH

A

BC

D

F G

EH

IJ

K LM

NPQ

R

X Y

X Y

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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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SplittingA B

C D

R

F GE

N P QL M X Y

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SplittingA B

C D

R

F GE

N P QL MX Y

E

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SplittingA B

C D

R

F GE

N P QL MX Y

E

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SplittingA B

C D

R

F GE

N P QL MX Y

E

B

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Deletion

Find the leaf node L containing the record E to bedeleted.

Remove E from L. Set Q = {}.

WHILE (E

is not the root and has