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SPATIAL INDEXING

PREDICTED TIME : 30 MIN SLIDE NOM:41

SUBJECT:

SPATIAL INDEXING EMAIL

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CONTENTS…

What is an index? Main index types… Point access methods(PAMS)… Spatial access methods(SAMS)… R-TREE issues Summary References

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What is an index?

Concept of an index: “auxiliary file to search a data file“

index records have •key value •address of relevant data sector (arrows) In general indices improve access time but may cause deletion And insertion data items can increase processing time!

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An index in general…

Assume we have some files… In computer science 4 ways are exist:

1-pile records 2-fixed size records 3-sequential records 4-indexed sequential

No meaningful sequence worst access time best insertion time

better access time(fixed size) Still best insertion time Ordered by a key sequence value good access time Low speed insertion time(to keep sequence order)

Include the primary data area and an indexed area good access time Good insertion time(may need to refer indexed area)

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Main index types

Point access methods(PAMs) i. Grid File ii. kd-tree based

iii. Z-ordering iv. B-tree

Spatial access methods(SAMS) i. R-TREE(R*-tree, Hilbert R-tree)

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Point access methods(PAMS)

PAM: index only point data Multidimensional Hashing(grid files)

Hierarchical (tree-based) structures (kd-tree)

Space filling curve(z-ordering or quad-tree)

The problem

Given a point set and a rectangular query, find the points enclosed in the query

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PAMS>Grid File

Idea: Use a grid to partition the space each cell is associated with one page

Exponential growth of the directory implementation

Grid array: 2 dimensional array with pointers to buckets G(0,…, nx-1, 0, …, ny-1) Linear scales: Two 1 dimensional arrays that used to access the grid array X(0, …, nx-1), Y(0, …, ny-1)

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PAMS>Grid File>Example

Linear scale X

Linear scale

Y

Grid Directory

Buckets/Disk

Blocks

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PAMS>KD-TREE

kd-tree is a main memory binary tree for indexing k-dimensional points

Storing in external memory is tricky At each level we use a different dimension

kd-tree is not necessarily balanced

A

B C

D E

x=5

y=6

x=6

Y=3

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X=5

y=5 y=6

x=3

y=2

x=8 x=7

X=5 X=8

X=7

Y=6

Y=2

Y=5

X=3

PAMS>KD-TREE>Example

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Map points from 2-dimensions to 1-dimension

Basic assumption: Finite precision in the representation of each co-ordinate, K bits

(2K values)

The address space is a square (image) and represented as a 2K x 2K array

Each element is called a pixel

PAMS> Z-ordering

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PAMS> Z-ordering

Impose a linear ordering on the pixels of the image 1 dimensional problem

00 01 10 11 00

01

10

11

A ZA = shuffle(xA, yA) = shuffle(“01”, “11”)

= 0111 = (7)10

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PAMS> Z-ordering for Regions

Break the space into 4 equal quadrants: level-1 blocks For a level-i block: all its pixels have the same prefix up to

2i bits; the z-value of the block

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Object is recursively divided into blocks until: Blocks are homogeneous Pixel level

Quadtree: ‘0’ stands for S and W ‘1’ stands for N and E

00 01 10 11 00

01

10

11

SW SE

NW

NE

11 00 10

01

11 1001 1011

PAMS> Quad tree

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Quad tree(2D)

00 01 10 11

00 01 10 11

00 01

10 11

00 01

10 10

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Quad-tree(3D) or Oc-tree

010 011 100 101 000 001 110 111

010 011 100 101 000 001 110 111

000 001 010 011 100 101 110 111

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PAMS>B-TREE

Good access time Reasonable sequential read on the sequence key Insertion and deletion do not damage the balance

of the tree

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Spatial access methods(SAMS)

Indexes for spatial data that have extend (not only point data)

Use only Minimum Bounding Rectangles –MBRs (filtering)

R-tree (Guttman, 1984) is the prominent SAM Implemented in Oracle, Postgres, Informix

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2-dimensional version of the B-tree!

SAMS>R-TREE

Can store: i. a set of polygons (regions of a subdivision) ii. a set of polygonal lines (or boundaries) iii. a set of points iv. a mix of the above

Stored objects may overlap

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SAMS>R-TREE

Originally by Guttman, 1984 Dozens of variations and optimizations since Suitable for windowing, point location and intersection

queries

Every internal node contains entries (rectangle, pointer to child node) All leaves contain entries (rectangle, pointer to object) in database or file Rectangles are minimal bounding rectangles (MBR)

Definition R-tree:

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SAMS>R-TREE>Grouping of objects

Objects close together in same leaves ⇒ small rectangles ⇒ queries descend in only few subtrees

Group the child nodes under a parent node such that small rectangles arise

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Heuristics for fast queries

Small area of rectangles Small perimeter of rectangles Little overlap among rectangles

Good access time

Reasonable amount of insertion and deletion does not cause tree reconstraction

Height number of deletion and insertion requires restruction of tree

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SAMS>R-TREE>Example

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SAMS>R-TREE>Example

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SAMS>R-TREE>Example

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SAMS>R-TREE>Example

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SAMS>R-TREE>Example

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SAMS>R-TREE>Example

point containment query

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SAMS>R-TREE>Example

point containment query

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SAMS>R-TREE>Searching

Q is query object (point, window, object) For each rectangle R in the current node,

if Q and R intersect,

search recursively in the subtree under the pointer at R (at an internal node)

get the object corresponding to R and test for intersection with R (at a leaf)

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SAMS>R-TREE>Inserting

Determine minimal bounding rectangle (MBR) of new object When not yet at a leaf (choose subtree):

i. determine rectangle whose area increment after insertion of R is smallest

ii. increase this rectangle if necessary and insert R

At a leaf: i. if there is space, insert, otherwise Split Node

New MBRs

Split Node

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SAMS>R-TREE>Deletion

Find the leaf (node) and delete object; determine new (possibly smaller) MBR

If the node is too empty (< m entries): i. delete the node recursively at its parent

ii. insert all entries of the deleted node into the R-tree

Note: Insertions of entries/sub-trees always occurs at the level where it came from

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SAMS>R-TREE>Deletion>Example

Should deleted www.geobook.ir

SAMS>R-TREE>Deletion>Example

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SAMS>R-TREE>Deletion>Example

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SAMS>R-TREE>Deletion>Example

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SAMS>R-TREE>Deletion>Example

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R*-TREES!

Is there any other property that can be optimized? R*-tree Yes!

Optimization Criteria: i. Area covered by an index MBR

ii. Overlap between directory MBRs

iii. Margin of a directory rectangle

iv. Storage utilization

Sometimes it is impossible to optimize all the above criteria at the same time!

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REFRENCES…

H. V. Jagadish: Linear Clustering of Objects with Multiple Atributes. ACM SIGMOD Conference 1990: 332-342

Walid G. Aref, Hanan Samet: A Window Retrieval Algorithm for Spatial Databases Using Quadtrees. ACM-GIS 1995: 69-77

A. Guttman (1984). R-trees: A dynamic index structure for spatial searching. Proc. A CM SIGMOD Int. Conf. on Management of Data, pages 47-57.

Oracle Spatial 10g White Paper (2006). Oracle Spatial Quadtree Indexing, 10g Release 1 (10.1).

بخش “ پايگاه داده مکانی“جزوه کالسی درس1388.حکيم پورفDATA INDEXING دوره .دانشگاه صنعتی کرمان GISکارشناسی ارشد

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