1. Spatial Databases - Indexing
Spring, 2015
Ki-Joune Li

2. What is Indexing ? Indexing : Fight against TIME Example
Suppose that you have a Hamlet, and
you want to know the name of Hamlet’s father.
Without Index : Full (Sequential) Scan of the book
With Index : Direct Access to the Page
Hamlet

3. Some Constraints Modern Database What should we do ?
Very Huge Volume : e.g. several peta bytes
Storage on Disk
Inevitable
But slow (cf. main memory) : msec. vs. nano sec.
Even in Main Memory Database System
What should we do ?
Minimize the number of Disk Access

4. The Objective of Indexing
Database
in Disk
Query Condition
Disk Address
(Block Number)
Indexing

5. Classification of Indexing
According to the type of query and data
Alphanumeric query
Image
Spatial
What is the nearest post office to the Louvre Museum ?
Spatial predicate
Spatial Index
Database
in Disk
Spatial Query
Disk Address
(Block Number)

6. Spatial Query Sophisticated Types of Spatial Query
One Scan Query
Region Query : Containment, Intersection
K-Nearest Neighbor Query
Multi-Scan Query : Join
Spatial Join
Distance Join
Spatial Query Processing
Tightly coupled with Spatial Indexing Method

7. Spatial Processing Strategy
Filtering and Refinement Strategy
Index
Verification of Geometry
Complete
Data
Candidates
Spatial Query
Result
Simplification of Geometry
Filtering
Refinement
1. More Light Index : e.g. < 1 M bytes
2. Remove Unnecessary Disk Accesses

8. Classification of Spatial Indexing Methods
Hashing and Indexing
Index (in wide sense)
Hashing, Indexing (in narrow sense)
Space Decomposition vs. MBR
Decomposition of a space : Whole Space
Bounding Rectangle : Only Interesting Area
Dimensionality
No Transformation
to Higher Dimension
To Lower Dimension : Linearization

9. Indexing vs. Hashing Hashing Indexing (in narrow sense)
1. b = h(r.key)
2. Store(r, b)
Block number is determined by hashing function or mechanism
Only for primary index
Search by a hashing function
Indexing (in narrow sense)
1. b = Store(r )
2. Insert(B, (r.key, b) )
Block number is independent from indexing mechanism
For primary or secondary index
Search by a data structure called index

10. Decomposition vs. Bounding Region

11. Decomposition Methods
Grid File : An Extension of Hashing to 2-D
Variation
Fixed Grid
Grid File
Multi-Level Grid File
Hierarchical Data Structure
KD-tree
Quadtree
skd-tree
etc.

12. Fixed Grid Most Simple Method Minimum Data for Hashing 1 Disk Page
Query Window 20 30 40 50 10
1. Find intersecting grids
2. Find corresponding blocks
3. Read objects from the blocks
4. Refinement

13. Problems of Fixed Grid Only for Point Object Large Dead Space
Object with measure : duplicated storage
Degrade performance
Large Dead Space
Causes Unnecessary Disk Accesses
Not very Flexible
On Distribution
Query Window 20 30 40 50 10

14. Grid File To overcome problems of Fixed Grid
Reduce Dead Space within a cell
Increase Blocking Factor
Query Window
Directory
Grid
Boundary
Block# A
(0,0),(15,20)
Page 0 B
(15,0),(30,20)
Page 1
. . . I
(30,28),(50,40)
Page 15 40 28 20 15 20 30 50

15. Blocking Factor A Key Factor on performance How to increase Bf ?
Number of Objects in a Disk Block
Number of Disk Accesses
How to increase Bf ?
Increase Block Size : not always possible
Packing

16. Problems of Fixed Grid Only for Point Object Still Large Dead Space
Large Size of Directory
Directory
Grid
Boundary
Block# A
(0,0),(15,20)
Page 0 B
(15,0),(30,20)
Page 1
. . . I
(30,28),(50,40)
Page 15

17. Hierarchical Decomposition
To overcome the size of directory in Grid File
Hierarchical Structure of Directory
Acceleration of Search

18. KD-tree : Index Extension of Binary Tree to K-Dimension (K=2 for us)
Example : suppose Bf =3
A Directory B E
x=20
y=20
y=10
x=30
=<
< 15 A B E 10 D
Each leaf node points to the disk page A C C D 20 30

19. KD-tree : Search B E x=20 =< < y=20 y=10 15 A A B x=30 E 10 D A

20. Weak Points of KD-tree Only for Point Objects Dead Space
How to Store Tree Structure on Disk Space
Blocking Problem
Widely used for main memory index
Rarely used for disk resident index
Unbalanced Tree
Zipf’s Law (or 80/20 law)
Most events are concentrated
Leads highly skewed tree B E D A C

21. Quadtree Extension of KD-tree : KD-tree : binary split
Quadtree 4-way equi-split instead
Example : Bf =3 C D F A F
Each leaf node points to the disk page B E B C D E G H I J H J G A I

22. Weak Points of Quadtree
Same Problems of KD-tree
In addition to the lack of flexibility
Only for Point Objects
Dead Space
How to Store Tree Structure on Disk Space
Blocking Problem
Widely used for main memory index
Rarely used for disk resident index
Unbalanced Tree
Zipf’s Law (or 80/20 law)
Most events are concentrated
Leads highly skewed tree

23. Point Quadtree A Simple Variation of Quadtree
Specification of Partition Point instead of equi-split
More Adaptive to the distribution of objects
Less Skewed
(10,20)
(5,25) A
(5,25) F
(35,10)
(10,20) B C D E G H I J
(35,10)

24. Linear Quadtree : Space-Filling Curve
Quadtree but another representation
Linearization by Space-Filling Curve 11 6 13
N-order
Hilbert
Column-wise
Linearize points(or cells) by their peano-key

25. Linear Quadtree Example : N-order curve
Computation of Peano-Key : Bit-Interleaving 11
1. Binary representation of coordinates (10,01) 10
2. Bit-Interleaving x = y = 01
Peano key = 00
= 9 00 01 10 11

26. MBR Methods MBR (Minimum Bounding Box) R-tree and its variants
Two dimensional geometric simplification of objects
Not the Whole space,
only in the region occupied by objects
R-tree and its variants
(X1max, X2max )
(X1min, X2min)

27. R-tree Construction of R-tree : Sequence of Insertion Upward SplitB C E A H F G I B C D D E F G H I J K J K A
Leaf node points to the disk page
2-D Objects
Construction of R-tree : Sequence of Insertion
Upward Split

28. Splitting in R-tree Split MBR in the case of overflow
Line sweeping : Compare Cost-X and Cost-Y 
New MBR
Splitting Line
Cost Measure
Area,
Perimeter
Overlapping Area

29. R-tree : Query ProcessingB C E A A H
Query
Region W F B B C C D D I G D E E F F G G H H I I J J K K J
Candidate K A
Read its exact geometry from databaseCandidate
Refinement
Sample :

30. Strength of R-tree For point and non-point Objects
Good for non-uniform distribution
Paged Tree
Hierarchical Structure but Balanced
Less Dead Space than Decomposition Methods A B C D E J K C D H I E F G

31. Weak Points of R-tree : Overlapping Area
Overlapping : False Matching
Query Region A B C J D E F K G H I L M A B G C L H K J D I K E F M
False Matching : Visit unnecessary node
Performance Degradation

32. Weak Points of R-tree : Dead Space
Query Region A B G C L H J D I E K F M
At least one visit at this node (K) even though there is nothing

33. Weak Points of R-tree : Bad Split
Good Split
Bad Split
1. Make them as COMPACT as possible
2. Preserve spatial proximity as possible

34. Improvement of R-tree Minimize Or Make it more COMPACT
Overlapping area
Dead Space
Or Make it more COMPACT
Preserve Spatial Proximity
Two approaches
Packing (or Bulk Loading)
Good Split or Insertion Strategies

35. R*-tree : An Improvement of R-tree
Re-Insertion Strategy on Overflow
Overflow
Newly Inserted Object
Delete and Re-Insert this

36. R*-tree : An Improvement of R-tree
Re-Insertion Strategy on Overflow
More Compact
Re-Inserted Object

37. R*-tree : An Improvement of R-tree
Compact
Small Overlapping Area
Small Sum of MBR area or perimeters
Small Dead Space
Stable : Not very affected by the order of insertions
The most widely used spatial indexing method

38. Packing R-tree : Improvement of R-tree
Preprocessing for making R-tree more compact
Hilbert R-tree
STR (Sort-Tile Recursive)
Uniformization
Instead of Sequential Insertions

39. Hilbert Packing Hilbert Curve A Space Filling Curve
Linearize spatial objects by their peano-key
N-order
Hilbert
Column-wise

40. Hilbert Packing Hilbert Packing Example: Bf =3
Sort objects by Hilbert key
Packing by round-robin way
Maximize storage utilization
Minimum Dead Space, and Sum of MBR area
Example: Bf =3

41. STR (Sort-Tile Recursive)
Basic idea : “tile” the data space using vertical slices
r : number of rectangles
n : blocking factor
P ( leaf node page ) =
Example
Suppose r = 25, n =3
nTile = 9,
nV = 3, nH = 3

42. Comparison : Hilbert Packing vs. STRHP
Large Objects
STR HP
Points
STR

43. Uniformization Non-Uniform Distribution Uniformization Technique
Negative Effect on the performance
But in real applications : Non-Uniform
Uniformization Technique
Step 1 : Transform Non-Uniform data to Uniform by STR
Step 2 : Apply R-tree (or Fixed Grid)
Step 3 : Transform Query Region
Strength
High Storage Utilization
Very Simple and Good Performance

44. Uniformization
Non Equi-Width
Equi-Width
1. Area of each cell : identical 2. Number of objects within each cell : almost identical

45. Uniformization : Example
By Delaunay Triangulation
By STR
Original

46. Uniformization : Example
Original
By STR

47. Query Processing by R-tree : Nearest Neighbor
Query Point
Searching Space
2nd Distances in 2-D
Minimum

48. Query Processing by R-tree : Nearest Neighbor
Branching
Branching
Pruning
Minimum

49. Transformation to Higher Space
Transformation to Higher Dimension
Transform non-point object to point object
Reuse of spatial indexing methods (e.g. Grid File) applicable only to point objects to non-point objects
Example
Max C B B A  A C
Amin
Amax
Min

50. Corner Transformation
From 2-D to 4-D
1. Simplification by MBR
2. MBR ((Xmin, Ymin), (Xmax, Ymax)) to Point (Xmin, Ymin, Xmax, Ymax)
(Xmax, Ymax)
(Xmin, Ymin)

51. Query Processing for Corner Transformation : 1-D ExampleW
Query : Find Contained Objects
Max VI IV
III A V II A  I
Min
Amin
Amax
Region I : Wmax < Amin
Region II : W  A
Region III : Amax < Wmin
Region IV : Amin < Wmin, Amax < Wmax
Region V : Wmin < Amin, Wmax < Amax
Region VI : A  W

52. Transformation to Lower Dimension : Linear Quadtree
1. Simplification of Geometry
(22, 0)
2. Compute Peano Key with lower-left corner
(28, 1)
(23, 0)
3. If necessary, divide it and give peano key to each
4. Define the size of each piece according to the number of quadrants
4. Insert them into B-tree
5. Query Processing by B-tree
(0, 2)