1. Spatial Indexing
Kristian Torp
Department of Computer Science
Aalborg University
people.cs.aau.dk/˜torp
torp@cs.aau.dk
November 19, 2015
daisy.aau.dk
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 1 / 54
3. Learning Goals
The Problem
Existing indexes good for scalar values, e.g., 42 and ’Hello, World’
Existing indexes not good complex data, e.g., geometries and time
series
Quiz
Which index structures do you know?
Goals
Overview of spatial indexing techniques
Understand how spatial indexing different from “plain” indexing
Understand the basic spatial indexing techniques
Be able to construct a spatial index on PostgreSQL
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 3 / 54
6. Spatial Indexing
Purposes of Spatial Index Structure
Should speed-up the most typical spatial queries
Should not take up too much space
Should efficiently support modifications
Assumptions
Spatial query only covers small part of the “world”
Spatial data is queried more often then modified
Note
Indexing necessary to make queries run faster
Hashing does not work well for spatial data types
You can index spatial data using conventional (B+-tree) indexes
It is often slow!
Will see how can be used with space-filling curves
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 6 / 54
7. Overview of Spatial Indexing Techniques
Techniques
Space-filling curves (a set of approaches)
Old wine, recycled bottles
R-trees (a large family of index structures)
A generalization of the B+
-tree
Quadtrees (a family of index structures)
Old wine, new bottles
Other Domains
Image processing, e.g., medical and biometrics
Image recognition, e.g., letters and books
Gaming, e.g., for the terrain, buildings, and roads
Chip design, e.g., integrated circuits, think AMD, ARM, and Intel
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 7 / 54
8. Create Spatial Index
Example (Table with Spatial Data Column)
create table landscape (
geo name varchar (20) primary key ,
geo geometry not n u l l
) ;
Example (Create B+
-tree Index on PostgreSQL)
create index landscape geo name
on landscape ( geo name ) ;
Example (Create R-tree Index on PostgreSQL)
create index landscape geo
on landscape using g i s t ( geo ) ;
Note
GIST = Generalized Search Trees = R-tree = spatial index
From SQL interface not a big think!
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 8 / 54
13. The Basic Idea: SFC
Example (Division of World)
0 1 2 3 4
0
1
2
3
4
0 1
23
4
5 6
7 8
9 10
11
1213
14 15
a
14. The Basic Idea: SFC
Example (Division of World)
0 1 2 3 4
0
1
2
3
4
0 1
23
4
5 6
7 8
9 10
11
1213
14 15
a
b
15. The Basic Idea: SFC
Example (Division of World)
0 1 2 3 4
0
1
2
3
4
0 1
23
4
5 6
7 8
9 10
11
1213
14 15
a
b
c
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 10 / 54
16. SFC for Indexing Spatial Objects
Example (Space-filling Curve)
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0
1
2
3
4
0 1
23
4
5 6
7 8
9 10
11
1213
14 15
a
b
c
ObjectName Idx Coor
a 10 (3.7, 3.7)
b 7 (1.2, 2.7)
c 1 (ref)
c 2 (ref)
c 13 (ref)
c 14 (ref)
Variations
This is a Hilbert space-filling curve
Index and spatial data typically split in two tables
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 11 / 54
17. Spatial Query using SFC
Example (Space-filling Curve)
0 1 2 3 4
0
1
2
3
4
0 1
23
4
5 6
7 8
9 10
11
1213
14 15
a
b
c
ObjectName Idx Coor
a 10 (3.7, 3.7)
b 7 (1.2, 2.7)
c 1 (ref)
c 2 (ref)
c 13 (ref)
c 14 (ref)
Example (Spatial Query is Plain SQL Query)
select d i s t i n c t ObjectName
from t a b s f c h i l b e r t −− table where s p a t i a l index stored
where idx between 0 and 3
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 12 / 54
18. Hilbert, Peano, and Sierpinski SFCs
Example
Hilbert Peano Sierpinsk
Order= 4 Order = 3 Order = 3
Note
There are several variations of these SFCs
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 13 / 54
19. SFC Indexing
Example
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0
1
2
3
4
0 1
23
4
5 6
7 8
9 10
11
1213
14 15
x
y
z
v
ObjectName Idx
x ?
y ?
z ?
v ?
Note
Point, linestring, and polygon supported
Best support for points
Why?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 14 / 54
20. Motivation for using SFCs
Why?
Reuse B+-tree index structure
Reuse B+-tree concurrency algorithm
Extremely complicated to make efficient
Some operations can be very efficiently implemented
When first in main memory
Add spatial index support to a “plain” DBMS
Note
Space-filling curves can be combined with quadtrees
Space-filling curves can be combined with R-tree
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 15 / 54
21. 2D Hilbert Curves of Different Orders
Example (Order 2 to 7)
Large Geometries
How many index values for a large geometry?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 16 / 54
23. Summary: SFC
Note
Space division
Reduces from nD/3D/2D to 1D (integers)
The numbers are indexed using a B+
-tree
There are many variations of space-filling curves
Hilbert, Moore, Peano, and Sierpinksi
Very simple to implement
Main disadvantages: Spatial close, index distant
Concurrency control already implemented
Not optimal when index must be stored on disk (block device)
Quiz
Did you notice anything about the dimension sizes?
Can you give a reason why?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 18 / 54
26. Examples of MBRs
Example (MBRs for Various Shapes)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
Note
Four coordinates to index all spatial objects in 2D
Leads to false positives
27. Examples of MBRs
Example (MBRs for Various Shapes)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
Note
Four coordinates to index all spatial objects in 2D
Leads to false positives
28. Examples of MBRs
Example (MBRs for Various Shapes)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
Note
Four coordinates to index all spatial objects in 2D
Leads to false positives
29. Examples of MBRs
Example (MBRs for Various Shapes)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
Note
Four coordinates to index all spatial objects in 2D
Leads to false positives
30. Examples of MBRs
Example (MBRs for Various Shapes)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
Note
Four coordinates to index all spatial objects in 2D
Leads to false positives
31. Examples of MBRs
Example (MBRs for Various Shapes)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
Note
Four coordinates to index all spatial objects in 2D
Leads to false positives
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 21 / 54
32. Insert
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1
2
A
Note
Assuming maximum 3 values per node
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 22 / 54
33. Insert
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1
2
3
A
Note
Assuming maximum 3 values per node
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 22 / 54
34. Insert
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1
2
3
4
A
B
Note
Assuming maximum 3 values per node
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 22 / 54
35. Insert
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1
2
3
45
A
B
Note
Assuming maximum 3 values per node
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 22 / 54
36. Insert
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1
2
3
45
6A
B
Note
Assuming maximum 3 values per node
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 22 / 54
37. Insert
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1
2
3
45
6
7
A
B C
Note
Assuming maximum 3 values per node
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 22 / 54
38. Quiz: Good MBR?
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1 2
3
A
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1 2
3 A
A B
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1 2
3
A
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1 2
3
A
C D
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 23 / 54
39. Delete
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
1
2
3
45
6
7
A
B C
Note
Assuming minimum 2 values and maximum 3 values per node
Delete 1 ⇒ shrink A
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 24 / 54
40. Delete
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
2
3
45
6
7
A
B C
Note
Delete 6
Underflow in A merge with other MBR at same level
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 24 / 54
41. Delete
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
2
3
45
7
B
C
Note
Delete 3
Underflow in B merge with other MBR at same level
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 24 / 54
42. Delete
Example (R-tree)
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
2
45
7
B
C
Note
There are many details
Picking good candidate for merge is hard
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 24 / 54
43. A Complete R-tree
Example (R-tree)
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0
1
2
3
4
5
6
7
8
9
10
1
2
3
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
44. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
45. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
46. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
47. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
48. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
49. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
50. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I
10
11
12
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
51. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I
10
11
12
D
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
52. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I
10
11
12
D
13
14 15
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
53. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I
10
11
12
D
13
14 15
E
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
54. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I
10
11
12
D
13
14 15
E
16
17
18
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
55. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I
10
11
12
D
13
14 15
E
16
17
18
F
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
56. A Complete R-tree
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
A
4
5
6
B
7
8 9
C
I
10
11
12
D
13
14 15
E
16
17
18
FII
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 25 / 54
57. Clustering of Rectangles
Example (Clustering One)
a
b
c
d
Example (Clustering Two)
a
b
c
d
Questions
Which clustering of rectangles is best?
What are the evaluation criteria?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 26 / 54
58. R-tree Node Properties
Parameters
M is maximum fanout and determined by page size
m is minimum fanout
Typically 0.4-0.5 of M
R-tree Properties
The root node has minimum two children, unless it is a leaf
All non-root nodes have m and M children
Leaf-node entries <MBR, oid> MBR for the data object
Non-leaf node <MBR, ptr> MBR contains all child nodes
All leaf nodes at the same level
Note
R-tree properties similar to B+-tree properties
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 27 / 54
59. R-tree: Online Demo
R-tree Online Demo Applet
gis.umb.no/gis/applets/rtree2/jdk1.1/
Questions
What are the m and M?
What happens when you insert within an MBR?
What happens when you insert outside an MBR?
What happens if you delete from an MBR such that there are less
than m objects?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 28 / 54
61. Filter and Refine
Two-Step Filtering
R-Tree
Intermediate
Filter
Geometry
Comparison
Result
False Hits
Primary Filter Secondary Filter
Two-Step Process
Use MBR for checking for potential overlap (fast)
Use actual geometry to check for real overlap (slow)
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 30 / 54
62. MBRs for Overlap
O1 O2
O3
O4
O5
Which objects are in the result?
Which objects are not in the result?
Which objects may be in the result?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 31 / 54
63. Range Search
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
64. Range Search
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
10
11
12
13
14 15
16
17
18
D E
FII
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 32 / 54
65. Range Search
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
10
11
12
13
14 15
16
17
18
D E
FII
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 32 / 54
66. Range Search, One
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
10
11
12
13
14 15
16
17
18
D E
FII
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
67. Range Search, One
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
10
11
12
13
14 15
16
17
18
D E
FIIQ1
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 33 / 54
68. Range Search, One
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
10
11
12
13
14 15
16
17
18
D E
FIIQ1
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 33 / 54
69. Range Search, Two
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
10
11
12
13
14 15
16
17
18
D E
FII
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
70. Range Search, Two
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
10
11
12
13
14 15
16
17
18
D E
FII
Q2
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 34 / 54
71. Range Search, Two
Example (R-tree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6 7
8 9
A
B
C
I
10
11
12
13
14 15
16
17
18
D E
FII
Q2
I II - -
A B C -
1 2 3 - 4 5 6 - 7 8 9 -
D E F -
10 11 12 - 13 14 15 - 16 17 18 -
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 34 / 54
72. Quiz: Range Query
Example (R-tree)
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0
1
2
3
4
5
6
7
8
9
10
1 2 3
4
5
6
7
8 9 10
11
12
13
14
15
16
17
18
19
20 21
22
23
24 25
26
27
28 2930
31 32
33
34
35
36 37
A B
C D
E F
G
H
I
J
K
L
M
I
II
III
IV
Q
Quiz
Which MBRs are examined?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 35 / 54
73. R-tree versus R+
-tree and R∗
-tree
R∗
-tree
Like R-tree just insert algorithm changed
Minimize area of each MBR
Minimize overlap between MBRs
Maximize storage usage
Maximize square-ness of MBRs
Overall: Slower modifications, faster search
R+
-tree
No overlap between MBRs for non-leaf node
Indexed object may be inserted more than once
Multi-path not needed for search
Overall: Higher tree / faster search
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 36 / 54
74. Summary: R-tree
Note
Data division
Multi-level index (a tree!)
There are many variations of the R-tree, e.g., R+-tree and R∗-tree
Very complicated to implement
Concurrency control very complicated to implement
Very good when index must be stored on disk (block device)
Good index structure for in-memory databases
Many (hundreds) of index structures derived from R-tree
Performance
The R-tree typically outperforms quadtrees and SFC!
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 37 / 54
82. The Quad and Tree in Quadtree
Quad
0 1
2 3
10 11
12 13
30 31
32 33
120 121
122 123
Tree
All
0 1
10 11 12
120 121 122 123
13
2 3
30 31 32 33
Note
One entry for each point
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 40 / 54
83. Rectangles in a Quadtree
The Data
0 1
2 3
10 11
12 13
120 121
122 123
A
B
C
D
E
F
G H
The Quadtree
All
0:A,E 1:C,D
10 11 12:F
120 121 122:G 123:H
13
2:A,B 3:B,D
Note
Each rectangle in multiple subtrees
Where is the worst place a rectangle can be placed?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 41 / 54
84. Quiz: Indexing Rectangles
The Data
0 1
2 3
10 11
12 13
30 31
32 33
120 121
122 123
130 131
132 133
A B
C
D
E
F
Which Index Values?
A?
B?
C?
D?
E?
F?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 42 / 54
85. Quiz: Indexing Rectangles
The Data
0 1
2 3
10 11
12 13
30 31
32 33
120 121
122 123
130 131
132 133
A B
C
D
E
F
Which Index Values?
A?
B?
C?
D?
E?
F?
Trade off
Accurate (smallest quads)
Space usage (biggest quads)
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 42 / 54
86. Complicated Polygon in a Quadtree, One
Data
0 1
2 3
20 21
22 23
30 31
32 33
10 11
12 13
120 121
122 123
210 211
212 213
Index
0 1
2 3
20 21
22 23
30 31
32 33
10 11
12 13
120 121
122 123
210 211
212 213
Note
Many entries for a single object
Assuming maximum three levels in index
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 43 / 54
87. Complicated Polygon in a Quadtree, Two
Data
0 1
2 3
10 11
12 13
20 21
22 23
30 31
32 33
120 121
122 123
Index
0 1
2 3
10 11
12 13
20 21
22 23
30 31
32 33
120 121
122 123
Note
Another example
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 44 / 54
89. SQL Server 2008 R2 Spatial Grid
[http://sqlskills.com/blogs/paul/post/SQL-Server-2008-Spatial-indexes.aspx]
Note
Maximum 4 levels
Small = 4x4, Medium = 8x8, Large 16x16
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 46 / 54
90. SQL Server 2008 R2: Complex Object
[http://social.technet.microsoft.com]
Questions
Why different granularities?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 47 / 54
91. Quadtree Demo
PR Quadtree Demo
donar.umiacs.umd.edu/quadtree/points/prquad.html
Questions
What can you note about the sizes of the regions when split?
Is the tree balanced?
PMR Quadtree Demo
donar.umiacs.umd.edu/quadtree/rectangles/PMR.html
Questions
What happens when you place a rectangle in the middle of an empty
region?
Is the tree balanced?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 48 / 54
92. Which Quadtree are Legal?
A B
C D
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 49 / 54
93. Quadtree: Comments
Note
Space division
Can be used for all types of spatial objects
There are many variations of the quad tree
The version here is called the PR quadtree
Variation of quadtree is used in SQL Server 2008 R2
Data structure is fairly simple to implement
Concurrency control is fairly simple to implement
Not optimal when index must be stored on disk (block device)
Quiz
Why are there multiple levels in a quadtree?
Did you notice anything about the dimension sizes?
Can you give a reason why?
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 50 / 54
95. Comparison of Indexing Methods
Support for Data Types
SFC Quadtree R-tree
Point Good Good Good
Linestring Okay Okay Good
Polygon Okay Okay Good
Support for Queries
SFC Quadtree R-tree
Point Good Good Good
Range Bad Okay Good
Join Okay Okay Good
NN Okay Okay Okay
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 52 / 54
96. Summary: Spatial Indexing
Note
The R-tree is the industry “work horse”
Good 2D and 3D
Breaks around∼ 10D, called the dimensional curse
Quadtree used because simple to implement
Particular good if all data is in main memory
Very good demo at donar.umiacs.umd.edu/quadtree/index.html
Space-filling curves may be exactly sufficient for your app!
Simple to implement
Much faster than full table scans
Important Concepts
Data and space division
Minimum bounding rectangle (MBR)
Split
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 53 / 54
97. More Information
Web Links
R-tree at Wikipedia at en.wikipedia.org/wiki/R-tree
Good overview with many links to additional material
R-tree demo at gis.umb.no/gis/applets/rtree2/jdk1.1/
Quadtree and R-Tree demo at
donar.umiacs.umd.edu/quadtree/index.html
What new spatial next SQL Server social.technet.microsoft.
com/wiki/contents/articles/4136.aspx
Research Related
The original R-tree paper
www-db.deis.unibo.it/courses/SI-LS/papers/Gut84.pdf
The first R∗-tree paper epub.ub.uni-muenchen.de/4256/1/31.pdf
Kristian Torp (Aalborg University) Spatial Indexing November 19, 2015 54 / 54
