1. File Processing : Multi-dimensional Index
2018, Spring
Pusan National University
Ki-Joune Li

2. Multi-Dimensional Index
Multi-Attributes Query vs. Single Attribute Query
Single Attribute : Only ONE attribute to specify query condition
Example : Find Students whose record is in [3.5, 4.5]
Multi-Attributes : Several attributes
Example : Find students whose height is greater than 180 cm and weight is less than 70 Kg
Each attribute corresponds to a dimension
Multi-Attribute Query : Multi-Dimensional Query

3. Processing Multi-dimensional Queries
Example : Find students whose height > 180 cm and weight < 70 Kg
Method 1 : Using a B+-tree
Step 1 : Apply B+-tree to search student taller than 180 cm
Step 2 : Search students lighter than 70 Kg from the result of step 1
Height and Weight or Weight and Height ?
< 70
Result
180

4. Processing Multi-dimensional Queries
Method 2 : Using Two B+-trees
Step 1 : Result1 ← Students taller than 180 cm by B+-tree
Step 2 : Result2 ← Students lighter than 70 Kg by B+-tree
Step 3 : Result ← Result1  Result2
Comparison of Method 1 and Method 2
180
< 70
Result = 

5. Processing Multi-dimensional Queries
Method 3 : Unified Index for Several Attributes
One index for several attributes
Multi-Dimensional Space
Two approaches
Extending B+-tree
Extending Dynamic Hashing
Index for Height and Weight
Weight
Height

6. Extending Hashing : Grid Approach
Weight
Height
block pointer
. . .
Block Pointer Array
Query
Fixed
Variable
Fixed Grid Method
Grid File

7. Extending Hashing : Grid File
Directory
(x1, y1)
(x2, y2)
Block Pointer
Query

8. Problem 1: Dead Space No objects in this query area 5 block accesses
Dead Space  Empty space with no objects
How to reduce dead space

9. Minimum Bounding Rectangle
MBR (Minimum Bounding Rectangle)
Query
Only 1 Disk Access

10. Problem 2: Non-Point Object
Where to store this object

11. Minimum Bounding Rectangle
MBR (Minimum Bounding Box)
Two dimensional geometric simplification of objects
Not the Whole space,
only in the region occupied by objects
(X1max , X2max )
(X1min , X2min)

12. Extending B+-tree : R-tree
B+-tree vs. R-tree
B+-tree : Interval (1-D rectangle)
R-tree : Multi-Dimensional Interval (Rectangle)
R-tree : Rectangle B+-tree
Each Node
MBR (Minimum Bounding Rectangle) instead of Interval (or Delimiter)
No Linked-List for External Nodes
A certain amount of overlapping is indispensable

13. Extending B+-tree : R-tree
Example
Root
Query

14. Upward Split like B-tree
Split MBR in the case of overflow
Line sweeping : Compare Cost-X and Cost-Y 
New MBR
Splitting Line

15. Splitting Strategy 50:50 Split
Instead of 50:50 split, other cost measures
Area,
Perimeter
Overlapping Area
Good Split
Bad Split
1. Make them as COMPACT as possible
2. Preserve spatial proximity as possible

16. R*-tree: An Improvement of R-tree
Re-Insertion Strategy on Overflow
Most Popular Index for Multi-Dimensional Index
Overflow
More Compact
Newly Inserted Object
Delete and Re-Insert this
Re-Inserted Object