1. The Quad tree The index is represented as a quaternary tree
Each internal node has four children, one per quadrant
NW, NE, SW, SE
Each leaf is associated a disk page, which stores the index entries

2. The kd-tree A binary tree
At each node, a dimension is selected to partition the space into two
Disk-based variants: KDB-tree, skd x1
X=x1
Y=y1 y1 y2

3. Mapping -- Indexing using PAMs
Map the MBR in 2-d into a point in 4-d:
[(x1, x2), (y1, y2)]  (x1, x2, y1, y2)
Transform the query into the new space
Use a 4-d PAM to answer queries. b Q Q a a b

4.        
Space Filling Curves
Assumption: att. values can be represented with some fixed # of bits
Space domain on each dimension: 2k values
Linearize the doman
Each point can be represented by a single dimensional value

5.        
Z-ordering 11 10 01 00 00 01 10 11

6. Z-ordering The z-value is obtained by interleaving the bits.
Eg. X=01, Y=11
z-value = 0111 = 7
Clustering effect on X-Y and z-values can be indexed using B+-trees
Range queries: problematic?

7. Z-ordering & Locational Keys11
142
144
120
140 10
141
143
100 01
110
130
Space filing curves techniques include peano curve, hilbert curve, gray ordering and z-ordering.
In z-ordering, the bits from each dimension are interleaved.
For example, …
For a region, it is decomposed into several sub-regions, where each region can be represented by the
A common prefix string. For example, …
An alternative to space filling curve is the method proposed Abel and Smith in 84, where a quad-tree
Representation is used to derive locational key. To some extent, the idea is similar to curve filling
In philosophy.
The problem of such method is the number of substrings need tobe used, and hence the number of
Subqueries. 00 00 01 10 11
120
140
011*
110*

8.        
Hilbert Curve
111
110
101
100
011
010
001
000
000
001
010
011
100
101
110
111

9. Grid Files Based on extendible hashing
       
Grid Files
Based on extendible hashing
Design principle: any point query can be answered in at most 2 disk accesses.
Two structures: k-dimensional array and k 1-dimensional array

10.        
Grid Files

11. Scales, Directory, Bucket
       
Scales, Directory, Bucket
Data structures:
Linear scales
directory: an array whose elements are one-to-one correspondence with the grid cells; each entry points to a data bucket
data buckets

12.        
Splitting and Merging

13.        
Splitting and Merging

14. Grid Files One page is associated with each cell
When a cell overflows, it is split into two cells and the points are assigned to the new cell
Two adjacent cells (buddies) can reference the same page

15. Grid Files... Repetitive splitting by halving
       
Grid Files...
Repetitive splitting by halving
Merging based on buddy system
Regions are represented as (cx, cy, dx, dy)
point queries: cx-dx <= qx <= cx+dx,
& cy-dy <= qy <= cy+dy

16. Grid Files... dx cy E A D F E F B C D B cx qx dy A C B D cx C E A F cy
       
Grid Files... dx cy E A D E F F B C D B cx qx A dy C B cx D C E A F cy qy