The Quad tree

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
Xx1
Yy1
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. X01, Y11
• 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 Keys
11
142
144
120
140
10
143
141
100
01
110
130
00
00
01
10
11
120 140
011 110
8
Hilbert Curve
111
110
101
100
011
010
001
000
100
011
001
010
000
101
110
111
9
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

• 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
• Merging based on buddy system
• Regions are represented as (cx, cy, dx, dy)
• point queries cx-dx lt qx lt cxdx,
• cy-dy lt qy lt cydy

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
qy