1. Multidimensional Indexing

2. More Dimensions
There are applications that require us to see data in two or more dimensions, e.g. Geographical Information System
Roughly, every attribute of data can be seen as a dimension

3. Example Queries
Partial Match: looking for a set data items with specific values for every dimension
Range: looking for a set of data items within a specific range for every dimension
Nearest-Neighbor: looking for the closets point to a given point, e.g. a city of over population closest to a given city
Where-am-I: finding out where a specific point is located, e.g. locating mouse pointer on the screen

4. Using Conventional Indexes
Suppose points are distributed randomly in a 2D space with x and y ranging from 0 to 1000.
If we are looking for points with 450<x<550 and 450<y<550, i.e. an area of 100x100
Using a B-Tree for x we find pointers having x within the range
One way is to retrieve all those points and verify their y value, in order to find the points at the intersection

5. Using Conventional Indexes
Almost any data structure allows us to execute nearest-neighbor query by specifying a range in each dimension, but which point is closer?

6. Multidimensional Indexes
Hash-Table like Structures
Tree Like Structures

7. Multidimensional Indexes
Hash-Table like Structures
Grid File, does not hash, partitions the dimensions by sorting the values along those dimensions
Partitioned Hashing, does hash the various dimensions, each dimension contributes to the bucket number

8. Grid File (Hash Table)
Each of the regions can be thought of as a bucket of a hash table
Each point in that region has its record placed in a block belonging to that bucket
For example: the central rectangle represents data items with
40 ≤ age < 55 and 90 ≤ salary < 225

9. Grid File Instead of one dimensional array of buckets
Grid file uses an array with number of dimensions same as the data file
Hashing is different from applying a hash function
The positions of the data item in each of the dimension together determine the bucket

10. Grid File

11. Grid File
Inserting:
If there is place in the block of the proper bucket, then we insert
If there is no place
Add overflow blocks to the bucket
Reorganize the structure by adding or moving grid lines

12. Grid File Reorganizing the structure:
Adding a grid line splits all the buckets along that line
It may not be possible to select a new line that does the best for all buckets
This may create for example too many empty buckets or leaving several very full buckets

13. Grid File Age = 51 Example: Inserting point (52, 200)
Vertical Line age = 51 doesn’t help,
Since it doesn’t split any other bucket,
It only create 3 empty buckets

14. Partitioned Hashing Example: Three bits used for bucket number
The left most bit is determined by first attribute
The two right most bits are determined by second attribute
h(25) = 25 % 2 = 110 = 12
h(60) = 60 % 4 = 010 = 02 = 002
Therefore h(25,60) = 100
h(45) = 45 % 2 = 110 = 12
h(350) = 350 % 4 = 210 = 102
Therefore h(45,350) = 110

15. Grid File <-> Partitioned Hash
Partial Match Query -> Partitioned Hash
Nearest Neighbor -> Grid File
Range Query -> Grid file
However with these methods we no longer have the advantage that the answer is in exactly one bucket, but still they limit our search to a subset of the buckets

16. Tree Like Structures
Multiple-key Indexes: a tree in which the nodes at each level are indexes for one attribute
kd-trees (k-dimensional search tree): a binary tree
Note: in these structures we are going to lose the advantage of having balanced trees

17. Multiple-key Indexes Very efficient for partial match query
Works quite well for range queries

18. kd-tree Index A binary tree
Interior nodes have an attributes, a dividing value for that attribute, and pointer to left and right children.
Leaves are blocks, with space for as many records as a block can hold.

19. kd-tree Index

20. kd-tree Index Inserting data item (35,500)
If there is no room in the proper block
We split the leaf node and create a new internal node