1. Multidimensional Search Structures
CSE © Leonidas Fegaras Multidimensional Indexing

2. Spatial Databases Spatial Objects Data Types Spatial Queries
Points: location (x,y)
Lines: pairs of points (roads, coastal lines)
Polygons: list of points (states, countries)
Data Types
Point: a data type with no extension (area)
Region: has location and boundary that defines the extension
Spatial Queries
Range queries
“Find all Italian restaurants within 20 miles from UTA”
Nearest neighbor queries
“Find the 10 Italian restaurants that are nearest to UTA”
“Find the nearest fire station to Neddermann Hall”
Spatial join queries
“Find pairs of cities within 20 miles of each other”
“Find restaurants that are adjacent to university campuses”
CSE © Leonidas Fegaras Multidimensional Indexing

3. Applications Geographical Information Systems (GIS)
Map systems
Resource management systems
Computer-aided design and manufacturing (CAD/CAM)
VLSI design (avoiding overlaps, routing wires)
Multimedia databases
Various dimensions (color, shape, texture)
CSE © Leonidas Fegaras Multidimensional Indexing

4. Representation of Spatial Objects
It is very expensive to work on real boundary lines
Minimum Bounding Rectangle (MBR)
Testing for intersection:
Test if MBRs intersect
If they do, test if boundary lines intersect
CSE © Leonidas Fegaras Multidimensional Indexing

5. Grid-Tree (G-Tree) Mainly for point data of any dimension
Based on hypercubes
insert point E E A C C F A B B D F D A C C B A B D D E E 1
0000
0010 10 00 11
0001
0011 01 10
1110
000
111
001 01
110
1111
110
0001
0000
0010
1111
0011
1110
CSE © Leonidas Fegaras Multidimensional Indexing

6. G-Tree Organization G-Trees are organized in B+-treesx
Rotate vertical/horizontal split
When split, add 0/1 at the end x0 x0 x1 x0 x1 x1
G-Trees are organized in B+-trees
Key is the binary string 01
1110 1 00 11 01
0000
0001
0010 01 10
110
1110
1111 10
000
111
001
110
0001
0000
0010
1111
0011
1110
CSE © Leonidas Fegaras Multidimensional Indexing

7. Searching G-Trees Search: find point P=(x1,x2,…,xn)
Let m be the number of bits of the largest bitstring
Find the m-bit region that contains P:
Start with si=0 and ti=1for all 1<=i<=n
For k=0 to m-1:
Slice over j=(k mod n)+1 dimension
The kth bit of the bitstring is 0 iff xj < (sj+tj)/2; then set tj= (sj+tj)/2
Otherwise, it is 1 and set sj= (sj+tj)/2
For n=2, m=5, and P=(0,1,0.7):
0.1 < ½ bit0 is 0
0.7 > ½ bit1 is 1
0.1 < ¼ bit2 is 0
0.7 < ½+¼ bit3 is 0
0.1 < 1/8 bit4 is 0
Bitstring is 00010
Use the bitstring as the key for the G-Tree
CSE © Leonidas Fegaras Multidimensional Indexing

8. Point Quadtrees NE NW SE SW A B C D E F G H SE NE SW NW
Divide the space into four quadrants
Represented as an unbalanced tree where each node has 4 children
Internal node: point
Leaf: empty
Node: (x,y,NE,NW,SW,SE) E B F A G H NE NW C D SE SW A
B C D
E F G H SE NE SW NW
CSE © Leonidas Fegaras Multidimensional Indexing

9. Spatial Operations on Quadtrees
Search for a point P=(x,y)
if current node T in quadtree is leaf, then not found
if T=P, then found
else find quadrant of T that includes P and search recursively:
e.g., if x>T.X and y>T.Y then search SE quadrant
Given a rectangle (x1,y1,x2,y2) find all points in the rectangle
Need to keep a set of points
The rectangle may intersect more than one quadrants
Insert a point P=(x,y)
If current node is leaf, then replace leaf with (x,y,nil,nil,nil,nil)
Else determine the quadrant and apply the algorithm recursively
CSE © Leonidas Fegaras Multidimensional Indexing

10. R*-Trees A A C E D B C D E F B F
It is like a B+-tree with key the MBR, but
There is no order in the rectangles in each node
Sibling rectangles may overlap
Restrictions on R*-tree nodes:
The root has at least two children (unless it is the only node)
Every non-root node has m children where M/4<=m<=M
The tree is balanced
CSE © Leonidas Fegaras Multidimensional Indexing

11. Spatial Operations on R*-Trees21
Find all objects that intersect rectangle 9: 2 9 1 3 22 5 4 6 7 8 23
Insert rectangle 9: 26 25 2 24
26 27 1 3 27 22 9 5 4 6 7 8 23
CSE © Leonidas Fegaras Multidimensional Indexing

12. Insert a New Object Rectangle S
At each node (starting from root) choose only one bounding rectangle R to insert the rectangle S so that the insertion of S into R has the least overlap enlargement.
(Overlap enlargement of a rectangle is the total overlapping
area of the rectangle with the other rectangles in the tree.)
If there are many with the same overlap enlargement, choose the one with the smallest area
If overflow, split the node into two sets of nodes using a split axis so that there is no underflow and the perimeters of the two bounding boxes are minimized.
CSE © Leonidas Fegaras Multidimensional Indexing