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Multidimensional Search StructuresPowerPoint Presentation

Multidimensional Search Structures

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**Jims** - Follow User

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Multidimensional Search Structures. Spatial Databases. Spatial Objects 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)

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Spatial Databases

- Spatial Objects
- 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”

- Range queries

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)

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

Grid-Tree (G-Tree)

- Mainly for point data of any dimension
- Based on hypercubes

insert point

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G-Tree Organization

- G-Trees are organized in B+-trees
- Key is the binary string

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- Rotate vertical/horizontal split
- When split, add 0/1 at the end

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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

Point Quadtrees

- 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)

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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

R*-Trees

- 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

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Spatial Operations on R*-Trees

Find all objects that intersect rectangle 9:

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Insert rectangle 9:

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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.

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