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R-Trees
• Extension of B+-trees.
Collection of d-dimensional rectangles.
A point in d-dimensions is a trivial rectangle.
Non-rectangular Data
• Non-rectangular data may be represented by
minimum bounding rectangles (MBRs).
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R-Trees - Advanced Data Structures - Lecture Slides, Slides of Computer Science
These are the Lecture Slides of Advanced Data Structures which includes Split Algorithm, Unbalanced Binary Search Trees, Forward Pass, Forward Pass Example, Backward Cleanup Pass, Retrace Path, Current Nodes, Roots of Respective Tries, Branch Nodes etc. Key important points are: R-Trees, Dimensional Rectangles, Trivial Rectangle, Non-Rectangular Data, Minimum Bounding Rectangles, Rectangle Collections, Data Nodes, Index Nodes, Parent Node, Possible Partitioning
Typology: Slides
2012/2013
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R-Trees
- Extension of B+-trees.
Collection of d-dimensional rectangles. A point in d-dimensions is a trivial rectangle.
Non-rectangular Data
• Non-rectangular data may be represented by
minimum bounding rectangles (MBRs).
Operations
• Insert
• Delete
• Find all rectangles that intersect a query
rectangle.
• Good for large rectangle collections stored
on disk.
R-Trees—Structure
• Data nodes (leaves) contain rectangles.
• Index nodes (non-leaves) contain MBRs for
data in subtrees.
• MBR for rectangles or MBRs in a non-root
node is stored in parent node.
Example
• Possible partitioning of our example data
into 12 leaves.
Example
• Possible R-tree of order 4 with 12 leaves.
a b c d e f g h i j k l
m n o p
Leaves are data nodes that contain 4 input rectangles each.
a-p are MBRs
Example
- Possible corresponding
grouping.
a
b
c d
m
a b cd e f g h i j k l
m n o p
Example
a
b
c d
m
e n f
- Possible corresponding
grouping.
a b cd e f g h i j k l
m n o p
Query
- Start at root and find all MBRs that overlap query.
- Search corresponding subtrees recursively.
Query
m
n
o p
a
a
a
x
a b cd e f g h i j k l
m n o p
Query
• Search m.
m
n
o p
a
b
c d
a
a
x x
a b cd e f g h i j k l
m n o p
Insert
• Similar to insertion into B+-tree but may
insert into any leaf; leaf splits in case
capacity exceeded.
Which leaf to insert into?
How to split a node?
Insert—Leaf Selection
• Insert into n.
n
Insert—Leaf Selection
• Insert into o.
o
Insert—Leaf Selection
• Insert into p.
p
Insert—Split A Node
- Split set of M+1 rectangles/MBRs into 2 sets A
and B.
A and B each have at least m rectangles/MBRs. Sum of areas of MBRs of A and B is minimum.
M = 8, m = 4
Insert—Split A Node
- Exhaustive search for best A and B.
Compute area(MBR(A)) + area(MBR(B)) for each possible A. Note—for each A, the B is unique. Select partition that minimizes this sum.
- When |A| = m = ceil(M/2), number of choices for
A is
(M+1)!
m!(M+1-m)!
Impractical for large M.
Insert—Split A Node
• Grow A and B using a clustering strategy.
Start with a seed rectangle a for A and b for B.
Grow A and B one rectangle at a time.
Stop when the M+1 rectangles have been
partitioned into A and B.
Insert—Split A Node
• Quadratic Method—seed selection.
Let S be the set of M+1 rectangles to be
partitioned.
Find a and b in S that maximize
area(MBR(a,b)) – area(a) – area(b) M = 8, m = 4
Insert—Split A Node
• Quadratic Method—seed selection.
Let S be the set of M+1 rectangles to be
partitioned.
Find a and b in S that maximize
area(MBR(a,b)) – area(a) – area(b) M = 8, m = 4
Insert—Split A Node
• Quadratic Method—assign remaining
rectangles/MBRs.
Assign c to partition whose area increases least.
M = 8, m = 4
Insert—Split A Node
- Quadratic Method—assign remaining
rectangles/MBRs.
Continue assigning in this way until all remaining rectangles must necessarily be assigned to one of the two partitions for that partition to have m rectangles.
M = 8, m = 4
Insert—Split A Node
• Linear Method—seed selection.
Choose a and b to have maximum normalized
separation.
M = 8, m = 4
Insert—Split A Node
• Linear Method—seed selection.
Choose a and b to have maximum normalized
separation.
M = 8, m = 4
Separation in x- dimension
Insert—Split A Node
• Linear Method—seed selection.
Choose a and b to have maximum normalized
separation.
M = 8, m = 4
Separation in y- dimension
Insert—Split A Node
• Linear Method—seed selection.
Choose a and b to have maximum normalized
separation.
M = 8, m = 4
Rectangles with max y-separation
Insert—Split A Node
• Linear Method—seed selection.
Choose a and b to have maximum normalized
separation.
M = 8, m = 4
Divide by y-width to normalize
Insert—Split A Node
- Linear Method—assign remainder.
Assign remaining rectangles in random order. Rectangle is assigned to partition whose MBR area increases least. Stop when all remaining rectangles must be assigned to one of the partitions so that the partition has its minimum required m rectangles. M = 8, m = 4