R-Trees: A Dynamic Index Structure for Spatial Data

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R-Trees A Dynamic Index Structure for Spatial
Data

• Antonin Guttman

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R-Tree Why, What ?

• Why do we need R-Trees?
• What are R-Trees?
• How do I perform operations?
• Alternatives? Why not a B tree?

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Properties of R-Trees

• Height Balanced
• 2 types of nodes
• Leaves point to disk pages
• Records in the leaves point to actual data
  objects
• For a max capacity of M, min occupancy should be
  M/2
• Completely dynamic
• Guaranteed Fan-out of M/2
• Every leaf record is a smallest bounding box.
• Root has at least two children

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R-Trees The Structure.

• Internal nodes ( rectangle, child pointer)
• N dimensional rectangle.
• Pointer to all rectangles that are cointained.
• Leaf Nodes (MBR , tuple-identifier)
• MBR is minimum bounding rectangle
• Tuple-identifier is a pointer to the data object.

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R-tree of order 4
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Example
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Example
a
b
c
d
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Example
e
f
a
b
c
d
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Example
e
f
a
b
g
h
c
d
i
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R-Trees Operations

• Inserts
• Deletes
• Updates ( delete and re-insert)
• Queries/Searches
• Names of all the roads in 1 sq km area?
• Which buildings would be encountered between
  Rogers Hall and Reitz Union?
• Give me all rectangles that are contained in the
  input rectangle.
• Give me all rectangles intersecting this
  rectangle.

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Insert

• Similar to insertion into B-tree but may insert
  into any leaf leaf splits in case capacity
  exceeded.
• Which leaf to insert into? (Choose Leaf)
• How to split a node? (Node Split)

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Insert Choose Leaf
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Insert Choose Leaf
m
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Insert Choose Leaf
n
15
Insert Choose Leaf
o
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Insert Choose leaf
p
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Node Splitting

• Quadratic method
• Select max area gradient in the nodes as seeds.
• Start clustering from the seeds
• Linear method
• Select seeds with max separation using max x, y
• Randomly assign rectangles to seeds

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Delete

• Search for the rectangle
• If the rectangle is found, remove it.
• If the node is deficient,
• Put the remaining entries in a re-insert queue.
• Adjust the parent rectangle if needed.
• Continue this till you reach the root.
• Re-insert in such a way that all internal nodes
  remain above the leaf nodes.
• Adjust the rectangles making them smaller.
• Alternative sibling combination like a B-tree.
• But re-insertion shows similar performance and is
  simple to implement.

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Performance Tests

• R-Trees in C under UNIX on VAX11/780 computer
  running on 2D data(1057) for 5 page sizes
• Linear node split was better than quadratic as
  expected.
• CPU time unchanged with page sizes, indicating
  that when one side became full all split
  algorithms simply put everything in the other
  side.
• Delete is affected by the fill factor.
• Search insensitive to the fill factor and split
  algorithm used.
• Storage space is a function of the fill factor,
  page size and split algorithm
• All split algorithms came in 10 of the best
  exhaustive search and split algorithm.

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Performance 2nd Innings

• Same configuration but on various data sizes
  1057, 2238, 3295 and 4559 rectangles.
• Low CPU cost, close to 150 micro seconds.
• Comparable performance of split algorithms
• Most space was used by the leaf nodes

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Conclusions from the paper.

• R-Tree perform well for spatial data with non
  zero node sizes.
• With smaller node structure can be used as an
  in-memory spatial data index.
• CPU performance of in-memory R-tree index is
  comparable and there is no IO cost.
• Linear split was almost as good as others.
• It was fast.
• Node split quality was a bit off-target, but it
  did not hurt the search performance noticeably.
• Possible use with abstract data types and
  abstract indexes to streamline handling of
  spatial data.