Chapter 3: Data Storage and Access Methods

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Chapter 3 Data Storage and Access Methods

• Title The R Tree An Efficient and Robust
  Access Method for Points and Rectangles
• Authors N. Beckmann, H. Kriegel, R. Schneider
  and B. Seeger
• Pages 207-216

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The R Tree An Efficient and Robust Access
Method for Points and Rectangles

• Problem
• Problem Statement
• Why is this problem important?
• Why is this problem hard?
• Approaches
• Approach description, key concepts
• Contributions (novelty, improved)
• Assumptions

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Problem Statement R Tree

• Given
• Data containing points and rectangles
• Spatial queries (point, range query, insert,
  delete)
• Find - An Access Method (Data Structure)
• A hierarchical organization of rectangles
• Example from wikipedia
• Objectives
• Efficiency of spatial queries
• Constraints
• Balanced tree
• Each node is a disk page and has gt m (min of
  entries) entries.
• Root has at least two children unless it is a
  leaf
• Efficiency metric number of disk-pages accessed

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Why is this problem important?

• Multi-dimensional Applications
• Large geographic data. e.g., Map objects like
  countries occupy regions of non-zero size in two
  dimension.
• Common real world usage Find all museums within
  2 miles of my current location".
• CAD
• Many DBMS servers support spatial indices
• Orcale, IBM DB2,

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Why is this problem Hard?

• B-tree split methods ineffective in 2-dimensions
• Ex. Sorting
• Size variation across data Rectangles
• Large rectangles limit split options!
• Non-uniform data distribution over space
• Dynamic Access Method
• Insertions and deletions
• Overlapping directory rectangles gt multiple
  search paths

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Novelty of Contribution

• Related Work
• Traditional one-dimensional indexing structures
  (e.g., hash, B-tree) are not appropriate for
  range search
• B tree
• Represents sorted data in a way that allows for
  efficient insertion and removal of elements.
• Dynamic, multilevel index with maximum and
  minimum bounds on the number of keys in each
  node.
• Leaf nodes are linked together as a linked list
  to make range queries easy.
• R-tree
• R-tree is a foundation for spatial access method
• A complex spatial object is represented by
  minimum bounding rectangles while preserving
  essential geometric properties
• Over-lapping regions
• Heuristic minimize the area of each enclosing
  rectangle in the inner nodes.

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Principles of R-tree

• Height-balanced tree similar to a B-tree with
  index records in its leaf nodes containing
  pointers to data objects.
• Heuristic Optimization minimize the area of each
  enclosing rectangle in the inner nodes.

Reference A Guttman R-tree a dynamic index
structure for spatial searching, 1984
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Performance Parameters beyond R-tree

• (Q1) The area covered by a directory rectangle
  should be minimized.
• (Q2) The overlap between directory rectangles
  should be minimized.
• (Q3) The margin of a directory rectangle should
  be minimized.
• (Q4) Storage utilization should be optimized.
• Intuitions
• Reduce overlap between sibling nodes.
• Reduce traversal of multiple branches for point
  query
• Reinsert old data changes entries between
  neighboring nodes and thus decreases overlap.
• Due to more restructuring, less splits occur

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Difference between R-tree and R-tree

• Minimization of area, margin, and overlap is
  crucial to the performance of R-tree / R-tree.
• The R-tree attempts to reduce the tree, using a
  combination of a revised node split algorithm and
  the concept of forced reinsertion at node
  overflow. This is based on the observation that
  R-tree structures are highly susceptible to the
  order in which their entries are inserted, so an
  insertion-built (rather than bulk-loaded)
  structure is likely to be sub-optimal. Deletion
  and reinsertion of entries allows them to "find"
  a place in the tree that may be more appropriate
  than their original location. ? Improve retrieval
  performance

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Example

Preferred by R-tree
R1
R2
R5
R4
R3
Preferred by R-tree
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Validation Methodology

• Methodology
• Experiments with simulated workloads
• Evaluation of design decisions
• Results
• R-tree outperforms variants of R-tree and
  2-level grid file.
• R-tree is robust against non-uniform data
  distributions.

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Summary

• Papers focus
• R-tree implementations and performance
• Ideas
• Heuristic Optimizations (pp. 208)
• Reduction of area, margin, and overlap of the
  directory rectangles
• Better Storage Utilization (pp 211)
• Forced Reinsertion (splits can be prevented)
• Experimental comparison
• Using many data distributions

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Assumptions, Rewrite today

• Assumptions
• Indexing data in two-dimensional space
• Bulk load and bulk reorganization not available
• Concurrency control and recovery costs are
  negligible
• Reinserts during split!
• Rewrite today
• Bulk-load of rectangles
• Compare with newer methods
• R tree (disjoint sibling), Hilbert-R-tree
• Analytical results
• Formally compare R-tree with alternatives