1. 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:

2. 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

3. 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 >= m (min # of entries) entries.
Root has at least two children unless it is a leaf
Efficiency metric = number of disk-pages accessed

4. 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, …

5. 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 => multiple search paths

6. 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.

7. 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

8. 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

9. 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

10. Example Preferred by R-tree Preferred by R*-tree R1 R2 R3 R5 R4 R1 R2

11. Validation 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.

12. Summary Paper’s focus Ideas Experimental comparison
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

13. Assumptions, Rewrite today
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