The Priority R-Tree: A Practically Efficient and Worst-Case Optimal R-Tree

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The Priority R-Tree A Practically Efficient and
Worst-Case Optimal R-Tree

• Lars Arge1, Mark de Berg2, Herman Haverkort3 and
  Ke Yi1
• Department of Computer Science
• Duke University
• Department of Computer Science
• TU Eindhoven
• Institute of Information and Computing Sciences
• Utrecht University

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Problem Definition

• Input
• N rectangles in the plane
• Window query Q
• Output
• All rectangles intersecting Q
• Applications
• Spatial databases
• GIS
• CAD
• Computer vision
• Robotics

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R-Tree

• Definition Guttman84
• Advantages
• Little redundancy
• Multi-purpose
• Easy to update

Fanout ?(B) B disk block size
G
F
E
B
A
H
I
A
B
C
D
E
F
G
H
I
C
D
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How to Build an R-Tree

• Repeated insertions
• Guttman84
• R-tree Sellis et al. 87
• R-tree Beckmann et al. 90
• Bulkloading
• Hilbert R-Tree Kamel and Faloutos 94
• Top-down Greedy Split Garcia et al. 98
• Advantages
• Much faster than repeated insertions
• Better space utilization
• Usually produce R-trees with higher quality

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R-Tree Variant Hilbert R-Tree
Hilbert Curve

• To build a Hilbert R-Tree (cost O(N/B logM/BN)
  I/Os)
• Sort the rectangles by the Hilbert values of
  their centers
• Build a B-tree on top
• 4D Hilbert R-tree

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R-Tree Variant TGS R-Tree
(Top-down Greedy Split)

• To build a TGS R-tree
• Start from the root and buildthe tree top-down
• To build one node, use binary cutsuntil the
  desired fan-out is reached
• To make a binary cut, consider4 orderings of the
  rectangles xmin, ymin, xmax, ymax
• In each ordering, consider the B cutting
  positions
• Choose the one that minimizes the sum of the
  areas of the two resulted bounding boxes
• Typical bulk-load cost O(N/B log2N) I/Os

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Our Results

• None of existing R-tree variants has worst-case
  query performance guarantee!
• In the worst-case, a query can visit all nodes in
  the tree even when the output size is zero
• Priority R-Tree
• The first R-tree variant that answers a query by
  visiting
  nodes in the worst case
• T Output size
• It is optimal!
• There exists a dataset such that for any R-tree,
  there is an empty query that visits
  nodes. Kanth and Singh 99, Agarwal et
  al. 02

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Roadmap

• Pseudo-PR-Tree
• Has the desired
  worst-case guarantee
• Not a real R-tree
• Transform a pseudo-PR-Tree into a PR-tree
• A real R-tree
• Maintain the worst-case guarantee
• Experiments
• PR-tree
• Hilbert R-tree (2D and 4D)
• TGS-R-tree

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Building a Pseudo-PR-Tree
root
priority leaves
Step 1 take out B extreme rectangles from each
direction and put them into priority leaves
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Building a Pseudo-PR-Tree
Step 2 Divide by the xmin coordinates and build
subtrees recursively. Division is performed
using xmin, ymin, xmax, ymax in a round-robin
fashion, like a 4D kd-tree
root
Analysis sketch nodes with at least one
priority leafcompletely reported O(T/B) nodes
with no priority leaf completely reported
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Pseudo-PR-Tree to a Real R-tree
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Query Complexity Remains Unchanged
Next level
nodes visited on leaf level
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PR-Tree Bulkload Updates

• Bulkload
• O(N/Blog2N) I/Os?O(N/BlogM/BN) I/Os, using
  grid method Agarwal et al. 01
• The same as Hilbert R-tree, but with a larger
  constant
• Updates
• Can use any previous heuristic to update in
  O(logBN) I/Os
• Without worst-case query guarantee
• Use logarithmic method
• Insert O(logBN 1/B logM/BN log2(N/M)) I/Os
• Delete O(logBN) I/Os
• Extending to d-dimensions
• Query bound O((N/B)1-1/d T/B), still optimal
• Bulkload update bounds remain the same

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Experiments

• Implemented with TPIE
• Priority R-tree
• Hilbert R-tree
• 4D Hilbert R-tree
• TGS R-tree
• Real-life data
• TIGER datasets
• 16 million rectangles
• Synthetic data
• Varying from normal to extreme data
• 10 million rectangles

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Experiments with Real-Life Data

• Query performance on the TIGER datasets

Shown I/Os spent in answering a query
T/B
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Experiments with Synthetic Data SIZE
Each side of a rectangle is uniformly
distributed in 0, max_side
Queries are squares with area 1
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Experiments with Synthetic Data ASPECT
Fix the area, vary aspect ratio
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Experiments with Synthetic Data SKEWED
Randomly place points, then do yyc on the
y-coordinates
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Experiments with Synthetic Data CLUSTER
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Conclusions

• In theory
• The PR-tree is the first R-tree variant that
  answers a window query in
  I/Os worst-case, which is optimal
• In practice
• Roughly the same as previous best R-trees on
  real-life and relatively nicely distributed data
• Outperforms them significantly on more extreme
  data
• Future work
• How previous heuristics may affect the
  performance of the PR-tree in the dynamic case

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Lower Bound Construction

• Each bounding box intersects at least
  queries
• N/B bounding boxes
• queries
• There exists a query that intersects at least
• bounding boxes

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Pseudo-PR-Tree Query Complexity

• Nodes v visited where all rectangles in at least
  one of the priority leaves of vs parent are
  reported O(T/B)
• Let v be a node visited but none of the priority
  leaves at its parent are reported completely,
  consider vs parent u

2D
4D
Q
ymin ymax(Q)
xmax xmin(Q)
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Pseudo-PR-Tree Query Complexity

• The cell in the 4D kd-tree of u is intersected by
  two different 3-dimensional hyper-planes
• The intersection of each pair of such
  3-dimensional hyper-planes is a 2-dimensional
  hyper-plane
• Lemma of cells in a d-dimensional kd-tree that
  intersect an axis-parallel f-dimensional
  hyper-plane is O((N/B)f/d)
• So, such cells in a 4D kd-tree
• Total nodes visited

u
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Experiments with Real-Life Data

• Datasets TIGER/Line data
• Bulk-loading