Indexing of Dynamic Abstract Regions

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Indexing of Dynamic Abstract Regions

• Joxan Jaffar, Roland Yap
• National Univerisity of Singapore
• Kenny Q. Zhu
• Microsoft
• ICDE 2006

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Outline

• What are we indexing?
• Existing Methods
• The RC-tree
• Clipping and Domain Reduction
• Partial Rebuilding
• Dynamic Segmentation
• Heuristics
• Technical Results
• Experimental Results
• Future Work
• Conclusion

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What are we indexing?

• Stock trading system
• Large number of triggers like these
• Trader A if price of IBM 50, then sell 100
  shares of IBM otherwise if price of Dell 35,
  then buy 200 shares of dell
• Trader B if the price of Dell 2 times the
  price of IBM, then buy 100 shares of IBM and sell
  100 shares of Dell
• Need an index for them!

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What are we indexing? (contd)

• Conjunctions of linear constraints, e.g.
• Abstract regions of interest or shapes
• Multi-dimensional spatial objects
• May overlap each other

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Existing Representations

• GIS applications usually rigid objects with no
  overlapping
• Minimum Bounding Rectangles (MBR) approximations
• Manual segmentation into smaller boxes (consider
  a freeway!)
• Segmentation either too coarse (dead space) or
  too fine (space and time cost)

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Existing Representations Highways in Germany
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Existing Data Structures

• R-tree object partitioning, height balanced
• R-tree space partitioning (clipping),
  height-balanced
• R-tree R-tree variant with a different object
  splitting method
• Kd-tree space partitioning, weight-balanced
• Quad-tree space partitioning, unbalanced
• CR-tree cache-conscious R-tree

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The RC-tree

• Reducible Clip-tree
• A general weight-balanced binary tree for dynamic
  search and update of k-dimensional spatial
  objects
• Designed for main-memory use
• Intermediate nodes space-partitioning
  hyper-planes as discriminators
• Leaf nodes (overflow nodes) indexed objects
• Objects that intersects with the discriminators
  are clipped and domain-reduced

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Clipping and Domain Reduction

• A Original objects
• B MBR-based indexing
• C RC-tree with discriminators d1, d2, and d3

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Benefits of Domain Reduction
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Dynamic Segmentation

• A pre-defined leaf capacity L
• A leaf is split only if number of (clipped)
  objects is larger than L
• Splitting of a leaf node may cause segmentation
  (clipping) of object if the objects are
  overlapped
• Compare with BSP
• Only segment an object when necessary (on demand)
  as opposed to static segmentation

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Dynamic Segmentation
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Dynamic Segmentation
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Dynamic Segmentation
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Partial Rebuilding

• Originally from Overmars
• Rebuild part of the tree that is out of balance
• Balancing criterion
• where T is any sub-tree rooted at node T
• Flush all objects (clipped) within T, merge them
  and rebuild the sub-tree
• Controls clipping and space usage

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Partial Rebuilding
partial rebuild
h
T
T
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Some Heuristics

• Additional MBRs at intermediate nodes to speed up
  ruling out false queries
• Finding the right discriminator
• RC-SWEEP sorts the objects and sweeps through to
  balance the weight and minimize clipping ? more
  optimized but slower
• RC-MID quickly partitions the objects in the
  middle of the sub-space and ignore the effects of
  clipping ? less optimized but fast

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Technical Results

• Given a constant scaling factor s which bounds
  the relative extents of objects and assuming a
  fixed dimensionality k
• The amortized insertion cost of a balanced
  RC-tree of n objects has O(sklog(n)) time
  complexity.
• A point query has average O((1.5)k(log(n)sk))
  time
• A RC tree has worst case O(skn) space
  complexity.

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Experimental Results

• Synthetic Datasets overlapping line segments and
  triangles
• Traditional GIS Datasets e.g. German roads

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Search cost synthetic data (log scale)
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Search cost GIS data
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Space Usage Synthetic Data
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Space usage GIS data
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Hit rates Synthetic Data (negative log scale)
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Search time GIS Data
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Insertion and Search Cost vs. log(n)
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Other Empirical Results

• Range queries better query accuracy
• Dynamic vs. static segmentation
• Cache oblivious RC-tree
• Some limited performance improvement
• Requires larger cache-line relative to the node
  size

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Future Work

• Indexing higher dimension regions
• Application of domain reduction techniques in
  height-balanced indexes
• More advanced cache study of RC-tree
• Using RC-tree and its techniques in triggering of
  active database and other business applications

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Conclusion

• A good weight-balanced clipping-based index for
  both overlapping and non-overlapping spatial data
• Allows the tuning of space-time trade off
• Combines dynamic segmentation, domain reduction
  and partial rebuilding
• Renews the interest in spacing partitioning and
  object clipping indexing methods