Bkdtree: A Dynamic Scalable kdtree

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Bkd-tree A Dynamic Scalable kd-tree

• Octavian Procopiuc, Pankaj K. Agarwal,
• Lars Arge, and Jeffrey Scott Vitter

Presenter Lei XIA
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Agenda

• Efficient index
• kd-tree
• kdB-tree
• Bkd-tree
• Experiment Platform
• Experimental Results
• Conclusion

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Efficient index

• High space utilization
• Query fast
• Update quickly

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Agenda

• Efficient index
• kd-tree
• kdB-tree
• Bkd-tree
• Experiment Platform
• Experimental Results
• Conclusion

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kd-tree

• kd-tree
• Recursive partitioning of point set into two
  equal subsets using vertical/horizontal line
• Horizontal line on even levels, vertical on
  uneven levels
• One point in each leaf
• Query I/Os

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kd-tree

• Insertion
• Search down the tree
• Split the leaf
• Problem
• For insertion the resulting tree is no longer a
  kd-tree.
• Reason the lines in the internal nodes no longer
  partition the points into equal sized set
• Result deteriorate query performance
• Conclusion kd-tree is a static data structure

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Agenda

• Efficient index
• kd-tree
• kdB-tree
• Bkd-tree
• Experiment Platform
• Experimental Results
• Conclusion

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kdB-tree

• kdB-tree
• Points stored in leaves
• Leaves and nodes stored in disk blocks
• Query

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kdB-tree

• Problem
• For insertion the split of an internal node may
  result in the need for splits of several of the
  subtrees
• Result Update inefficient and bad space
  utilization

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Agenda

• Efficient index
• kd-tree
• kdB-tree
• Bkd-tree
• Experiment Platform
• Experimental Results
• Conclusion

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Bkd-tree

• Bulk Loading kd-trees

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Bkd-tree

• Classically (binary) O((N/B)log2(N/B)) I/Os
  (plus the cost of sorting O((N/B)logM/B(N/B))
  I/Os, which is still the same)

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Bkd-tree

• Improved (grid) O((N/B)logM/B(N/B)) I/Os
• Sort
• Build the log2t kd-tree
• tt grid, each strip contains N/t points
• Scan each grid cell and stored in tt matrix A
• Create the upper subtree

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Bkd-tree(cont)

• Improved (grid) O((N/B)logM/B(N/B)) I/Os
• Build the log2t kd-tree
• 4. Scan input points and distribute the input
  points into rectangle
• Build the bottom levels either in main memory or
  by recursing step
• O((N/B)logM/B(N/B)) I/Os (better than
  O((N/B)log2(N/B)) I/Os )

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Dynamic Updates

• A Bkd-tree consists of log2(N/M) kd-trees, the
  ith kd-tree is either empty or contains exactly
  2iM points

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Dynamic Updates

• Deletion O(logB(N/B)log2(N/M)) I/Os
• Insertion O((logM/B(N/B)log2(N/M))/B) I/Os

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Query

• To answer a window query, we simply query all
  log2(N/M) kd-tree
• I/O

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Agenda

• Efficient index
• kd-tree
• kdB-tree
• Bkd-tree
• Experiment Platform
• Experimental Results
• Conclusion

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Experiment Platform

• Three different types of point sets
• Real points from the TIGER
• Uniform data
• Points along a diagonal

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Agenda

• Efficient index
• kd-tree
• kdB-tree
• Bkd-tree
• Experiment Platform
• Experimental Results
• Conclusion

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Experimental Results

• Space Utilization

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Experimental Results

• Bulk Loading Performance

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Experimental Results

• Bulk Loading Performance

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Experimental Results

• Insertion Performance

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Experimental Results

• Insertion Performance

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Experimental Results

• Query Performance

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Experimental Results

• Query Performance

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Agenda

• Efficient index
• kd-tree
• kdB-tree
• Bkd-tree
• Experiment Platform
• Experimental Results
• Conclusion

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Conclusion

• For space utilization, KDBTree has poor
  utilizatin, while BkdTree is above 99.
• For insertion, a BkdTree is 100 times faster than
  KDBTree.
• For query, BkdTree is excellent

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QA

• Thank you!