P-Tree%20Implementation

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P-Tree Implementation

• Anne Denton

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So far Logical Definition

• C.f. Dr. Perrizos slides
• Logical definition
• Defines node information
• Representation of structure open
• Wide variety of implementations has been tried

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Tree Representation Options

• Pointers
• Tree-walks
• Depth-first
• Breadth-first
• Node addresses (P-trees qids)
• Note Any one tree representation will make the
  tree loss-less!

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Issues

• Storage requirements
• Suitability to distributed processing
• (e.g., avoiding pointer swizzling)
• Ease of access to particular nodes
• Main issue
• Data structure must optimize anding speed at each
  node

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Main Desired Property

• Anding through Bit-vector operations
• New node information
• New structural information
• Why?
• Parallelism up-to 32 or 64 bits processed in
  parallel for single processor CPU

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QID-based P-Vector representation (Example P1V)

• 1001
• 01 0010
• 10 1101
• 01.00 1110
• 01.11 0010
• 10.10 1101
• Node information stored as bit-vector
• Structural information
• Traditional relation of degree 2
• Address is key

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Can We Convert Address to Bit-Vectors?

• 1001 0110
• 01 0010 01 1001
• 10 1101 ltgt 10 0010
• 01.00 1110
• 01.11 0010
• 10.10 1101
• We know this PMV!
• Claim qid is now redundant
• Standard conversion to bit-vectors

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Does this Define Structure?

• Yes!
• Concept
• Similar to Depth-First Search
• Mixed vector specifies existing children
• Slight modification
• Store all children to one node sequentially
• Reason address can be computed through counts on
  mixed

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Representation of Standard Example
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P-Tree Anding

• Start at root
• Pursue new (potentially) mixed children
• Deriving new mixed (m) and pure1 (u)
• u is AND of all ui
• m is AND of all (mi OR ui) AND NOT u
• Cannot be done with either u or m alone

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Fast Counting using Table Look-up

• How many bits are set in 01100110?
• Look-up table stores 4 for index 102
• Works up-to sequences of 8 bit
• 00000000 0
• 00000001 1
• 00000010 1
• 00000011 2
• 00000100 1
• 00000101 2

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Finding the next 1

• Which is the first bits set in 01100110?
• Look-up table stores 1 for index 102
• Works up-to sequences of 8 bit
• (00000000 8)
• 00000001 7
• 00000010 6
• 00000011 6
• 00000100 5
• 00000101 5

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Finding a child

• Assume children are stored in sequence
• For mixed vector 01100110 where is the child with
  index 5 (part of qid)?
• Count the children in 01100
• Storage location calculated with one table look-up

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Potential problems

• Eliminating large sub-trees slow
• Speeding up and
• Introduce additional access structure
• Array indices as pointers
• Note
• No lowest level due to adjacent storage of
  children
• Reduces storage by about 1/fanout (e.g., 1/16)
• Access structure does not need to be stored
  (P-tree loss-less without it)

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Full Example
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Summary

• PV1 node values stored as bit-vectors
• Now tree structure stored as bit-vectors as well
• Benefits Several fast bit-vector algorithms can
  be used
• Description of structure
• Modified depth-first tree-walk
• Additional access structure efficient