1. P-Tree Implementation Anne Denton

2. So far: Logical Definition C.f. Dr. Perrizo’s slides Logical definition Defines node information Representation of structure open Wide variety of implementations has been tried

3. 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!

4. 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

5. 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

6. 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

7. Can We Convert Address to Bit-Vectors? [ ] 1001[ ]0110 [01] 0010[01]1001 [10] 1101 [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

8. 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

9. Representation of Standard Example

10. P-Tree Anding Start at root Pursue new (potentially) mixed children Deriving new mixed (m) and pure1 (u): u is AND of all u i m is AND of all (m i OR u i ) AND NOT u Cannot be done with either u or m alone

11. 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 000000000 000000011 000000101 000000112 000001001 000001012

12. 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 (000000008) 000000017 000000106 000000116 000001005 000001015

13. 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

14. 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)

15. Full Example

16. 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