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B+-trees

B+-trees. Model of Computation. CPU. Memory. Data stored on disk(s) Minimum transfer unit: a page = b bytes or B records (or block) N records -> N/B = n pages I/O complexity: in number of pages. Disk. I/O complexity.

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B+-trees

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  1. B+-trees

  2. Model of Computation CPU Memory • Data stored on disk(s) • Minimum transfer unit: a page = b bytes or B records (or block) • N records -> N/B = n pages • I/O complexity: in number of pages Disk

  3. I/O complexity • An ideal index has space O(N/B), update overhead O(1) or O(logB(N/B)) and search complexity O(a/B) or O(logB(N/B) + a/B) where a is the number of records in the answer • But, sometimes CPU performance is also important… minimize cache misses -> don’t waste CPU cycles

  4. B+-tree • Records must be ordered over an attribute, SSN, Name, etc. • Queries: exact match and range queries over the indexed attribute: “find the name of the student with ID=087-34-7892” or “find all students with gpa between 3.00 and 3.5”

  5. B+-tree:properties • Insert/delete at log F (N/B) cost; keep tree height-balanced.(F = fanout) • Minimum 50% occupancy (except for root). Each node contains d <= m <= 2d entries/pointers. • Two types of nodes: index nodes and data nodes; each node is 1 page (disk based method)

  6. Example 100 Root 120 150 180 30 3 5 11 120 130 180 200 100 101 110 150 156 179 30 35

  7. Index node 57 81 95 to keys to keys to keys to keys < 57 57£ k<81 81£k<95 95£

  8. Data node Struct { Key real; Pointr *long; } entry; Node entry[B]; From non-leaf node to next leaf in sequence 57 81 95 To record with key 57 To record with key 81 To record with key 85

  9. Insertion • Find correct leaf L. • Put data entry onto L. • If L has enough space, done! • Else, must splitL (into L and a new node L2) • Redistribute entries evenly, copy upmiddle key. • Insert index entry pointing to L2 into parent of L. • This can happen recursively • To split index node, redistribute entries evenly, but push upmiddle key. (Contrast with leaf splits.) • Splits “grow” tree; root split increases height. • Tree growth: gets wider or one level taller at top.

  10. Deletion • Start at root, find leaf L where entry belongs. • Remove the entry. • If L is at least half-full, done! • If L has only d-1 entries, • Try to re-distribute, borrowing from sibling (adjacent node with same parent as L). • If re-distribution fails, mergeL and sibling. • If merge occurred, must delete entry (pointing to L or sibling) from parent of L. • Merge could propagate to root, decreasing height.

  11. Characteristics • Optimal method for 1-d range queries: Space: O(N/B), Updates: O(logB(N/B)), Query:O(logB(N/B) + a/B) • Space utilization: 67% for random input • Original B-tree: index nodes store also pointers to records

  12. Example 100 Range[32, 160] Root 120 150 180 30 3 5 11 120 130 180 200 100 101 110 150 156 179 30 35

  13. Other issues • Internal node architecture [Lomet01]: • Reduce the overhead of tree traversal. • Prefix compression: In index nodes store only the prefix that differentiate consecutive sub-trees. Fanout is increased. • Cache sensitive B+-tree • Place keys in a way that reduces the cache faults during the binary search in each node. • Eliminate pointers so a cache line contains more keys for comparison.

  14. References [Lomet01]:David B. Lomet: The Evolution of Effective B-tree: Page Organization and Techniques: A Personal Account. SIGMOD Record 30(3): 64-69 (2001) http://www.acm.org/sigmod/record/issues/0109/a1-lomet.pdf

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