Chapter 13: Indexing. Chapter 13. ? value. Topics. Indexing & Hashing. value. Conventional indexes B-trees Hashing schemes (self-study) record

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1 Chapter 13: Indexing (Slides by Hector Garcia-Molina, Chapter 13 1 Chapter 13 Indexing & Hashing value record? value Chapter 13 2 Topics Conventional indexes B-trees Hashing schemes (self-study) Chapter

2 Sequential File 0 Chapter 13 4 Dense Index Sequential File 0 Chapter 13 5 Sparse Index Sequential File 0 Chapter

3 Sparse 2nd level Sequential File 0 Chapter 13 7 Question: Can we build a dense, 2nd level index for a dense index? Chapter 13 8 Notes on pointers: (1) Block pointer (sparse index) can be smaller than record pointer BP RP Chapter

4 Notes on pointers: (2) If file is contiguous, then we can omit pointers (i.e., compute them) Chapter 13 R1 K1 K2 K3 K4 R2 R3 R4 say: 24 B per block if we want K3 block: get it at offset (3-1)24 = 48 bytes Chapter Sparse vs. Dense Tradeoff Sparse: Less index space per record can keep more of index in memory Dense: Can tell if any record exists without accessing file (Also: sparse better for insertions dense needed for secondary indexes) Chapter

5 Terms Index sequential file Search key ( primary key) Primary index (on Sequencing field) Secondary index Dense index (all Search Key values in) Sparse index Multi-level index Chapter Next: Duplicate keys Deletion/Insertion Secondary indexes Chapter Duplicate keys 45 Chapter

6 Duplicate keys Dense index, one way to implement? 45 Chapter Duplicate keys Dense index, better way? 45 Chapter Duplicate keys Sparse index, one way? careful if looking for or! 45 Chapter

7 Duplicate keys Sparse index, another way? place first new key from block should this be? 45 Chapter Summary Duplicate values, primary index Index may point to first instance of each value only File Index a a a.. b Chapter 13 Deletion from sparse index Chapter

8 Deletion from sparse index delete record Chapter Deletion from sparse index delete record Chapter Deletion from sparse index delete records & Chapter

9 Deletion from dense index Chapter Deletion from dense index delete record Chapter Insertion, sparse index case Chapter

10 Insertion, sparse index case insert record 34 our lucky day! we have free space where we need it! 34 Chapter Insertion, sparse index case insert record 15 Illustrated: Immediate reorganization Variation: insert new block (chained file) update index Chapter Insertion, sparse index case insert record overflow blocks (reorganize later...) Chapter 13

11 Insertion, dense index case Similar Often more expensive... Chapter Secondary indexes Sequence field 0 Chapter Secondary indexes Sparse index 0... does not make sense! 0 Sequence field Chapter

12 Secondary indexes Dense index Sequence field... sparse high level... 0 Chapter With secondary indexes: Lowest level is dense Other levels are sparse Also: Pointers are record pointers (not block pointers; not computed) Chapter Summary so far Conventional index Basic Ideas: sparse, dense, multi-level Duplicate Keys Deletion/Insertion Secondary indexes Chapter

13 Conventional indexes Advantage: - Simple - Index is sequential file good for scans Disadvantage: - Inserts expensive, and/or - Lose sequentiality & balance Chapter Outline: Conventional indexes B-Trees NEXT Hashing schemes (self-study) Chapter NEXT: Another type of index Give up on sequentiality of index Try to get balance Chapter

14 B+Tree Example n=3 Root Chapter 13 Sample non-leaf to keys to keys to keys to keys < k<81 81 k<95 95 Chapter Sample leaf node: From non-leaf node 57 to next leaf in sequence To record with key 57 To record with key 81 To record with key Chapter

15 Size of nodes: n+1 pointers (fixed) n keys Chapter Don t want nodes to be too empty Use at least Non-leaf: Leaf: (n+1)/2 pointers (n+1)/2 pointers to data Chapter n=3 Full node min. node Non-leaf Leaf counts even if null Chapter

16 B+tree rules: tree of order n (1) All leaves are at the same lowest level (balanced tree) (2) Pointers in leaves point to records, except for sequence pointer Chapter (3) Number of pointers/keys for B+tree Max Max Min ptrs keys ptrs data Min keys Non-leaf (non-root) n+1 n (n+1)/2 (n+1)/2-1 Leaf (non-root) n+1 n (n+1)/2 (n+1)/2 Root n+1 n 1 1 Chapter Insert into B+tree (a) simple case space available in leaf (b) leaf overflow (c) non-leaf overflow (d) new root Chapter

17 (a) Insert key = 32 n= Chapter (a) Insert key = 7 n= Chapter 13 (c) Insert key = 1 n= Chapter

18 (d) New root, insert 45 n=3 new root Chapter Deletion from B+tree (a) Simple case - no example (b) Coalesce with neighbor (sibling) (c) Re-distribute keys (d) Cases (b) or (c) at non-leaf Chapter (b) Coalesce with sibling Delete n=4 0 Chapter

19 (c) Redistribute keys Delete n= Chapter (d) Non-leaf coalese Delete 37 n=4 new root Chapter B+tree deletions in practice Often, coalescing is not implemented Too hard and not worth it! Chapter

20 Variation on B+tree: B-tree (no +) Idea: Avoid duplicate keys Have record pointers in non-leaf nodes Chapter K1 P1 K2 P2 K3 P3 to record to record to record with K1 with K2 with K3 to keys to keys to keys to keys < K1 K1<x<K2 K2<x<k3 >k3 Chapter B-tree example n=2 sequence pointers not useful now! (but keep space for simplicity) Chapter 13

21 Tradeoffs: B-trees have faster lookup than B+trees in B-tree, non-leaf & leaf different sizes in B-tree, deletion more complicated B+trees preferred! Chapter But note: If blocks are fixed size (due to disk and buffering restrictions) Then lookup for B+tree is actually better!! Chapter So... 8 records B+ B ooooooooooooo ooooooooo 156 records 8 records Total = 116 Conclusion: For fixed block size, B+ tree is better because it is bushier Chapter

22 Outline/summary Conventional Indexes Sparse vs. dense Primary vs. secondary B trees B+trees vs. B-trees B+trees vs. indexed sequential Hashing schemes (self-study) Chapter