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Multilevel Indexing and B+ Trees

Multilevel Indexing and B+ Trees. Indexed Sequential Files. Provide a choice between two alternative views of a file: Indexed : the file can be seen as a set of records that is indexed by key; or

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Multilevel Indexing and B+ Trees

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  1. Multilevel Indexing and B+ Trees

  2. Indexed Sequential Files • Provide a choice between two alternative views of a file: • Indexed: the file can be seen as a set of records that is indexed by key; or • Sequential: the file can be accessed sequentially (physically contiguous records), returning records in order by key.

  3. Example of applications • Student record system in a university: • Indexed view: access to individual records • Sequential view: batch processing when posting grades • Credit card system: • Indexed view: interactive check of accounts • Sequential view: batch processing of payments

  4. The initial idea • Maintain a sequence set: • Group the records into blocks in a sorted way. • Maintain the order in the blocks as records are added or deleted through splitting, concatenation, and redistribution. • Construct a simple, single level index for these blocks. • Choose to build an index that contain the key for the last record in each block.

  5. Maintaining a Sequence Set • Sorting and re-organizing after insertions and deletions is out of question. We organize the sequence set in the following way: • Records are grouped in blocks. • Blocks should be at least half full. • Link fields are used to point to the preceding block and the following block (similar to doubly linked lists) • Changes (insertion/deletion) are localized into blocks by performing: • Block splitting when insertion causes overflow • Block merging or redistribution when deletion causes underflow.

  6. Example: insertion • Block size = 4 • Key : Last name Block 1 • Insert “BAIRD …”: Block 1 Block 2

  7. Example: deletion Block 1 • Delete “DAVIS”, “BYNUM”, “CARTER”, Block 2 Block 3 Block 4

  8. Add an Index set KeyBlock BERNE 1 CAGE 2 DUTTON 3 EVANS 4 FOLK 5 GADDIS 6

  9. Tree indexes • This simple scheme is nice if the index fits in memory. • If index doesn’t fit in memory: • Divide the index structure into blocks, • Organize these blocks similarly building a tree structure. • Tree indexes: • B Trees • B+ Trees • Simple prefix B+ Trees • …

  10. Separators BlockRange of KeysSeparator 1 ADAMS-BERNE BOLEN 2 BOLEN-CAGE CAMP 3 CAMP-DUTTON EMBRY 4 EMBRY-EVANS FABER 5 FABER-FOLK FOLKS 6 FOLKS-GADDIS

  11. root EMBRY Index set BOLEN CAMP FABER FOLKS ADAMS-BERNE CAMP-DUTTON EMBRY-EVANS FOLKS-GADDIS 1 3 4 6 BOLEN-CAGE FABER-FOLK 2 5

  12. B Trees • B-tree is one of the most important data structures in computer science. • What does B stand for? (Not binary!) • B-tree is a multiway search tree. • Several versions of B-trees have been proposed, but only B+ Trees has been used with large files. • A B+tree is a B-tree in which data records are in leaf nodes, and faster sequential access is possible.

  13. Formal definition of B+ Tree Properties • Properties of a B+ Tree of order v: • All internal nodes (except root) has at least v keys and at most 2v keys . • The root has at least 2 children unless it’s a leaf.. • All leaves are on the same level. • An internal node with k keys has k+1 children

  14. B+ tree: Internal/root node structure P0 K1 P1 K2 ……………… Pn-1 Kn Pn Each Pi is a pointer to a child node; each Ki is a search key value # of search key values= n, # of pointers = n+1 • Requirements: • K1 < K2 < … < Kn • For any search key value K in the subtree pointed by Pi, If Pi = P0, we requireK < K1 If Pi = Pn, Kn  K If Pi = P1, …, Pn-1, Ki < K  Ki+1

  15. B+ tree: leaf node structure L K1 r1 K2 ……………… Kn rn R • Pointer L points to the left neighbor; R points to the right neighbor • K1 < K2 < … < Kn • v  n  2v (v is the order of this B+ tree) • We will use Ki* for the pair <Ki, ri> and omit L and R for simplicity

  16. Root 40 20 33 51 63 46* 55* 10* 15* 20* 27* 33* 37* 40* 51* 97* 63* Example: B+ tree with order of 1 • Each node must hold at least 1 entry, and at most 2 entries

  17. Root 30 13 17 24 39* 3* 5* 19* 20* 22* 24* 27* 38* 2* 7* 14* 15* 29* 33* 34* Example: Search in a B+ tree order 2 • Search: how to find the records with a given search key value? • Begin at root, and use key comparisons to go to leaf • Examples: search for 5*, 16*, all data entries >= 24* ... • The last one is a range search, we need to do the sequential scan, starting from the first leaf containing a value >= 24.

  18. B+ Trees in Practice • Typical order: 100. Typical fill-factor: 67%. • average fanout = 133 (i.e, # of pointers in internal node) • Can often hold top levels in buffer pool: • Level 1 = 1 page = 8 Kbytes • Level 2 = 133 pages = 1 Mbyte • Level 3 = 17,689 pages = 133 MBytes • Suppose there are 1,000,000,000 data entries. • H = log133(1000000000/132) < 4 • The cost is 5 pages read

  19. How to Insert a Data Entry into a B+ Tree? • Let’s look at several examples first.

  20. 8* 8* 3* 5* 2* 7* You overflow 30 13 17 24 5* 7* 3* 2* Inserting 16*, 8* into Example B+ tree Root 30 13 17 24 16* 14* 15* One new child (leaf node) generated; must add one more pointer to its parent, thus one more key value as well.

  21. You overflow! 5 13 17 24 30 Inserting 8* (cont.) 13 17 24 30 • Copyup the middle value (leaf split) Entry to be inserted in parent node. (Note that 5 is s copied up and 5 continues to appear in the leaf.) 3* 5* 2* 7* 8*

  22. Entry to be inserted in parent node. 17 5 13 17 24 30 appears once in the index. Contrast this with a leaf split.) 5 13 24 30 Insertion into B+ tree (cont.) • Understand difference between copy-up and push-up • Observe how minimum occupancy is guaranteed in both leaf and index pg splits. We split this node, redistribute entries evenly, and push up middle key.  (Note that 17 is pushed up and only

  23. Example B+ Tree After Inserting 8* Root 17 24 5 13 30 39* 2* 3* 5* 7* 8* 19* 20* 22* 24* 27* 38* 29* 33* 34* 14* 15* Notice that root was split, leading to increase in height.

  24. Inserting a Data Entry into a B+ Tree: Summary • 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, put middle key in L2 • 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.

  25. Deleting a Data Entry from a B+ Tree • Examine examples first …

  26. 22* 27* 29* 22* 24* Delete 19* and 20* Root 17 24 5 13 30 39* 2* 3* 5* 7* 8* 19* 20* 22* 24* 27* 38* 29* 33* 34* 14* 16* You underflow Have we still forgot something?

  27. Deleting 19* and 20* (cont.) • Notice how 27 is copied up. • But can we move it up? • Now we want to delete 24 • Underflow again! But can we redistribute this time? Root 17 27 5 13 30 39* 2* 3* 5* 7* 8* 22* 24* 27* 29* 38* 33* 34* 14* 16*

  28. Deleting 24* You underflow Merge with sibling! • Observe the two leaf nodes are merged, and 27 is discarded from their parent, but … • Observe `pull down’ of index entry (below). 30 39* 22* 27* 38* 29* 33* 34* New root 5 13 17 30 3* 39* 2* 5* 7* 8* 22* 38* 27* 33* 34* 14* 16* 29*

  29. Deleting a Data Entry from a B+ Tree: Summary • 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.

  30. 2* 3* 5* 7* 8* 39* 17* 18* 38* 20* 21* 22* 27* 29* 33* 34* 14* 16* Example of Non-leaf Re-distribution • Tree is shown below during deletion of 24*. (What could be a possible initial tree?) • In contrast to previous example, can re-distribute entry from left child of root to right child. Root 22 30 17 20 5 13

  31. After Re-distribution • Intuitively, entries are re-distributed by `pushingthrough’ the splitting entry in the parent node. • It suffices to re-distribute index entry with key 20; we’ve re-distributed 17 as well for illustration. Root 17 22 30 5 13 20 2* 3* 5* 7* 8* 39* 17* 18* 38* 20* 21* 22* 27* 29* 33* 34* 14* 16*

  32. Terminology • Bucket Factor: the number of records which can fit in a leaf node. • Fan-out : the average number of children of an internal node. • A B+tree index can be used either as a primary index or a secondary index. • Primary index: determines the way the records are actually stored (also called a sparse index) • Secondary index: the records in the file are not grouped in buckets according to keys of secondary indexes (also called a dense index)

  33. Summary • Tree-structured indexes are ideal for range-searches, also good for equality searches. • B+ tree is a dynamic structure. • Inserts/deletes leave tree height-balanced; High fanout (F) means depth rarely more than 3 or 4. • Almost always better than maintaining a sorted file. • Typically, 67% occupancy on average. • If data entries are data records, splits can change rids! • Most widely used index in database management systems because of its versatility. One of the most optimized components of a DBMS.

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