1 / 20

B+-Tree Index

B+-Tree Index. Chapter 10. Modified by Donghui Zhang Nov 9, 2005. Motivation. Suppose every disk page holds 133 records. You are given 133 4 = 0.3 billion records. They occupy 133 3 = 2.3 million disk pages. You can utilize a small memory buffer of 134 pages.

macon-knapp
Download Presentation

B+-Tree Index

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. B+-Tree Index Chapter 10 Modified by Donghui Zhang Nov 9, 2005

  2. Motivation • Suppose every disk page holds 133 records. • You are given 1334 = 0.3 billion records. They occupy 1333 = 2.3 million disk pages. • You can utilize a small memory buffer of 134 pages. • You can build an index structure. • Given the key of a record, what is the minimum guaranteed number of disk I/Os to find the record?

  3. Content • B+-tree index • Structure • Search • Insert • Delete • Bulk-loading a B+-tree

  4. Index Entries (Direct search) Data Entries ("Sequence set") B+ Tree Structure • Insert/delete at log F N cost; keep tree height-balanced. (F = fanout, N = # leaf pages) • Minimum 50% occupancy (except for root). Each node contains d <= m <= 2d entries. The parameter d is called the order of the tree. • Supports equality and range-searches efficiently.

  5. B+ Tree Equality Search • Search begins at root, and key comparisons direct it to a leaf. • Search for 15*… Root 30 13 17 24 39* 3* 5* 19* 20* 22* 24* 27* 38* 2* 7* 14* 16* 29* 33* 34* • Based on the search for 15*, we know it is not in the tree!

  6. B+ Tree Range Search • Search all records whose ages are in [15,28]. • Equality search 15*. • Follow sibling pointers. Root 30 13 17 24 39* 3* 5* 19* 20* 22* 24* 27* 38* 2* 7* 14* 16* 29* 33* 34*

  7. B+ Trees in Practice • Typical order: 100. Typical fill-factor: 67%. • average fanout = 133 • Can often hold top levels in buffer pool: • Level 1 = 1 page = 8 KB • Level 2 = 133 pages = 1 MB • Level 3 = 17,689 pages = 145 MB • Level 4 = 2,352,637 pages = 19 GB • With 1 MB buffer, can locate one record in 19 GB (or 0.3 billion records) in two I/Os!

  8. Inserting a Data Entry into a B+ Tree • 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.

  9. Inserting 8* into Example B+ Tree • Find leaf, in the same way as the Search algorithm. • Handle overflow by splitting. Root 30 13 17 24 39* 3* 5* 19* 20* 22* 24* 27* 38* 2* 7* 14* 16* 29* 33* 34*

  10. Entry to be inserted in parent node. (Note that 17 is pushed up and only 17 this with a leaf split.) 5 13 24 30 Inserting 8* into Example B+ Tree Entry to be inserted in parent node. (Note that 5 is s copied up and • Observe how minimum occupancy is guaranteed in both leaf and index pg splits. • Note difference between copy-upand push-up; be sure you understand the reasons for this. 5 continues to appear in the leaf.) 3* 5* 2* 7* 8* appears once in the index. Contrast

  11. 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* 16* • Notice that root was split, leading to increase in height. • In this example, we can avoid split by re-distributing entries; however, this is usually not done in practice.

  12. Deleting a Data Entry from a B+ Tree • 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.

  13. Deleting 19* and 20* Root • Deleting 19* is easy. • Deleting 20* is done with re-distribution. 17 24 5 13 30 39* 2* 3* 5* 7* 8* 19* 20* 22* 24* 27* 38* 29* 33* 34* 14* 16*

  14. After Deleting 19* and 20* • Notice, in re-distribution, how middle key is copied up. • If delete 24*… Root 17 27 5 13 30 39* 2* 3* 5* 7* 8* 22* 24* 27* 29* 38* 33* 34* 14* 16*

  15. ... And Then Deleting 24* • Must merge. • Observe `toss’ of index entry (on right), and `pull down’ of index entry (below). 30 39* 22* 27* 38* 29* 33* 34* Root 5 13 17 30 3* 39* 2* 5* 7* 8* 22* 38* 27* 33* 34* 14* 16* 29*

  16. 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*. • In contrast to previous example, can re-distribute entry from left child of root to right child. Root 22 30 17 20 5 13

  17. 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*

  18. 3* 6* 9* 10* 11* 12* 13* 23* 31* 36* 38* 41* 44* 4* 20* 22* 35* Bulk Loading of a B+ Tree • If we have a large collection of records, and we want to create a B+ tree on some field, doing so by repeatedly inserting records is very slow. • Bulk Loadingcan be done much more efficiently. • Initialization: Sort all data entries, insert pointer to first (leaf) page in a new (root) page. Root Sorted pages of data entries; not yet in B+ tree

  19. Bulk Loading (Contd.) Root • Index entries for leaf pages always entered into right-most index page just above leaf level. • Assume pages in the rightmost path to have double page size. • Split when double plus one. Data entry pages 10 12 20 6 not yet in B+ tree 3* 6* 9* 10* 11* 12* 13* 23* 31* 36* 38* 41* 44* 4* 20* 22* 35* Root 12 Data entry pages not yet in B+ tree 6 20 10 23 3* 6* 9* 10* 11* 12* 13* 23* 31* 36* 38* 41* 44* 4* 20* 22* 35*

  20. Summary of Bulk Loading • Option 1: multiple inserts. • Slow. • Does not give sequential storage of leaves. • Option 2:Bulk Loading • Has advantages for concurrency control. • Fewer I/Os during build. • Leaves will be stored sequentially (and linked, of course). • Can control “fill factor” on pages.

More Related