Tree-Structured IndexesReview: Files, Pages, RecordsTree-Structured Indexes: IntroductionA Note of CautionRange SearchesISAMExample ISAM TreeISAM is a STATIC StructureExample: Insert 23*, 48*, 41*, 42*PowerPoint PresentationISAM ---- Issues?B+ Tree: The Most Widely Used IndexExample B+ TreeB+ Trees in PracticeInserting a Data Entry into a B+ TreeSlide 17Data vs. Index Page Split (from previous example of inserting “8*”)Deleting a Data Entry from a B+ TreeSlide 20Prefix Key CompressionSuffix Key CompressionBulk Loading of a B+ TreeBulk Loading (Contd.)Summary of Bulk LoadingA Note on `Order’SummarySummary (Contd.)Tree-Structured IndexesR & G Chapter 10“If I had eight hours to chop down a tree, I'd spend six sharpening my ax.” Abraham LincolnReview: Files, Pages, Records•Abstraction of stored data is “files” with “pages” of “records”.–Records live on pages–Physical Record ID (RID) = <page#, slot#>•Variable length data requires more sophisticated structures for records and pages. (why?)–Fields in Records: offset array in header–Records on Pages: Slotted pages w/internal offsets & free space area•Often best to be “lazy” about issues such as free space management, exact ordering, etc. (why?)•Files can be unordered (heap), sorted, or kinda sorted (i.e., “clustered”) on a search key.–Tradeoffs are update/maintenance cost vs. speed of accesses via the search key.–Files can be clustered (sorted) at most one way.•Indexes can be used to speed up many kinds of accesses. (i.e., “access paths”)Tree-Structured Indexes: Introduction•Selections of form: field <op> constant•Equality selections (op is =)–Either “tree” or “hash” indexes help here.•Range selections (op is one of <, >, <=, >=, BETWEEN)–“Hash” indexes don’t work for these.•More complex selections (e.g. spatial containment)–There are fancier trees that can do this… more on this soon!•Tree-structured indexing techniques support both range selections and equality selections.•ISAM: static structure; early index technology.–(Indexed Sequential Access Method)•B+ tree: dynamic, adjusts gracefully under inserts and deletes.A Note of Caution•ISAM is an old-fashioned idea–B+-trees are usually better, as we’ll see•Though not always•But, it’s a good place to start–Simpler than B+-tree, but many of the same ideas•Upshot–Don’t brag about being an ISAM expert on your resume–Do understand how they work, and tradeoffs with B+-treesRange Searches•``Find all students with gpa > 3.0’’–If data is in sorted file, do binary search to find first such student, then scan to find others.–Cost of binary search in a database can be quite high. Q: Why???•Simple idea: Create an `index’ file. Can do binary search on (smaller) index file!Page 1Page 2Page NPage 3Data Filek2kNk1Index FileISAM•Index file may still be quite large. –But we can apply the idea repeatedly! Leaf pages contain data entries.P0K1P1K2P2KmPmindex entryNon-leafPagesPagesLeafOverflow pagePrimary pagesExample ISAM Tree•Index entries:<search key value, page id> they direct search for data entries in leaves.•Example where each node can hold 2 entries;10* 15* 20* 27* 33* 37*40*46*51*55*63*97*20 3351 6340RootISAM is a STATIC Structure•File creation: Leaf (data) pages allocated •sequentially, sorted by search key; then •index pages allocated, then overflow pgs.•Search: Start at root; use key comparisons to go to leaf. –Cost = log F N ; F = # entries/pg (i.e., fanout), N = # leaf pgs– no need for `next-leaf-page’ pointers. (Why?)•Insert: Find leaf that data entry belongs to, and put it there. Overflow page if necessary.•Delete: Find and remove from leaf; if empty page, de-allocate. Static tree structure: inserts/deletes affect only leaf pages.Data PagesIndex PagesOverflow pages23*48*Example: Insert 23*, 48*, 41*, 42*10* 15* 20* 27* 33* 37*40*46*51*55*63*97*20 3351 6340RootOverflowPagesLeafIndexPagesPagesPrimary41*42*48*10* 15* 20* 27* 33* 37*40*46*51*55*63*97*20 3351 6340RootOverflowPagesLeafIndexPagesPagesPrimary23*41*42* ... then Deleting 42*, 51*, 97* Note that 51* appears in index levels, but not in leaf!ISAM ---- Issues?•Pros–????•Cons–????•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 m entries where d <= m <= 2d entries. “d” is called the order of the tree.•Supports equality and range-searches efficiently.•As in ISAM, all searches go from root to leaves, but structure is dynamic.B+ Tree: The Most Widely Used IndexIndex EntriesData Entries("Sequence set")(Direct search)Example B+ Tree•Search begins at root, and key comparisons direct it to a leaf (as in ISAM).•Search for 5*, 15*, all data entries >= 24* ... Based on the search for 15*, we know it is not in the tree!Root1724302*3* 5*7* 14* 16*19* 20*22* 24* 27*29* 33* 34*38*39*13B+ Trees in Practice•Typical order: 100. Typical fill-factor: 67%.–average fanout = 133•Typical capacities:–Height 2: 1333 = 2,352,637 entries–Height 3: 1334 = 312,900,700 entries•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 MBytesInserting a Data Entry into a B+ Tree•Find correct leaf L. •Put data entry onto L.–If L has enough space, done!–Else, must split L (into L and a new node L2)•Redistribute entries evenly, copy up middle key.•Insert index entry pointing to L2 into parent of L.•This can happen recursively–To split index node, redistribute entries evenly, but push up middle key. (Contrast with leaf splits.)•Splits “grow” tree; root split increases height. –Tree growth: gets wider or one level taller at top.Example B+ Tree - Inserting 8* 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.Root1724302*3* 5*7* 14* 16*19* 20*22* 24* 27*29* 33* 34*38*39*132* 3*Root17243014* 16*19* 20* 22* 24* 27*29* 33* 34*38*39*1357*5* 8*Data vs. Index Page Split (from previous example of inserting “8*”)•Observe how minimum occupancy is guaranteed in both leaf and index pg splits.•Note difference between copy-up and push-up; be sure you understand the reasons for this.2*3*
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