Tree-Structured Indexes Lecture 5 R & G Chapter 10 If I had eight hours to chop down a tree, I'd spend six sharpening my ax. Abraham Lincoln Review: Files, Pages, Records Abstraction of stored data is files of records. Records live on pages Physical Record ID (RID) = Variable length data requires more sophisticated structures for records and pages. (why?) Records: offset array in header 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 constant Equality selections (op is =) Either tree or hash indexes help here. Range selections (op is one of <, >, <=, >=, BETWEEN) Hash indexes dont work for these. Tree-structured indexing techniques support both range selections and equality selections. ISAM: static structure; early index technology. B+ tree: dynamic, adjusts gracefully under inserts and deletes. ISAM =Indexed Sequential Access Method A Note of Caution ISAM is an old-fashioned idea B+-trees are usually better, as well see Though not always But, its a good place to start Simpler than B+-tree, but many of the same ideas Upshot Dont brag about being an ISAM expert on your resume Do understand how they work, and tradeoffs with B+-trees Range 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. Page 1 Page 2 Index File kN k1 k2 Page 3 Page N Data File Can do binary search on (smaller) index file! index entry ISAM P
0 K 1 P 1 K 2 P K m 2 Index file may still be quite large. But we can apply the idea repeatedly! Non-leaf Pages Leaf Pages Overflow page Leaf pages contain data entries. Primary pages
Pm Example ISAM Tree Index entries: they direct search for data entries in leaves. Example where each node can hold 2 entries; Root 40 10* 15* 20 33 20* 27* 51 33* 37* 40*
46* 51* 63 55* 63* 97* Data Pages ISAM is a STATIC Structure File creation: Leaf (data) pages allocated sequentially, sorted by search key; then pages allocated, then overflow pgs. Search: Start at root; use key comparisons to go to leaf. Cost = log F N ; # entries/pg (i.e., fanout), N = # leaf pgs index Index Pages Overflow pages F=
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. c tree structure: inserts/deletes affect only leaf pa Example: Insert 23*, 48*, 41*, 42* Root 40 Index Pages 20 33 20* 27* 51 63 51* 55*
33* 37* 40* 46* 48* 41* Pages Overflow 23* 63* 97* Pages 42* Note that 51* appears in index levels, but not in leaf! ISAM ---- Issues? Pros ????
Cons ???? B+ Tree: The Most Widely Used Index Insert/delete at log balanced. F N cost; keep tree height- 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. Index Entries (Direct search) Data Entries ("Sequence set")
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* ... Root 13 2* 3* 5* 7* 14* 16* 17 24 19* 20* 22* 30 24* 27* 29* 33* 34* 38* 39*
Based on the search for 15*, we know it is not in the tree! B+ 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 MBytes 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 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* Root Root 17 13 5 2* 3* 2* 3* 7* 24 30 24 13
5* 5* 17 7* 8* 14* 16* 14* 16* 19* 20* 22* 19* 20* 22* 30 33* 34* 38* 39* 24* 27* 29* 24* 27* 29* 33* 34* 38* 39* Notice that root was split, leading to increase in height. In this example, we can avoid split by redistributing entries; however, this is usually not done in practice. 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 copyup and pushup; be sure you understand the reasons for this. Data Page Split 2* 3* Index Page Split 5 13 2* 3* 5*
7* 8* Entry to be inserted in parent node. (Note that 5 is s copied up and continues to appear in the leaf.) 5 5* 7* 8* 5 17 24 13 17 24
30 Entry to be inserted in parent node. (Note that 17 is pushed up and only appears once in the index. Contrast this with a leaf split.) 30 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, merge L and sibling. If merge occurred, must delete entry (pointing to L or sibling) from parent of L. Merge could propagate to root, decreasing height. Example Tree (including 8*) Delete 19* and 20* ... Root Root 13 5
5* 5* 7* 7* 8* 8* 14* 14* 16* 16* 22* 19* 24* 20* 22* 30 30 27* 24* 29* 27* 29* 38* 39* 33* 33* 34* 34* 38* 39* Deleting 19* is easy. Deleting 20* is done with re-distribution. Notice how middle key is copied up.
... And Then Deleting 24* Must merge. Observe `toss of index entry (on right), and `pull down of index entry (below). 30 22* 27* 29* 33* 34* 38* 39* Root 5 2* 3*
5* 7* 8* 13 14* 16* 17 30 22* 27* 29* 33* 34* 38* 39* Example of Non-leaf Redistribution Tree is shown below during deletion of 24*. (What could be a possible initial tree?) In contrast to previous example, can redistribute entry from left child of root to right child. Root 22 5 2* 3* 5* 7* 8*
13 14* 16* 17 30 20 17* 18* 20* 21* 22* 27* 29* 33* 34* 38* 39* After Re-distribution Intuitively, entries are re-distributed by `pushing through the splitting entry in the parent node. It suffices to re-distribute index entry with key 20; weve re-distributed 17 as well for Root illustration. 17 5
2* 3* 5* 7* 8* 13 14* 16* 20 17* 18* 20* 21* 22 30 22* 27* 29* 33* 34* 38* 39* Prefix Key Compression Important to increase fan-out. (Why?) Key values in index entries only `direct traffic; can often compress them. E.g., If we have adjacent index entries with search key values Dannon Yogurt, David Smith and Devarakonda Murthy, we can abbreviate David Smith to Dav. (The other keys can be compressed too ...) Is this correct? Not quite! What if there is a data entry Davey Jones?
(Can only compress David Smith to Davi) In general, while compressing, must leave each index entry greater than every key value (in any subtree) to its left. Insert/delete must be suitably modified. 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. Also leads to minimal leaf utilization --- why? Bulk Loading can be done much more efficiently. Initialization: Sort all data entries, insert pointer to first (leaf) page in a new (root) page. Root 3* 4* Sorted pages of data entries; not yet in B+ tree 6* 9* 10* 11* 12* 13* 20* 22* 23* 31* 35* 36* 38* 41* 44* Bulk Loading (Contd.) Root
Index entries for leaf pages always entered into rightmost index page just above leaf level.3* When this fills up, it splits. (Split may go up right-most path to the root.) Much faster than repeated inserts, especially when one considers locking! 10 12 6 4* 3* 4* 6* 9* 20 23 20
10 6* 9* not yet in B+ tree 10* 11* 12* 13* 20*22* 23* 31* 35* 36* 38*41* 44* Root 6 Data entry pages 35 12 Data entry pages not yet in B+ tree 35 23 38 10* 11* 12* 13* 20*22* 23* 31* 35* 36* 38*41* 44* 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. A Note on `Order Order (d) concept replaced by physical space criterion in practice (`at least half-full). Index pages can typically hold many more entries than leaf pages. Variable sized records and search keys mean different nodes will contain different numbers of entries. Even with fixed length fields, multiple records with the same search key value (duplicates) can lead to variablesized data entries (if we use Alternative (3)). Many real systems are even sloppier than this --only reclaim space when a page is completely empty. Summary Tree-structured indexes are ideal for rangesearches, also good for equality searches. ISAM is a static structure. Only leaf pages modified; overflow pages needed. Overflow chains can degrade performance unless size of data set and data distribution stay constant.
B+ tree is a dynamic structure. Inserts/deletes leave tree height-balanced; log F N cost. High fanout (F) means depth rarely more than 3 or 4. Almost always better than maintaining a sorted file. Summary (Contd.) Typically, 67% occupancy on average. Usually preferable to ISAM, modulo locking considerations; adjusts to growth gracefully. If data entries are data records, splits can change rids! Key compression increases fanout, reduces height. Bulk loading can be much faster than repeated inserts for creating a B+ tree on a large data set. 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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