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Indexes and B-Trees Lecture 9 R & G Chapters 8 & 9 “If I had eight hours to chop down a tree, I'd spend six sharpening my ax.” Abraham Lincoln.

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Presentation on theme: "Indexes and B-Trees Lecture 9 R & G Chapters 8 & 9 “If I had eight hours to chop down a tree, I'd spend six sharpening my ax.” Abraham Lincoln."— Presentation transcript:

1 Indexes and B-Trees Lecture 9 R & G Chapters 8 & 9 “If I had eight hours to chop down a tree, I'd spend six sharpening my ax.” Abraham Lincoln

2 Administrivia Homework 1 Due Tonight, 10pm Homework 2 Available Today, Due 3 weeks from today Midterm Exam 1 will be a week from Thursday –It will be in class, at the usual time –Next Tuesday’s class will be a review More SQL Exercises on the Class Website

3 Review Last two weeks: –Formal Query Languages: Rel. Algebra & Calculus –Actual Query Language: SQL This week: Indexes –Tree Indexes (HW2) –Hash Indexes Next week: –Review –Midterm 1

4 Review: Division A/B A B1 B2 B3 A/B1 A/B2A/B3

5 Review: Use NOT Exists for Division SELECT S.sname FROM Sailors S WHERE NOT EXISTS (SELECT B.bid FROM Boats B WHERE NOT EXISTS (SELECT R.bid FROM Reserves R WHERE R.bid=B.bid AND R.sid=S.sid)) Find Sailors S such that... there is no boat B... without a reservation by Sailor S Find sailors who’ve reserved all boats. X = set of sailors and Y = set of all boats with reservations. Recall: X/Y means only give me X tuples that have a match in Y.

6 2 Division sidbidday 11039/ / / / /13 sidsnameratingage 1Frodo722 2Bilbo239 3Sam827 Sailors Reserves S S S SELECT S.sname FROM Sailors S WHERE NOT EXISTS (SELECT B.bid FROM Boats B WHERE NOT EXISTS (SELECT R.bid FROM Reserves R WHERE R.bid=B.bid AND R.sid=S.sid)) bidbnamecolor 101Ninared 103Pintablue Boats R R R B B R R

7 Null Values Values are sometimes –unknown (e.g., a rating has not been assigned) or –inapplicable (e.g., no spouse’s name). –SQL provides a special value null for such situations. The presence of null complicates many issues. E.g.: –Special operators needed to check if value is/is not null. –“rating>8” - true or false when rating is null? What about AND, OR and NOT connectives? –Need a 3-valued logic (true, false and unknown). –Meaning of constructs must be defined carefully. (e.g., WHERE clause eliminates rows that don’t evaluate to true.) –New operators (in particular, outer joins) possible/needed.

8 Null Values – 3 Valued Logic ANDTFNull T F NULL (null > 0) (null + 1) (null = 0) null AND true is null ORTFNull T F NULL T FNull TF FF F F TT T T

9 Null Values in SQL “Where” clause must evaluate to true “IS NULL” operator, e.g. “where name is null” “IS NOT NULL” operator Outer Joins: Left, Right, Full

10 SELECT s.sid, s.name, r.bid FROM Sailors s LEFT OUTER JOIN Reserves r ON s.sid = r.sid

11 SELECT r.sid, b.bid, b.name FROM Reserves r RIGHT OUTER JOIN Boats b ON r.bid = b.bid

12 Review – Buffer Management and Files Storage of Data –Fields, either fixed or variable length... –Stored in Records... –Stored in Pages... –Stored in Files If data won’t fit in RAM, store on Disk –Need Buffer Pool to hold pages in RAM –Different strategies decide what to keep in pool

13 Today: File Organization How to keep pages of records on disk but must support operations: –scan all records –search for a record id “RID” –search for record(s) with certain values –insert new records –delete old records

14 Alternative File Organizations Many alternatives exist, tradeoffs for each: –Heap files: Suitable when typical access is file scan of all records. –Sorted Files: Best for retrieval in search key order Also good for search based on search key – Indexes: Organize records via trees or hashing. Like sorted files, speed up searches for search key fields Updates are much faster than in sorted files.

15 Indexes Often want to get records byvalues in one or more fields, e.g., –Find all students in the “CS” department –Find all students with a gpa > 3 An index on a file is a: –Disk-based data structure –Speeds up selections on the search key fields for the index. –Any subset of the fields of a relation can be index search key –Search key is not the same as key (e.g. doesn’t have to be unique ID). An index –Contains a collection of key/data entry pairs –Supports efficient retrieval of all records with a given search key value k.

16 Index Classification What selections does it support? What does index actually store? –3 alternatives: Data record with key value k Clustered vs. Unclustered Indexes Single Key vs. Composite Indexes Tree-based, hash-based, other Can have multiple (different) indexes per file. –E.g. file sorted by age, with a hash index on salary and a B+tree index on name.

17 First Question to Ask About an Index What kinds of selections does it support? –Selections of form field constant –Equality selections (op is =) –Range selections (op is one of, =, BETWEEN) –More exotic selections: 2-dimensional ranges (“east of Berkeley and west of Truckee and North of Fresno and South of Eureka”) –Or n-dimensional 2-dimensional distances (“within 2 miles of Soda Hall”) –Or n-dimensional Ranking queries (“10 restaurants closest to VLSB”) Regular expression matches, genome string matches, etc. One common n-dimensional index: R-tree

18 What data is held by the index? Alternative 1: Actual data record (with key value k) –Index structure is file organization for data records (like Heap files or sorted files). –At most one index on a table can use Alternative 1. –Saves pointer lookups –Can be expensive to maintain with insertions and deletions.

19 What data is held by the index? (Contd.) Alternative 2 and Alternative 3 –Easier to maintain than Alt 1. –At most one index can use Alternative 1; any others must use Alternatives 2 or 3. –Alternative 3 more compact than Alternative 2, but leads to variable sized data entries even if search keys are of fixed length. –Even worse, for large rid lists the data entry might have to span multiple pages!

20 Clustered and Unclustered Clustered vs. unclustered: –If order of data records is the same as, or `close to’, order of index data entries, then called clustered index. A file can be clustered on at most one search key. Cost to retrieve data records with index varies greatly based on whether index clustered or not! Alternative 1 implies clustered, but not vice-versa.

21 Clustered vs. Unclustered Index Suppose that Alternative (2) is used for data entries, and that the data records are stored in a Heap file. –To build clustered index, first sort the Heap file (with some free space on each block for future inserts). –Overflow blocks may be needed for inserts. (Thus, order of data recs is `close to’, but not identical to, the sort order.) Index entries Data entries direct search for (Index File) (Data file) Data Records data entries Data entries Data Records CLUSTERED UNCLUSTERED

22 Unclustered vs. Clustered Indexes What are the tradeoffs???? Clustered Pros –Efficient for range searches –May be able to do some types of compression –Possible locality benefits (related data?) Clustered Cons –Expensive to maintain (on the fly or sloppy with reorganization)

23 Hash-Based Indexes Good for equality selections. Index is a collection of buckets. Bucket = primary page plus zero or more overflow pages. Hashing function h: –h(r) = bucket in which record r belongs. –h looks at the search key fields of r. If Alternative (1) is used, the buckets contain the data records; otherwise, they contain or pairs.

24 B+ Tree Indexes  Leaf pages contain data entries, and are chained (prev & next)  Non-leaf pages contain index entries and direct searches : P 0 K 1 P 1 K 2 P 2 K m P m index entry Non-leaf Pages Leaf

25 Comparing File Organizations Heap files (random order; insert at eof) Sorted files, sorted on Clustered B+ tree file, Alternative (1), search key Heap file with unclustered B + tree index on search key Heap file with unclustered hash index on search key

26 Operations to Compare Scan: Fetch all records from disk Fetch all records in sorted order Equality search Range selection Insert a record Delete a record

27 Cost Model for Analysis I/O cost 150,000 times more than hash function –We ignore CPU costs, for simplicity B: The number of data pages R: Number of records per page F: Fanout of B-tree Average-case analysis; based on several simplistic assumptions. * Good enough to show the overall trends!

28 Assumptions in Our Analysis Heap Files: – Equality selection on key; exactly one match. Sorted Files: – Files compacted after deletions. Indexes: –Alt (2), (3): data entry size = 10% size of record – Hash: No overflow buckets. 80% page occupancy => File size = 1.25 data size – Tree: 67% occupancy (this is typical). Implies file size = 1.5 data size

29 I/O Cost of Operations B: Number of data pages (packed) R: Number of records per page S: Time required for equality search Heap File Scan all records B Get all in sort order 4B Equality Search 0.5 B Range Search B Insert 2 Delete 0.5B + 1

30 I/O Cost of Operations B: Number of data pages (packed) R: Number of records per page S: Time required for equality search Sorted File Scan all records B Get all in sort order B Equality Search log 2 B Range Search S + # matching pages Insert S + B Delete S + B

31 I/O Cost of Operations B: Number of data pages (packed) R: Number of records per page F: Fanout of B-Tree S: Time required for equality search Clustered Tree Scan all records 1.5 B Get all in sort order 1.5 B Equality Search log F (1.5 B) Range Search S + #matching pages Insert S + 1 Delete 0.5B + 1

32 I/O Cost of Operations B: Number of data pages (packed) R: Number of records per page F: Fanout of B-Tree S: Time required for equality search Unclustered Tree Scan all records B (ignore index) Get all in sort order 4B (ignore index) Equality Search log F (.15 B) + 1 Range Search S + #matching records Insert S + 2 Delete S + 2

33 I/O Cost of Operations B: Number of data pages (packed) R: Number of records per page S: Time required for equality search Hash Index Scan all records B (ignore index) Get all in sort order 4B (ignore index) Equality Search 2 Range Search B (ignore index) Insert 4 Delete S + 2

34 I/O Cost of Operations B: The number of data pages R: Number of records per page F: Fanout of B-Tree S: Time required for equality search * Don’t Use Index Heap FileSorted FileClustered TreeUnclustered TreeHash Index Scan all records BB1.5 BB* Get all in sort order 4BB1.5 B4B* Equality Search 0.5 Blog 2 Blog F (1.5 B)log F (.15 B) Range Search BS + #matching pages S + #matching records B* Insert 2S + BS + 1S + 24 Delete 0.5B + 1S + B0.5B + 1S + 2

35 Index Selection Guidelines Attributes in WHERE clause are candidates for index keys. –Exact match condition suggests hash index. –Range query suggests tree index. Clustering is especially useful for range queries; can also help on equality queries if there are many duplicates. Multi-attribute search keys should be considered when a WHERE clause contains several conditions. – Order of attributes is important for range queries. – Such indexes sometimes enable index-only strategies For index-only strategies, clustering is not important! Choose indexes that benefit as many queries as possible. Since only one index can be clustered per table, choose it based on important queries that would benefit the most from clustering.

36 B+ Tree: The Most Widely Used Index Supports equality and range-searches efficiently. 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. Index Entries Data Entries ("Sequence set") (Direct search)

37 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! Root * 3*5* 7*14*16* 19*20* 22*24*27* 29*33*34* 38* 39* 13

38 B+ Trees in Practice Typical order: 100. Typical fill-factor: 67%. –average fanout = 133 Typical capacities: –Height 4: = 312,900,700 records –Height 3: = 2,352,637 records 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

39 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.

40 Example B+ Tree - Inserting 8* Root * 3*5* 7*14*16* 19*20* 22*24*27* 29*33*34* 38* 39* 13

41 Example B+ Tree - Inserting 8* v Notice that root was split, leading to increase in height. v In this example, we can avoid split by re-distributing entries; however, this is usually not done in practice. 2*3* Root *16* 19*20*22*24*27* 29*33*34* 38* 39* 135 7*5*8*

42 Inserting 8* into Example B+ Tree 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* 5* 7* 8* 5 Entry to be inserted in parent node. (Note that 5 is continues to appear in the leaf.) s copied up and appears once in the index. Contrast Entry to be inserted in parent node. (Note that 17 is pushed up and only this with a leaf split.) … …

43 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.

44 B-Tree Demo

45 Example Tree (including 8*) Delete 19* and 20*... Deleting 19* is easy. 2*3* Root *16* 19*20*22*24*27* 29*33*34* 38* 39* 135 7*5*8*

46 Example Tree (including 8*) Delete 19* and 20*... Deleting 19* is easy. Deleting 20* is done with re-distribution. Notice how middle key is copied up. 2*3* Root *16* 33*34* 38* 39* 135 7*5*8* 22*24* 27 27*29*

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

48 Summary Alternative file organizations, tradeoffs for each If selection queries are frequent, sorting the file or building an index is important. – Hash-based indexes only good for equality search. – Sorted files and tree-based indexes best for range search; also good for equality search. (Files rarely kept sorted in practice; B+ tree index is better.) Index is a collection of data entries plus a way to quickly find entries with given key values.

49 Summary (Contd.) Data entries can be actual data records, pairs, or pairs. – Choice orthogonal to indexing technique used to locate data entries with a given key value. Can have several indexes on a given file of data records, each with a different search key. Indexes can be –clustered, unclustered –B-tree, hash table, etc.

50 Summary (Contd.) Understanding the nature of the workload for the application, and the performance goals, is essential to developing a good design. – What are the important queries and updates? What attributes/relations are involved? Indexes must be chosen to speed up important queries (and perhaps some updates!). – Index maintenance overhead on updates to key fields. – Choose indexes that can help many queries, if possible. – Build indexes to support index-only strategies. – Clustering is an important decision; only one index on a given relation can be clustered! – Order of fields in composite index key can be important.


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