1. B-Trees, Part 2 Hash-Based Indexes R&G Chapter 10 Lecture 10

2. Administrivia The new Homework 3 now available –Due 1 week from Sunday –Homework 4 available the week after Midterm exams available here

3. Review Last time discussed File Organization –Unordered heap files –Sorted Files –Clustered Trees –Unclustered Trees –Unclustered Hash Tables Indexes –B-Trees – dynamic, good for changing data, range queries –Hash tables – fastest for equality queries, useless for range queries

4. Review (2) For any index, 3 alternatives for data entries k*: – Data record with key value k – – Choice orthogonal to the indexing technique

5. Today: Indexes –Composite Keys, Index-Only Plans B-Trees –details of insertion and deletion Hash Indexes –How to implement with changing data sets

6. 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.

7. Indexes with Composite Search Keys Composite Search Keys: Search on a combination of fields. – Equality query: Every field value is equal to a constant value. E.g. wrt index: age=20 and sal =75 – Range query: Some field value is not a constant. E.g.: age =20; or age=20 and sal > 10 Data entries in index sorted by search key to support range queries. – Lexicographic order, or – Spatial order. sue1375 bob cal joe12 10 20 8011 12 nameagesal 12,20 12,10 11,80 13,75 20,12 10,12 75,13 80,11 11 12 13 10 20 75 80 Data records sorted by name Data entries in index sorted by Data entries sorted by Examples of composite key indexes using lexicographic order.

8. Composite Search Keys To retrieve Emp records with age=30 AND sal=4000, an index on would be better than an index on age or an index on sal. – Choice of index key orthogonal to clustering etc. If condition is: 20<age<30 AND 3000<sal<5000: – Clustered tree index on or is best. If condition is: age=30 AND 3000<sal<5000: – Clustered index much better than index! Composite indexes are larger, updated more often.

9. Index-Only Plans A number of queries can be answered without retrieving any tuples from one or more of the relations involved if a suitable index is available. SELECT D.mgr FROM Dept D, Emp E WHERE D.dno=E.dno SELECT D.mgr, E.eid FROM Dept D, Emp E WHERE D.dno=E.dno SELECT E.dno, COUNT (*) FROM Emp E GROUP BY E.dno SELECT E.dno, MIN (E.sal) FROM Emp E GROUP BY E.dno SELECT AVG (E.sal) FROM Emp E WHERE E.age=25 AND E.sal BETWEEN 3000 AND 5000 Tree index! Tree index! or Tree!

10. Index-Only Plans (Contd.) Index-only plans are possible if the key is or we have a tree index with key –Which is better? –What if we consider the second query? SELECT E.dno, COUNT (*) FROM Emp E WHERE E.age=30 GROUP BY E.dno SELECT E.dno, COUNT (*) FROM Emp E WHERE E.age>30 GROUP BY E.dno

11. B-Trees: Insertion and Deletion Insertion –Find leaf where new record belongs –If leaf is full, redistribute* –If siblings too full, split, copy middle key up –If index too full, redistribute* –If index siblings full, split, push middle key up Deletion –Find leaf where record exists, remove it –If leaf is less than 50% empty, redistribute* –If siblings too empty, merge, remove key above –If index node above too empty, redistribute* –If index siblings too empty, merge, move above key down

12. B-Trees: For Homework 2 Insertion –Find leaf where new record belongs –If leaf is full, redistribute* –If siblings too full, split, copy middle key up –If index too full, redistribute* –If index siblings full, split, push middle key up Deletion –Find leaf where record exists, remove it –If leaf is less than 50% empty, redistribute* –If siblings too empty, merge, remove key above –If index node above too empty, redistribute* –If index siblings too empty, merge, move above key down This means that after deletion, nodes will often be < 50% full

13. B-Trees: For Homework 2 (cont) Splits –When splitting nodes, choose the middle key –If there are even number of keys, choose  middle  Your code must Handle Duplicate Keys –We promise that there will never be more than 1 page of duplicate values. –Thus, when splitting, if the middle key is identical to the key to the left, you must find the closest splittable key to the middle.

14. B-Tree Demo

15. Hashing Static and dynamic hashing techniques exist; trade-offs based on data change over time Static Hashing –Good if data never changes Extendable Hashing –Uses directory to handle changing data Linear Hashing –Avoids directory, usually faster

16. Static Hashing # primary pages fixed, allocated sequentially, never de-allocated; overflow pages if needed. h(k) mod N = bucket to which data entry with key k belongs. (N = # of buckets) h(key) mod N h key Primary bucket pages Overflow pages 2 0 N-1

17. Static Hashing (Contd.) Buckets contain data entries. Hash fn works on search key field of record r. Must distribute values over range 0... N-1. – h(key) = (a * key + b) usually works well. – a and b are constants; lots known about how to tune h. Long overflow chains can develop and degrade performance. – Extendible and Linear Hashing: Dynamic techniques to fix this problem.

18. Extendible Hashing Situation: Bucket (primary page) becomes full. Why not re-organize file by doubling # of buckets? – Reading and writing all pages is expensive! Idea: Use directory of pointers to buckets, – double # of buckets by doubling the directory, – splitting just the bucket that overflowed! – Directory much smaller than file, doubling much cheaper. – No overflow pages! – Trick lies in how hash function is adjusted!

19. Extendible Hashing Details Need directory with pointer to each bucket Need hash function to incrementally double range – can just use increasingly more LSBs of h(key) Must keep track of global “depth” – how many times directory doubled so far Must keep track of local “depth” – how many time each bucket has been split

20. Example Directory is array of size 4. To find bucket for r, take last `global depth’ # bits of h(r); we denote r by h(r). – If h(r) = 5 = binary 101, it is in bucket pointed to by 01.  Insert : If bucket is full, split it ( allocate new page, re-distribute ).  If necessary, double the directory. (As we will see, splitting a bucket does not always require doubling; we can tell by comparing global depth with local depth for the split bucket.) 13* 00 01 10 11 2 2 2 2 2 LOCAL DEPTH GLOBAL DEPTH DIRECTORY Bucket A Bucket B Bucket C Bucket D DATA PAGES 10* 1*21* 4*12*32* 16* 15*7*19* 5*

21. Insert h(r)=20 (Causes Doubling) 00 01 10 11 2 2 2 2 LOCAL DEPTH 2 DIRECTORY GLOBAL DEPTH Bucket A Bucket B Bucket C Bucket D 1* 5*21*13* 32*16* 10* 15*7*19* 4*12* 19* 2 2 2 000 001 010 011 100 101 110 111 3 3 3 DIRECTORY Bucket A Bucket B Bucket C Bucket D Bucket A2 (`split image' of Bucket A) 32* 1*5*21*13* 16* 10* 15* 7* 4* 20* 12* LOCAL DEPTH GLOBAL DEPTH

22. Now Insert h(r)=9 (Causes Split Only) 19* 3 2 2 000 001 010 011 100 101 110 111 3 3 3 DIRECTORY Bucket A Bucket B Bucket C Bucket D Bucket A2 32* 1*9* 16* 10* 15* 7* 4* 20* 12* LOCAL DEPTH GLOBAL DEPTH 3 Bucket B2 5*21*13* 19* 2 2 2 000 001 010 011 100 101 110 111 3 3 3 DIRECTORY Bucket A Bucket B Bucket C Bucket D Bucket A2 (`split image' of Bucket A) 32* 1*5*21*13* 16* 10* 15* 7* 4* 20* 12* LOCAL DEPTH GLOBAL DEPTH

23. Points to Note 20 = binary 10100. Last 2 bits (00) tell us r belongs in A or A2. Last 3 bits needed to tell which. – Global depth of directory: Max # of bits needed to tell which bucket an entry belongs to. – Local depth of a bucket: # of bits used to determine if splitting bucket will also double directory When does bucket split cause directory doubling? – If, before insert, local depth of bucket = global depth. Insert causes local depth to become > global depth; directory is doubled by copying it over and `fixing’ pointer to split image page. (Use of least significant bits enables efficient doubling via copying of directory!)

24. Directory Doubling Why use least significant bits in directory? ó Allows for doubling via copying! 13* 00 01 10 11 2 2 2 2 2 10* 1*21* 4*12*32* 16* 15*7*19* 5* 13* 000 001 010 011 3 2 2 2 2 10* 1*21* 4*12*32* 16* 15*7*19* 5* 100 101 110 111

25. Deletion Delete: If removal of data entry makes bucket empty, can be merged with `split image’. If each directory element points to same bucket as its split image, can halve directory. 13* 00 01 10 11 2 2 2 2 2 10* 1*21* 4*12*32* 16* 15* 5* 13* 00 01 10 11 2 1 2 2 1*21* 4*12*32* 16* 15* 5* Delete 10

26. Deletion (cont) Delete: If removal of data entry makes bucket empty, can be merged with `split image’. If each directory element points to same bucket as its split image, can halve directory. 13* 00 01 10 11 2 1 1 1*21* 4*12*32* 16* 5* Delete 15 13* 00 01 10 11 2 1 2 2 1*21* 4*12*32* 16* 15* 5* 13* 0 1 1 1 1 1*21* 4*12*32* 16* 5*

27. Comments on Extendible Hashing If directory fits in memory, equality search answered with one disk access; else two. – 100MB file, 100 bytes/rec, 4K pages contains 1,000,000 records (as data entries) and 25,000 directory elements; chances are high that directory will fit in memory. – Directory grows in spurts, and, if the distribution of hash values is skewed, directory can grow large. Biggest problem: – Multiple entries with same hash value cause problems! – If bucket already full of same hash value, will keep doubling forever! So must use overflow buckets if dups.

28. Linear Hashing This is another dynamic hashing scheme, an alternative to Extendible Hashing. LH handles the problem of long overflow chains without using a directory, and handles duplicates. Idea: Use a family of hash functions h 0, h 1, h 2,... – h i (key) = h(key) mod(2 i N); N = initial # buckets – h is some hash function (range is not 0 to N-1) – If N = 2 d0, for some d0, h i consists of applying h and looking at the last di bits, where di = d0 + i. – h i+1 doubles the range of h i (similar to directory doubling)

29. Let’s start with N = 4 Buckets –Start at “round” 0, “next” 0, have 2 round buckets –Each time any bucket fills, split “next” bucket –If (O ≤ h round (key) < Next), use h round+1 (key) instead Linear Hashing Example 13* 10* 1*21* 4*12*32* 16* 15* 5* Start 13* 10* 1*21* 4*12* 32* 16* 15* 5* Add 9 Next 9* Next 13* 10* 1*21* 4*12* 32* 16* 15* 5* Add 20 9* Next 20* N round

30. Overflow chains do exist, but eventually get split Instead of doubling, new buckets added one-at-a-time Linear Hashing Example (cont) 13* 10* 1*21* 4*12* 32* 16* 15* 5* Add 6 9* Next 20* 6* 13* 10* 1* 21* 4*12* 32* 16* 15* 5* Add 17 9* Next 20* 6* N round

31. Linear Hashing (Contd.) Directory avoided in LH by using overflow pages, and choosing bucket to split round-robin. – Splitting proceeds in `rounds’. Round ends when all N R initial (for round R) buckets are split. Buckets 0 to Next- 1 have been split; Next to N R yet to be split. – Current round number also called Level. – Search: To find bucket for data entry r, find h round (r): If h round (r) in range `Next to N R ’, r belongs here. Else, r could be h round (r) or h round (r) + N R ; –must apply h round+1 (r) to find out.

32. Overview of LH File In the middle of a round. Level h Buckets that existed at the beginning of this round: this is the range of Next Bucket to be split of other buckets) in this round Level hsearch key value) ( )( Buckets split in this round: If is in this range, must use h Level+1 `split image' bucket. to decide if entry is in created (through splitting `split image' buckets:

33. Linear Hashing (Contd.) Insert: Find bucket by applying h round / h round+1 : – If bucket to insert into is full: Add overflow page and insert data entry. (Maybe) Split Next bucket and increment Next. Can choose any criterion to `trigger’ split. Since buckets are split round-robin, long overflow chains don’t develop! Doubling of directory in Extendible Hashing is similar; switching of hash functions is implicit in how the # of bits examined is increased.

34. Another Example of Linear Hashing On split, h Level+1 is used to re-distribute entries. 0 h h 1 ( This info is for illustration only!) Round=0, N=4 00 01 10 11 000 001 010 011 (The actual contents of the linear hashed file) Next=0 PRIMARY PAGES Data entry r with h(r)=5 Primary bucket page 44* 36* 32* 25* 9*5* 14*18* 10* 30* 31*35* 11* 7* 0 h h 1 Round=0 00 01 10 11 000 001 010 011 Next=1 PRIMARY PAGES 44* 36* 32* 25* 9*5* 14*18* 10* 30* 31*35* 11* 7* OVERFLOW PAGES 43* 00 100

35. Example: End of a Round 0 h h 1 22* 00 01 10 11 000 001 010 011 00 100 Next=3 01 10 101 110 Round=0 PRIMARY PAGES OVERFLOW PAGES 32* 9* 5* 14* 25* 66* 10* 18* 34* 35*31* 7* 11* 43* 44*36* 37*29* 30* 0 h h 1 37* 00 01 10 11 000 001 010 011 00 100 10 101 110 Next=0 Round=1 111 11 PRIMARY PAGES OVERFLOW PAGES 11 32* 9*25* 66* 18* 10* 34* 35* 11* 44* 36* 5* 29* 43* 14* 30* 22* 31*7* 50*

36. LH Described as a Variant of EH The two schemes are actually quite similar: – Begin with an EH index where directory has N elements. – Use overflow pages, split buckets round-robin. – First split is at bucket 0. (Imagine directory being doubled at this point.) But elements,,... are the same. So, need only create directory element N, which differs from 0, now. When bucket 1 splits, create directory element N+1, etc. So, directory can double gradually. Also, primary bucket pages are created in order. If they are allocated in sequence too (so that finding i’th is easy), we actually don’t need a directory! Voila, LH.

37. Summary Hash-based indexes: best for equality searches, cannot support range searches. Static Hashing can lead to long overflow chains. Extendible Hashing avoids overflow pages by splitting a full bucket when a new data entry is to be added to it. (Duplicates may require overflow pages.) – Directory to keep track of buckets, doubles periodically. – Can get large with skewed data; additional I/O if this does not fit in main memory.

38. Summary (Contd.) Linear Hashing avoids directory by splitting buckets round-robin, and using overflow pages. – Overflow pages not likely to be long. – Duplicates handled easily. – Space utilization could be lower than Extendible Hashing, since splits not concentrated on `dense’ data areas. Can tune criterion for triggering splits to trade-off slightly longer chains for better space utilization. For hash-based indexes, a skewed data distribution is one in which the hash values of data entries are not uniformly distributed!