1 Database Systems ( 資料庫系統 ) November 8, 2004 Lecture #9 By Hao-hua Chu ( 朱浩華 )

2 Announcement Midterm exam: November 20 (Sat): 2:30 PM in CSIE 101/103 Assignment #6 is available on the course homepage. –It is due on 11/24 –It is very difficult –Suggest you do it before midterm exam Assignment #7 will be available on the course homepage later this afternoon. –It is due 11/16 (next Tuesday). –It is easy. –It will help you prepare midterm exam.

3 Cool Ubicomp Project Counter Intelligence (MIT) Smart kitchen & kitchen wares Talking Spoon –Salty, sweet,hot? Talking Cultery –Bacteria? Smart fridge & counters –RFID tags –Tracking food from fridge to your month

4 Hash-Based Indexing Chapter 11

5 Introduction Recall that Hash-based indexes are best for equality selections. –Cannot support range searches. –Equality selections are useful for join operations. Static and dynamic hashing techniques exist –Trade-offs similar to ISAM vs. B+ trees. –Static hashing technique –Two dynamic hashing techniques Extendible Hashing Linear Hashing

6 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

7 Static Hashing (Contd.) Buckets contain data entries. Hash function works on search key field of record r. –Ideally uniformly 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. Cost for insertion/delete/search – two/two/one disk page I/Os (no overflow chains). Long overflow chains can develop and degrade performance. – Why poor performance? Scan through overflow chains linearly. – Extendible and Linear Hashing: Dynamic techniques to fix this problem.

8 Simple Solution Avoid creating overflow pages: –When a bucket (primary page) becomes full, double # of buckets & re-organize the file. What’s wrong with this simple solution? – High cost concern: reading and writing all pages is expensive!

9 Extendible Hashing The basic Idea (another level of abstraction): – Use directory of pointers to buckets (hash to the directory entry) – Double # of buckets by doubling the directory – Splitting just the bucket that overflowed! Directory much smaller than file, so doubling it is much cheaper. Only one page of data entries is split – The page that overflows, rehash that page to two pages. Trick lies in how hash function is adjusted! – Before doubling directory, h(r) -> 0..N-1 buckets. – After doubling directory, h(r) -> 0.. 2N-1

10 Example Directory is array of size 4. To find bucket for r, take last global depth # bits of h(r); –Example: If h(r) = 5 = binary 101, it is in bucket pointed to by 01. Global depth: # of bits used for hashing directory entries. Local depth of a bucket: # bits for hashing a bucket. When can global depth be different from local depth? 13* 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*

11 Insert h(r)=20 (Causes Doubling) 20* 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* 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 4: : : : :

Extensible Hashing Insert Check if the bucket is full. –If no, done! Otherwise, check if local depth = global depth –if no, rehash the entries and distribute them into two buckets + increment the local depth –if yes, double the directory -> rehash the entries and distribute into two buckets Directory is doubled by copying it over and `fixing’ pointer to split image page. – You can do this only by using the least significant bits in the directory.

13 Insert 9 19* 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 1: : : : : * DIRECTORY Bucket A Bucket B Bucket C Bucket D Bucket B2 (`split image' of Bucket B) 32* 1*9* 16* 10* 15* 7* 5* 21* 13* LOCAL DEPTH GLOBAL DEPTH 3 Bucket A2 (`split image' of Bucket A) 4* 20* 12*

14 Directory Doubling Why use least significant bits in directory? ó Allows for doubling via copying! 3 vs * * 6 = * 6 = 110 Least SignificantMost Significant *

15 Comments on Extendible Hashing If directory fits in memory, equality search answered with one disk access; else two. – 100MB file, 100 bytes/rec, you have 1M data entries. – A 4K page (a bucket) can contain 40 data entries. You need about 25,000 directory elements; chances are high that directory will fit in memory. – If the distribution of hash values is skewed (concentrates on a few buckets), directory can grow large. 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.

16 Linear Hashing (LH) This is another dynamic hashing scheme, an alternative to Extendible Hashing. –LH fixes the problem of long overflow chains (in static hashing) without using a directory (in extendible hashing). Basic Idea: Use a family of hash functions h 0, h 1, h 2,... –Each function ’ s range is twice that of its predecessor. –Pages are split when overflows occur – but not necessarily the overflowing page. (Splitting occurs in turn, in a round robin fashion.) –Buckets are added gradually (one bucket at a time). –When all the pages at one level (the current hash function) have been split, a new level is applied. –Primary pages are allocated consecutively.

17 Levels of Linear Hashing Initial Stage. –The initial level distributes entries into N 0 buckets. –Call the hash function to perform this h 0. Splitting buckets. –If a bucket overflows its primary page is chained to an overflow page (same as in static hashing). –Also when a bucket overflows, some bucket is split. The first bucket to be split is the first bucket in the file (not necessarily the bucket that overflows). The next bucket to be split is the second bucket in the file … and so on until the Nth. has been split. When buckets are split their entries (including those in overflow pages) are distributed using h 1. –To access split buckets the next level hash function (h 1 ) is applied. –h 1 maps entries to 2N 0 (or N 1 )buckets.

18 Levels of Linear Hashing (Cnt) Level progression: –Once all Ni buckets of the current level (i) are split, the hash function h i is replaced by h i+1. –The splitting process starts again at the first bucket, and h i+2 is applied to find entries in split buckets.

19 Linear Hashing Example Initially, the index level equal to 0 and N 0 equals 4 (three entries fit on a page). h 0 maps index entries to one of four buckets. h 0 is used and no buckets have been split. Now consider what happens when 9 (1001) is inserted (which will not fit in the second bucket). Note that next indicates which bucket is to split next. (Round Robin) next h0h0

20 Linear Hashing Example 2 An overflow page is chained to the primary page to contain the inserted value. If h 0 maps a value from zero to next – 1 (just the first page in this case) h 1 must be used to insert the new entry. Note how the new page falls naturally into the sequence as the fifth page. h1h1 next 64 h0h0 next h0h0 6 h0h h1h1 36 The page indicated by next is split (the first one). Next is incremented

21 Linear Hashing Example 3 Assume inserts of 8, 7, 18, 14, 11 1, 32, 16 2, 10, 13, 23 3 After the 2 nd. split the base level is 1 (N 1 = 8), use h 1. Subsequent splits will use h 2 for inserts between the first bucket and next h1h1 h1h1 next h1h1 h1h h1h1 h0h0 next h0h0 h0h0 next h1h1 h1h1 36 h1h1 h1h1 513 h1h

22 Linear Hashing vs. Extendable Hashing What is the similarity? –One round of RR of splitting in LH is the same as 1- step doubling of directory in EH What are the differences? –Directory overhead vs. none –Overflow pages vs. none –Gradual splitting (of pages) vs. one-step doubling (of directory) –Pages are allocated in order vs. not in order –Splitting non-overflowing pages vs. splitting overflowing pages

23 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. –a skewed data distribution is one in which the hash values of data entries are not uniformly distributed!

24 Summary (Contd.) Linear Hashing avoids directory by splitting buckets round-robin, and using overflow pages. – Overflow pages not likely to be long. – 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.