1. 1 Hash-Based Indexes Chapter 11

2. 2 Introduction  Hash-based indexes are best for equality selections. Cannot support range searches.  Static and dynamic hashing techniques exist: Trade-offs similar to ISAM vs. B+ trees.

3. 3 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

4. 4 Static Hashing  Buckets contain data entries (alternative 1 to 3 possible).  Hash function works on search key field of record r.  h ( k ) mod N = bucket to which data entry with key k belongs with N = # of buckets  h() must distribute values over range 0... N-1.  h ( key ) = (a * key + b) works well.  a and b are constants;  lots known about how to tune h.

5. 5 Static Hashing Cons  Primary pages fixed space  static structure.  Long overflow chains can develop and degrade performance.  Fixed # bucket is the problem!  Rehashing can be done  Not good for search.  In practice, instead use overflow chains.  Or, employ dynamic techniques to fix this problem  Extendible hashing, or  Linear Hashing

6. 6 Extendible Hashing  Situation: Bucket (primary page) becomes full.  Solution: Why not re-organize file by doubling # of buckets?  Reading and writing all pages is expensive!  Idea : Use directory of pointers to buckets instead of buckets  double # of buckets by doubling the directory  splitting just the bucket that overflowed!  Discussion :  + Directory much smaller than file, so doubling it is much cheaper. + Only one page of data entries is split. + No overflow pages ever.  Trick lies in how hash function is adjusted!

7. 7 Example  Directory is array of size 4.  To find bucket for r, take last ` global depth ’ # bits of function h ( r );  Examples:  If h ( r ) = 5 = binary 101, it is in bucket pointed to by 01.  If h ( r ) = 4 = binary 100, it is in bucket pointed to by 00. 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*

8. 8 Example Insertion  To find bucket for r, take last ` global depth ’ # bits of h ( r )  If h ( r ) = 5 = binary 101, r is in bucket pointed to by 01.  Insert : If bucket is full, split it ( allocate new page, re-distribute ).  Splitting may :  double the directory, or  simply link in a new page.  To tell compare global depth with local depth for 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*

9. 9 Insert h (r)= 6 = binary 110 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* 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* 6*

10. 10 Insert h (r)=20 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* 20* 3 Bucket A2 Split Bucket A into Two buckets A1 and A2. 4*12* 3 Bucket A1 32* 16*

11. 11 Insert h (r)=20 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* 20* 3 Bucket A24*12* 3 Bucket A1 32* 16*

12. 12 Insert h (r)=20 20* 00 01 10 11 2 2 2 2 LOCAL DEPTH 2 2 DIRECTORY GLOBAL DEPTH Bucket A Bucket B Bucket C Bucket D Bucket A2 (`split image' of Bucket A) 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

13. 13 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 an entry belongs to this bucket.

14. 14 More Points to Note  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 an entry belongs to this bucket.  When does bucket split cause directory doubling?  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.  Why not first several bits?  Use of least significant bits enables efficient doubling via copying of directory!

15. 15 Directory Doubling 0 0101 1010 1 2 Why use least significant bits in directory? ó Allows for doubling via copying! 000 001 010 011 3 100 101 110 111 vs. 0 1 1 6* 6 = 110 0 1010 0101 1 2 3 0 1 1 6* 6 = 110 000 100 010 110 001 101 011 111 Least SignificantMost Significant

16. 16 Extendible Hashing : Delete  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.

17. 17 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 contain 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.  If the distribution of hash values is skewed, directory can grow large.

18. 18 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.  Large Number of 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.