Hash-Based Indexes Chapter 11 1
Introduction : Hash-based Indexes Best for equality selections. Cannot support range searches. Static and dynamic hashing techniques exist: Trade-offs similar to ISAM vs. B+ trees.
Static Hashing h(k) mod N = bucket to which data entry with key k belongs. (N = # of buckets) h(key) mod N key h 0 N-1 Primary bucket pages Overflow pages 3
Static Hashing : h(k) mod N # primary pages fixed (N = # of buckets) allocated sequentially never de-allocated overflow pages if needed. h(key) mod N key h 0 N-1 Primary bucket pages Overflow pages 4
Static Hashing h(k) mod N = bucket to which data entry with key k belongs with N = # of buckets Hash function works on search key of record r. h() must distribute values over range [0... N-1]. For example, h(key) = (a * key + b) a and b are constants lots known about how to tune h. 5
Static Hashing Cons Primary pages fixed space static structure. Fixed # buckets is the problem: Rehashing can be done Not good for search. In practice, instead use overflow chains. Long overflow chains degrade performance. Solution: Employ dynamic techniques : Extendible hashing, or Linear Hashing 6
Extendible Hashing Problem: Bucket (primary page) becomes full. Solution: Re-organize file by doubling # of buckets? But Reading and writing all pages is expensive! 8
Extendible Hashing Ideas : Use directory of pointers to buckets instead of buckets Details : Double # of buckets by doubling the directory Split just the bucket that overflowed! Trick : How hash function is adjusted! 9
Example Directory : array[4] LOCAL DEPTH GLOBAL DEPTH 4* 1* 3* 16* Bucket A To find bucket for r, take last `global depth # bits of function h(r) 00 1* 5* 1* 13* Bucket B 01 10 11 10* Bucket C DIRECTORY 15* 7* 19* Bucket D DATA PAGES 10
Example LOCAL DEPTH GLOBAL DEPTH 4* 1* 3* 16* Bucket A If h(r) = 5 = 101, it is in bucket pointed to by 01. 00 01 1* 5* 1* 13* Bucket B If h(r) = 4 = 100, 10 11 10* Bucket C it is in bucket pointed to by 00. DIRECTORY 15* 7* 19* Bucket D DATA PAGES 11
Insertion LOCAL DEPTH GLOBAL DEPTH 4* 1* 3* 16* Bucket A Insert: If bucket is full, split it (allocate new page, re-distribute content). 00 01 10 11 1* 5* 1* 10* 13* Bucket B Bucket C DIRECTORY 15* 7* 19* Bucket D Splitting may : double the directory, or simply link in a new page. DATA PAGES To tell what to do : Compare global depth with local depth for split bucket. 1
Insert h(r)= 6 = binary 110 LOCAL DEPTH GLOBAL DEPTH Bucket A 4* 1* 3* 16* Bucket B 00 1* 5* 1* 13* 01 10 11 10* Bucket C DIRECTORY 15* 7* 19* Bucket D DATA PAGES 13
Insert h(r)= 6 = binary 110 LOCAL DEPTH Bucket A LOCAL DEPTH Bucket A GLOBAL DEPTH 4* 1* 3* 16* GLOBAL DEPTH 4* 1* 3* 16* Bucket B Bucket B 00 1* 5* 1* 13* 00 1* 5* 1* 13* 01 01 10 11 10* Bucket C 10 11 Bucket C 10* 6* DIRECTORY 15* 7* 19* Bucket D DIRECTORY 15* 7* 19* Bucket D DATA PAGES DATA PAGES 14
Insert h(r)=0 = binary 10100 LOCAL DEPTH GLOBAL DEPTH Bucket A 4* 1* 3* 16* Bucket B 00 1* 5* 1* 13* 01 10 11 10* Bucket C DIRECTORY 15* 7* 19* Bucket D DATA PAGES 15
Insert h(r)=0 LOCAL DEPTH GLOBAL DEPTH Bucket A 4* 1* 3* 16* Split Bucket A into two buckets A1 and A Bucket B 00 01 10 1* 5* 1* 13* Bucket C 3 4* 1* 0* Bucket A 11 10* 3 DIRECTORY Bucket D 15* 7* 19* 16* 3* Bucket A1 DATA PAGES 16
Insert h(r)=0 3 LOCAL DEPTH Bucket A 4* 1* 0* Bucket A GLOBAL DEPTH 4* 1* 3* 16* 3 Bucket B 3* 16* Bucket A1 00 1* 5* 1* 13* 01 10 11 10* Bucket C DIRECTORY 15* 7* 19* Bucket D DATA PAGES 17
Insert h(r)=0 LOCAL DEPTH GLOBAL DEPTH 3*16* Bucket A LOCAL DEPTH GLOBAL DEPTH 3 3* 16* Bucket A 00 01 10 11 1* 5* 1*13* 10* Bucket B Bucket C 000 001 010 011 3 1* 5* 1* 13* 10* Bucket B Bucket C DIRECTORY 15* 7* 19* Bucket D 100 101 110 15* 7* 19* Bucket D 4* 1* 0* Bucket A (`split image' of Bucket A) 111 DIRECTORY 3 4* 1* 0* Bucket A (`split image' of Bucket A) 18
Points to Note 0 = binary 10100. Last bits (00) tell us if r belongs in A Last 3 bits needed to tell if r belongs into A1 or A 19
Bits: 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: Actual # 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 the one over-full page. 0
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.
Comments on Extendible Hashing If directory fits in memory, then equality search answered with one disk access; else with two. 100MB file, 100 bytes/rec, 4K pages contain 1,000,000 records (as data entries) and 5,000 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. 3
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 full bucket when new data to be added Directory to keep track of buckets, doubles periodically 4