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Hash-based IndexesIntroductionStatic HashingStatic Hashing (Contd.)Extendible HashingExampleHandling InsertsExample: Insert 21, then 19, 15Insert h(r)=20 (Causes Doubling)Points to NoteDirectory DoublingComments on Extendible HashingAdministrivia - Exam Schedule ChangeLinear HashingLinear Hashing – The Main IdeaLinear Hashing (Contd.)LH Search AlgorithmExample: Search 44 (11100), 9 (01001)Linear Hashing - InsertExample: Insert 43 (101011)PowerPoint PresentationExample: End of a RoundSummarySummary (Contd.)Slide 26Hash-based IndexesCS 186, Spring 2006Lecture 7R &G Chapter 11HASH, x. There is no definition for this word -- nobody knows what hash is. Ambrose Bierce, "The Devil's Dictionary", 1911Introduction•As for any index, 3 alternatives for data entries k*: Data record with key value k <k, rid of data record with search key value k> <k, list of rids of data records with search key k>–Choice orthogonal to the indexing technique•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.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 NhkeyPrimary bucket pagesOverflow pages10N-1Static Hashing (Contd.)•Buckets contain data entries.•Hash fn works on search key field of record r. Use its value MOD N to 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.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, so doubling it is much cheaper. Only one page of data entries is split. No overflow page!–Trick lies in how hash function is adjusted!Example13*000110112212LOCAL DEPTHGLOBAL DEPTHDIRECTORYBucket ABucket BBucket C10*1* 7*4* 12* 32*16*5*we denote r by h(r).•Directory is array of size 4.•Bucket for record r has entry with index = `global depth’ least significant bits of h(r);–If h(r) = 5 = binary 101, it is in bucket pointed to by 01.–If h(r) = 7 = binary 111, it is in bucket pointed to by 11.Handling Inserts•Find bucket where record belongs.•If there’s room, put it there.•Else, if bucket is full, split it:–increment local depth of original page–allocate new page with new local depth–re-distribute records from original page.–add entry for the new page to the directoryExample: Insert 21, then 19, 1513*0001101122LOCAL DEPTHGLOBAL DEPTHDIRECTORYBucket ABucket BBucket C2Bucket DDATA PAGES10*1* 7*24* 12* 32*16*15*7*19*5*we denote r by h(r).•21 = 10101•19 = 10011•15 = 011111221*24* 12* 32*16*Insert h(r)=20 (Causes Doubling)000110112222LOCAL DEPTHGLOBAL DEPTHBucket ABucket BBucket CBucket D1*5* 21*13*10*15* 7* 19*(`split image'of Bucket A)20*3Bucket A24* 12*of Bucket A)3Bucket A2(`split image'4*20*12*2Bucket B1* 5* 21* 13*10*219*2Bucket D15*7*332*16*LOCAL DEPTH0000010100111001011101113GLOBAL DEPTH332*16*Bucket CBucket APoints to Note•20 = binary 10100. Last 2 bits (00) tell us r belongs in either 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.•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.Directory Doubling000110112Why use least significant bits in directory?Allows for doubling by copying the directory and appending the new copy to the original.vs.011011Least SignificantMost Significant0, 21, 3110, 21, 3110, 12, 3110001101120, 12, 311Comments 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.–Multiple entries with same hash value cause problems!•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.Administrivia - Exam Schedule Change•Exam 1 will be held in class on Tues 2/21 (not on the previous thurs as originally scheduled).•Exam 2 will remain as scheduled Thurs 3/23 (unless you want to do it over spring break!!!).Linear Hashing•A dynamic hashing scheme that handles the problem of long overflow chains without using a directory.•Directory avoided in LH by using temporary overflow pages, and choosing the bucket to split in a round-robin fashion.•When any bucket overflows split the bucket that is currently pointed to by the “Next” pointer and then increment that pointer to the next bucket.Linear Hashing – The Main Idea•Use a family of hash functions h0, h1, h2, ...•hi(key) = h(key) mod(2iN)–N = initial # buckets–h is some hash function •hi+1 doubles the range of hi (similar to directory doubling)Linear Hashing (Contd.)•Algorithm proceeds in `rounds’. Current round number is “Level”.•There are NLevel (= N * 2Level) buckets at the beginning of a round•Buckets 0 to Next-1 have been split; Next to NLevel have not been split yet this round.•Round ends when all initial buckets have been split (i.e. Next = NLevel). •To start next round:Level++; Next = 0;LH Search Algorithm•To find bucket for data entry r, find hLevel(r):–If hLevel(r) >= Next (i.e., hLevel(r) is a bucket that hasn’t been involved in a split this round) then r belongs in that bucket for sure. –Else, r could belong to bucket hLevel(r) or bucket hLevel(r) + NLevel must apply hLevel+1(r) to find out.Example: Search 44 (11100), 9


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