w5ch11 hash index
TRANSCRIPT
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CPSC 461Instructor: Marina Gavrilova
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Goal Goal of todays lecture is to introduce concept of
hash-index. Hashing is an alternative to treestructure that allows near constant access time to ANYrecord in a very large database.
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Presentation Outline Introduction
Did you know that?
Static hashing definition and methods
Good hash function Collision resolution (open addressing, chaining)
Extendible hashing
Linear hashing Useful links and current market trends
Summary
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4
Approaches to Search
1. Sequential and list methods
(lists, tables, arrays).
2. Direct access by key value (hashing)
3. Tree indexing methods.
Introduction to Hashing
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DefinitionHashing is the process of mapping a key value to a
position in a table.
Ahash function maps key values to positions.
Ahash table is an array that holds the records.
Searching in a hash table can be done in O(1) regardless of the
hash table size.
Introduction to Hashing
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Introduction to Hashing
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Example of Usefullness
10 stock details, 10 table positions
Stock numbers are between 0 and 1
1000.
Using the whole stock numbers may
require 1000 storage locations and
this is an obvious waste of memory.
Introduction to Hashing
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Applications of Hashing
Compilers use hash tables to keep track of declared variables
A hash table can be used for on-line spelling checkersif
misspelling detection (rather than correction) is important, an entiredictionary can be hashed and words checked in constant time
Game playing programs use hash tables to store seen positions,
thereby saving computation time if the position is encountered
again
Hash functions can be used to quickly check for inequalityif
two elements hash to different values they must be different
Storing sparse data
Introduction of Hashing
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Did you know that? Cryptography was once known only to the key people in the theNational Security Agency and a few academics.
Until 1996, it was illegal to export strong cryptography from theUnited States.
Fast forward to 2006, and the Payment Card Industry Data Security
Standard (PCI DSS) requires merchants to encrypt cardholderinformation. Visa and MasterCard can levy fines of up to $500,000for not complying!
Among methods recommended are:
Strong one-way hash functions (hashed indexes)
Truncation
Index tokens and pads (pads must be securely stored)
Strong cryptography[Hashing for fun and profit: Demystifying encryption for PCI DSS
Roger Nebel]
http://www.nsa.gov/http://searchsecurity.techtarget.com/topics/0,295493,sid14_tax303586,00.htmlhttp://searchsecurity.techtarget.com/topics/0,295493,sid14_tax303586,00.htmlhttp://searchsecurity.techtarget.com/topics/0,295493,sid14_tax303586,00.htmlhttp://searchsecurity.techtarget.com/topics/0,295493,sid14_tax303586,00.htmlhttp://www.nsa.gov/ -
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Did you know that? Transport Layer Security protocol on networks (TLS) uses the Rivest,
Shamir, and Adleman (RSA) public key algorithm for the TLS key
exchange and authentication, and the Secure Hashing Algorithm 1
(SHA-1) for the key exchange and hashing.
[System cryptography: Use FIPS compliant algorithms for encryption,
hashing, and signing, Microsoft TechNews, 2005]
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Did you know that? Spatial hashing studies performed at Microsoft Research, Redmond
combine hashing with computer graphics to create a new set of tools for
rendering, mesh reconstruction, and collision optimization (see publicposter by Hugues Hoppe on the next slide)
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Perfect Spatial Hashing Sylvain Lefebvre Hugues Hoppe(Microsoft Research)
Hash table Offset table Hash table Offset table
Vector images Sprite maps
Alpha compression
3D textures 3D painting
Simulation Collision detection
2D 3D
1282 382 182 1283 353 193
ApplicationsHash function
p
p S
( )s h p
modq p r
[ ]q
Domain Hash
table H
Offset
table
( )h p p p
Perfect hash on multidimensional data
No collisions ideal for GPU
Single lookup into a small offset table
Offsets only ~4 bits per defined data
Access only ~4 instructions on GPU
Optimized spatial coherence
10243, 46MB, 530fps 20483, 56MB, 200fps
10243, 12MB, 140fps2563, 100fps
10242, 500KB, 700fps +900KB, 200fps
(modulo table sizes)
0.9bits/pixel, 800fps
1.8%
24372
83333
We design a perfect hash function to losslessly packsparse data while retaining efficient random access:
Simply:
453
nearest: 7.5MB, 370fps
11632
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Did you know that? Combining hashing and encryption provides a much
stronger tool for database and password protection.
http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/
[Security Briefs, SMDN Magazine]
http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/http://msdn.microsoft.com/msdnmag/issues/03/08/SecurityBriefs/ -
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Hash Functions
Hashing is the process of chopping up the key andmixing it up in various ways in order to obtain an index
which will be uniformly distributed over the range of
indices - hence the hashing.
There are several common ways of doing this:
Truncation Folding
Modular Arithmetic
Hash Functions
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Hash FunctionsTruncation
Truncation is a method in which parts of the key are ignored andthe remaining portion becomes the index.
- For this, we take the given key and produce a hash location bytaking portions of the key (truncating the key).
ExampleIf a hash table can hold 1000 entries and an 8-digit
number is used as key, the 3rd, 5thand 7th digits starting
from the left of the key could be used to produce the index.
- e.g. .. Key is 62538194 and the hash location is 589.
-
Advantage: Simple and easy to implement.
Problems: Clustering and repetition.
Hash Functions
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Hash FunctionsFolding
Folding breaks the key into several parts and combines the parts to forman index.
- The parts may be recombined by addition, subtraction, multiplications and may have
to be truncated as well.- Such a process is usually better than truncation by itself since it produces a better
distribution: all of the digits in the key are considered.
- Using a key 62538194 and breaking it into 3 numbers using the first 3 and the last 2
digits produced 625, 381 and 94. These could be added to get 1100 which could be truncated
to 100.
They could be also be multiplied together and then three digits chosen
from the middle of the number produced.
Hash Functions
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Hash Functions(Modular Arithmetic)
Modular Arithmeticprocess essentially assures that the indexproduced is within a specified range. For this, the key is converted to
an integer which is divided by the range of the index with the resulting
function being the value of the remainder.
Uses: biometrics, encryption, compression
- If the value of the modulus is a prime number, the distribution ofindices
obtained is quite uniform.
- A table whose size is some number which has many factors provides thepossibility of many indices which are the same, so the size should be a prime
number.
Hash Functions
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Good Hash Functions
Hash functions which use all of the key are almost always betterthan those which use only some of the key.
- When only portions are used, information is lost and therefore thenumber of possibilities for the final key are reduced.
- If we deal with the integer its binary form, then the number of
pieces that can be manipulated by the hash function is greatly
increased.
Hash Functions
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Collision
It is obvious that no matter what function is used, the possibilityexists that the use of the function will produce an index which is a
duplicate of an index which already exists. This is a Collision.
Collision resolution strategy:
- Open addressing: store the key/entry in a different position
- Chaining: chain together several keys/entries in each position
Collision Resolution
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Collision - Example- - Hash table size 11
- - Hash function: key mod hash size
So, the new positions in the hash table are:
Some collisions occur with this hash function.
Collision Resolution
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Collision ResolutionOpen Addressing
Resolving collisions by open addressing is resolving the problem bytaking the next open space as determined by rehashing the key
according to some algorithm.
Two main open addressing collision resolution techniques:
- - Linear probing: increase by 1 each time [mod table size!]
- - Quadratic probing:to the original position, add 1, 4, 9, 16,
also in some cases key-dependent increment technique is used.
Collision Resolution
Probing
If the table position given by the hashed key is already
occupied, increase the position by some amount, until an empty
position is found
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Collision ResolutionOpen Addressing
In order to try to avoid clustering, a method which does not look forthe first open space must be used.
Two common methods are used
- - Quadratic Probing
- - Key-dependent Increments
Collision Resolution
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Collision ResolutionOpen Addressing
Quadratic Probingnew position = (collision position + j2) MOD hash size
{ j = 1, 2, 3, 4, }
Example
Before quadratic probing:
After quadratic probing:
ProblemOverflowmay occurs when there is still space in the
hash table.
Collision Resolution
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Collision ResolutionOpen Addressing
Key-dependent Increments
This technique is used to solve the overflow problem of thequadratic probing method.
These increments vary according to the key used for the hashfunction.
If the original hash function results in a good distribution, then key-
dependent functions work quite well for rehashing and all locations in the
table will eventually be probed for a free position.
Key dependent increments are determined by using the key to
calculate a new value and then using this as an increment to determine
successive probes.
Collision Resolution
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Collision ResolutionOpen Addressing
Key-dependent IncrementsFor example, since the original hash function was key Mod 11,we might choose a function
of key MOD 7 to find the increment. Thus the hash function becomes - -
new position = current position + ( key DIV11) MOD 11
ExampleBefore key-dependent increments:
After key-dependent increments:
Collision Resolution
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Collision ResolutionOpen Addressing
Key-dependent Increments In all of the closed hash functions it is important to ensure that an
increment of 0 does not arise.
If the increment is equal to hash size the same position will be probed all the
time, so this value cannot be used.
If we ensure that the hash size is prime and the divisors for the open and
closed hash are prime, the rehash function does not produce a 0 increment,
then this method will usually access all positions as does the linear
probe.
- Using a key-dependent method usually result reduces clustering and therefore
searches for an empty position should not be as long as for the linear method.
Collision Resolution
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Collision ResolutionChaining Each table position is a linked list Add the keys and entries anywhere in the
list (front easiest)
Advantagesover open addressing:
- Simpler insertion and removal
- Array size is not a limitation (but should
still minimize collisions: make table size
roughly equal to expected number of keys
and entries)
Disadvantage- Memory overhead is large if entries are
small
Collision Resolution
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Collision ResolutionChaining
Example:
Before chaining:
After chaining:
Collision Resolution
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In analyzing search efficiency, the average is usually used. Searching withhash tables is highly dependent on how full the table is since as the tableapproaches a full state, more rehashes are necessary. The proportion of the
table which is full is called the Load Factor.
- When collisions are resolved using open addressing, the maximum load
factor is 1.
- Using chaining, however, the load factor can be greater than 1 when thetable is full and the linked list attached to each hash address has more than
one element.
- Chaining consistently requires fewer probes than open addressing.-Traversal of the linked list is slow and if the records are small, it may be just
as well to use open addressing.
- Chaining is the best under two conditions --- when the number of
unsuccessful searches is large or when the records are large.
- Open addressing would likely be a reasonable choice when most searches are
likely to be successful, the load factor is moderate and the records are
relatively small.
Analysis of Searching using Hash Tables
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Average number of probes for different collision resolution
methods:[ The values are for large hash tables, in this case larger than
500]
Analysis of Searching using Hash Tables
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When are other representations more suitable than hashing:
Hash tables are very good if there is a need for many searches in a
reasonably stable database
Hash tables are not so good if there are many insertions and
deletions, or if table traversals are neededin this case, trees arebetter for indexing
Also, hashing is very slow for any operations which require the
entries to be sorted (e.g. query to Find the minimum key)
Analysis of Searching using Hash Tables
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Aperfect hashing functionmaps a key into a unique address. If the
range of potential addresses is the same as the number of keys, the
function is a minimal (in space) perfect hashing function.
What makes perfect hashing distinctive is that it is a process for
mapping a key space to a unique address in a smaller address space, that is
hash (key) unique address
Not only does a perfect hashing function improve retrieval performance,but a minimal perfect hashing function would provide 100 percent storage
utilization.
Perfect Hashing
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Process of creating a perfect hash function
A general form of a perfect hashing function is:
p.hash (key) =(h0(key) + g[h
1(key)] + g[h
2(key)] mod N
Perfect Hashing
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In Cichellis algorithm, the component functions are:
h0 = length (key)
h1 = first_character (key)
h2 = second_character (key)
and g = T (x)
where Tis the table of values associated with individual characters xwhich
may apply in a key.
The time consuming part of Cichellis algorithm is determining T.
Cichellis Algorithm
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When we apply the Cichellis perfect hashing functionto the keyword
beginusing table 1, we can get
The keyword begin would be stored in location 33. Since the hash values
run from 2 through 37 for this set of data, the hash function is a minimal
perfect hashing function.
Cichellis Algorithm
Table 1: Values associated with the characters of the Pascal
reserved words
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Links for interactive hashing examples:
http://www.engin.umd.umich.edu/CIS/course.des/cis350/hashing/WEB/HashApplet.htm
http://www.cs.auckland.ac.nz/software/AlgAnim/hash_tables.html
http://www.cse.yorku.ca/~aaw/Hang/hash/Hash.html
http://www.cs.pitt.edu/~kirk/cs1501/animations/Hashing.html
Some Links to Hashing Animation
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Hashing as Database Index
The basic idea is to use hashing function, which maps asearch key value(of a field) into a record or bucket ofrecords.
As for any index, 3 alternatives for data entries k*: Data record with key value k
Hash-basedindexes are best for equalityselections.Cannot support range searches.
Static and dynamic hashing techniques exist; trade-offssimilar to ISAM vs. B+ trees.
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Static Hashing # primary pages fixed, allocated sequentially, never
de-allocated; overflow pages allowed if needed.
h(k) mod M = bucket to which data entry withkeykbelongs. (M = # of buckets)
h(key) mod N
hkey
Primary bucket pages Overflow pages
2
0
N-1
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Static Hashing (Contd.) Buckets contain data entries.
Hash function depends on search key field of record r.Must distribute values over range 0 ... M-1 (table size).
h(key) = (a * key + b) mod M usually works well. a and b are constants; lots known about how to tune h.
Long overflow chains can develop and degradeperformance.
Extendible and LinearHashing: Dynamic techniques to fixthis problem.
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Rule of thumb Try to keep space utilization between 50% and 80%
If 80%, overflow significantDepends on how good hashing function is
&
On # keys/bucket
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Extendible Hashing (Fagin et. al. 1979)
Expandable hashing (Knott 1971)
Dynamic Hashing (Larson 1978)
Hash Functions for Extendible Hashing
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Assume that a hashing technique is applied to a dynamicallychanging file composed of buckets, and each bucket can holdonly a fixed number of items.
Extendible hashing accesses the data stored in bucketsindirectly through an index that is dynamically adjusted toreflect changes in the file.
The characteristic feature of extendible hashing is theorganization of the index, which is an expandable table.
Extendible Hashing
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A hash function applied to a certain key indicates a position in the indexand not in the file (or table or keys). Values returned by such a hash
function are calledpseudokeys.
The database/file requires no reorganization when data are added to or
deleted from it, since these changes are indicated in the index.
Only one hash function hcan be used, but depending on the size of the
index, only a portion of the added h(K)is utilized.
A simple way to achieve this effect is by looking at the address into the
string of bits from which only the ileftmost bits can be used.
The number i is the depth of the directory.
In figure 1(a) (in the next slide), the depth is equal to two.
Extendible Hashing
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Figure 1. An example of extendible
hashing (Drozdek Textbook)
Example
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Extendible Hashing as Index
Situation: Bucket (primary page) becomes full. Why notre-organize file bydoubling # of buckets? Reading and writing all pages is expensive!
Idea: Use directory of pointers to buckets, double # ofbuckets bydoubling the directory, splitting just the bucketthat overflowed!
Directory much smaller than file, so doubling it is muchcheaper. Only one page of data entries is split. No
overflowpage! Trick lies in how hash function is adjusted!
2LOCAL DEPTH
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Example Directory is array of size 4.
To find bucket for r, take lastglobal depth # bits ofh(r); wedenote rbyh(r).
Ifh(r) = 5 = binary 101, it isin bucket pointed to by 01.
Insert
: If bucket is full, splitit (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 bycomparingglobal 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*
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Insert h(r)=20 (Causes Doubling)
20*
00
01
10
11
2 2
2
2
LOCAL DEPTH 2
2
DIRECTORY
GLOBAL DEPTHBucket 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
000001
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
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Points to Note 20 = binary 10100. Last 2 bits (00) tell us rbelongs in A or
A2. Last3 bits needed to tell which.
Global depth of directory: Max # of bits needed to tell whichbucket an entry belongs to.
Local depth of a bucket: # of bits used to determine if anentry belongs to this bucket.
When does bucket split cause directory doubling?
Before insert, local depth of bucket =global depth. Insertcauses local depth to become >global depth; directory isdoubled bycopying it overand `fixing pointer to split imagepage. (Use of least significant bits enables efficient doubling
via copying of directory!)
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Directory Doubling
00
01
10
11
2
Why use least significant bits in directory? Allows for doubling via copying!
000
001
010
011
3
100
101110
111
vs.
0
1
1
6*6*
6*
6 = 110
00
10
01
11
2
3
0
1
1
6* 6*6*
6 = 110000
100
010
110
001
101
011
111
Least Significant Most Significant
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Comments on Extendible Hashing
If directory fits in memory, equality search answered withone disk access; else two.
100MB file, 100 bytes/rec, 4K pages contains 1,000,000records (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 directoryelement points to same bucket as its split image, canhalve directory.
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Expandable Hashing
Similar idea to an extendible hashing.But binary tree is used to store an index on the buckets.
Dynamic Hashing
multiple binary trees are used.
Outcome:- To shorten the search.- Based on the key --- select what tree to search.
Hybrid methods
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Linear Hashing
This is another dynamic hashing scheme, an alternative toExtendible Hashing.
LH handles the problem of long overflow chains without
using a directory, and handles duplicates. Idea: Use a family of hash functions h0, h1, h2, ...
hi(key) = h(key) mod(2iN); N = initial # buckets
h is some hash function (range is not 0 to N-1)
If N = 2d0, for some d0, hi consists of applying h and looking atthe last di bits, where di = d0 + i.
hi+1 doubles the range ofhi (similar to directory doubling)
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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 allNRinitial (for round R) buckets are split. Buckets 0 toNext-1have been split; Next toNR yet to be split.
Current round number is Level.
Search: To find bucket for data entryr, findhLevel(r):
IfhLevel
(r) in range Next toNR
, rbelongs here.
Else, r could belong to bucket hLevel(r) or bucket hLevel(r)+NR; must applyhLevel+1(r) to find out.
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Overview of LH File
In the middle of a round.
Levelh
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
Levelh search key value )(
search 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:
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Linear Hashing (Contd.) Insert: Find bucket by applying hLevel / hLevel+1:
If bucket to insert into is full:
Add overflow page and insert data entry.
(Maybe) SplitNext bucket and incrementNext.
Can choose any criterion to `trigger split. Since buckets are split round-robin, long overflow chains
dont develop!
Doubling of directory in Extendible Hashing is similar;switching of hash functions is implicit in how the # of bitsexamined is increased.
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Example of Linear Hashing On split, hLevel+1 is used to
re-distribute entries.
0hh1
(This info
is for illustration
only!)
Level=0, N=4
00
01
10
11
000
001
010
011
(The actual contents
of the linear hashed
file)
Next=0PRIMARYPAGES
Data entry rwith h(r)=5
Primarybucket page
44* 36*32*
25*9* 5*
14* 18*10*30*
31* 35* 11*7*
0hh1
Level=0
00
01
10
11
000
001
010
011
Next=1
PRIMARYPAGES
44* 36*
32*
25*9* 5*
14* 18*10*30*
31* 35* 11*7*
OVERFLOWPAGES
43*
00100
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Example: End of a Round
0hh1
22*
00
01
10
11
000
001
010
011
00100
Next=3
01
10
101
110
Level=0
PRIMARYPAGES
OVERFLOW
PAGES
32*
9*
5*
14*
25*
66* 10*18* 34*
35*31* 7* 11* 43*
44*36*
37*29*
30*
0hh1
37*
00
01
10
11
000
001
010
011
00100
10
101
110
Next=0
Level=1
111
11
PRIMARY
PAGES OVERFLOWPAGES
11
32*
9* 25*
66* 18* 10* 34*
35* 11*
44* 36*
5* 29*
43*
14* 30* 22*
31* 7*
50*
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LH Described as a Variant of EH
The two schemes are actually quite similar: Begin with an EH index where directory hasNelements.
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 elementN, which differs from 0,now.
When bucket 1 splits, create directory elementN+1, etc.
So, directory can double gradually. Also, primary bucketpages are created in order. If they are allocatedin sequencetoo (so that finding ith is easy), we actually dont need adirectory!
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Useful Links http://www.cs.ucla.edu/classes/winter03/cs143/l1/han
douts/hash.pdf
http://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htm
http://www.smckearney.com/adb/notes/lecture.exten
dible.hashing.pdf
http://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.smckearney.com/adb/notes/lecture.extendible.hashing.pdfhttp://www.smckearney.com/adb/notes/lecture.extendible.hashing.pdfhttp://www.smckearney.com/adb/notes/lecture.extendible.hashing.pdfhttp://www.smckearney.com/adb/notes/lecture.extendible.hashing.pdfhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htmhttp://www.ecst.csuchico.edu/~melody/courses/csci151_live/Static_hash_course_notes.htm -
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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 afull 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.
Directoryless schemes (linear dynamic hashing) available Can get large with skewed data; additional I/O if this does
not fit in main memory.
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Summary 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.
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Check ListWhat is the intuition behind hash-structured indexes?
Why are they especially good for equality searches butuseless for range selections?
What is Extendible Hashing? How does it handle
search, insert, and delete?
What is Linear Hashing?
What are the similarities and differences betweenExtendible and Linear Hashing?
How does perfect hash function works?