Linear hashing: Difference between revisions

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'''Linear hashing''' ('''LH''') is a dynamic data structure which implements a [[hash table]] algorithmand grows or shrinks one bucket at a time. It was invented by Witold Litwin (1980)in 1980.<ref> name=WL80>{{Citation | first1=Witold | last1=Litwin | title=Linear hashing: A new tool for file and table addressing | journal=Proc. 6th Conference on Very Large Databases | pages=212–223 | year=1980 | url=httphttps://www.cs.cmu.edu/afs/cs.cmu.edu/user/christos/www/courses/826-resources/PAPERS+BOOK/linear-hashing.PDF|format=PDF}}</ref>, and later popularized by [[Paul Larson]]. Linear hashing allows for the expansion of the hash table one slot at a time.

<ref name=Ellis>

The frequent single slot expansion can very effectively control the length of

{{Citation | first1 = Carla Schlatter | last1=Ellis | title=Concurrency in Linear Hashing | journal=ACM Transactions on Database Systems | volume=12 | number=2 | pages=195–217 | date=June 1987| doi=10.1145/22952.22954 | s2cid=14260177 | doi-access=free }}

the collision chain. The cost of hash table expansion is spread out across each

</ref> It has been analyzed by Baeza-Yates and Soza-Pollman.<ref name=BS>{{Citation | first1=Ricardo | last1=Baeza-Yates | first2=Hector | last2=Soza-Pollman | title=Analysis of Linear Hashing Revised | journal=Nordic Journal of Computing | pages=70–85 | year=1998 | s2cid=7497598 | url=http://pdfs.semanticscholar.org/e6cd/667fef7cd377ed8d417cc648d3d578675ad5.pdf | archive-url=https://web.archive.org/web/20190307204217/http://pdfs.semanticscholar.org/e6cd/667fef7cd377ed8d417cc648d3d578675ad5.pdf | url-status=dead | archive-date=2019-03-07 }}</ref> It is the first in a number of schemes known as dynamic hashing<ref name=BS/>

hash table insertion operation, as opposed to be incurred all at once. <ref> {{Citation | first1=Per-Åke | last1=Larson | title=Dynamic Hash Tables | journal=Communications of the ACM | pages=446–457 | date=April 1988 | volume=31 | Number=4 | doi=10.1145/42404.42410 | url=http://doi.acm.org/10.1145/42404.42410}}</ref> Therefore

<ref name=RD>{{Citation | first1=Richard | last1=Enbody | first2 = HC | last2 = Du | title=Dynamic hashing schemes | journal=ACM Computing Surveys | pages=85–113 | date=June 1988 | volume=20 | number=2 | doi=10.1145/46157.330532 | s2cid=1437123 | doi-access=free }}</ref> such as Larson's Linear Hashing with Partial Extensions, <ref name=AL>{{Citation | first1=Per-Åke | last1=Larson | title=Dynamic Hash Tables | journal=Communications of the ACM | pages=446–457 | date=April 1988 | volume=31 | number=4 | doi=10.1145/42404.42410| s2cid=207548097 | doi-access=free }}</ref> Linear Hashing with Priority Splitting,<ref name=ruchte>

linear hashing is well suited for interactive applications.

{{Citation | first1 = Willard | last1 = Ruchte | first2 = Alan | last2 = Tharp |

title = Linear hashing with Priority Splitting: A method for improving the retrieval performance of linear hashing |

journal = IEEE Third International Conference on Data Engineering |

date = Feb 1987 | pages=2–9}}

</ref> Linear Hashing with Partial Expansions and Priority Splitting,<ref>

{{Citation | first1 = Yannis | last1 = Manolopoulos | first2 = N. | last2 = Lorentzos |

title = Performance of linear hashing schemes for primary key retrieval |

journal = Information Systems | volume = 19 | number = 5 |date = 1994 | pages =433–446| doi = 10.1016/0306-4379(94)90005-1 }}

</ref> or Recursive Linear Hashing.<ref name=RS>

{{Citation | first1 = K. |last1 = Ramamohanarao | first2 = R. | last2 = Sacks-Davis |

title = Recursive linear hashing |

journal = ACM Transactions on Database Systems|

volume=9 | number=3 |date=Sep 1984 | pages=369–391|doi = 10.1145/1270.1285 |s2cid = 18577730 |doi-access = free }}

</ref>

The file structure of a dynamic hashing data structure adapts itself to changes in the size of the file, so expensive periodic file reorganization is avoided.<ref name=RD/> A Linear Hashing file expands by splitting a predetermined bucket into two and shrinks by merging two predetermined buckets into one. The trigger for a reconstruction depends on the flavor of the scheme; it could be an overflow at a bucket or [[load factor (computer science)|load factor]] (i.e., the number of records divided by the number of buckets) moving outside of a predetermined range.<ref name=WL80 /> In Linear Hashing there are two types of buckets, those that are to be split and those already split. While extendible hashing splits only overflowing buckets, [[spiral Hashing|spiral hashing]] (a.k.a. spiral storage) distributes records unevenly over the buckets such that buckets with high costs of insertion, deletion, or retrieval are earliest in line for a split.<ref name=AL/>

==Algorithm Details==

As usual, a [[hash function]] controls the address calculation of linear hashing.

In linear hashing, the address calculation is always bounded by a size that

is a [[power of two] * N, where N is the chosen original number of buckets.

The number of buckets is given by N * 2 ^ Level e.g. Level 0, N; Level 1, 2N; Level 2 , 4N.

Linear Hashing has also been made into a scalable distributed data structure, '''LH*'''. In LH*, each bucket resides at a different server.<ref name=WL93>

address(level,key) = hash(key) mod N * (2^level)

{{Citation | first1=Witold | last1=Litwin | first2=Marie-Anne |last2 = Neimat |

first3 = Donavan A. | last3 = Schneider | title=LH: Linear Hashing for distributed files | journal=ACM SIGMOD Record |

pages = 327–336 | date = 1993| volume=22 | issue=2 | doi=10.1145/170036.170084 | s2cid=259938726 }}

</ref> LH* itself has been expanded to provide data availability in the presence of failed buckets.<ref name=LMS>

{{Citation | first1=Witold | last1=Litwin | first2 = Rim | last2 = Moussa |

first3 = Thomas | last3 = Schwarz | title=LH*RS - a highly-available scalable distributed data structure |journal=ACM Transactions on Database Systems | volume=30 | number=3 | pages=769–811 | date=Sep 2005| doi=10.1145/1093382.1093386 | s2cid=1802386 | url=https://basepub.dauphine.fr/handle/123456789/15124 }}

</ref> Key based operations (inserts, deletes, updates, reads) in LH and LH* take maximum constant time independent of the number of buckets and hence of records.<ref name = WL80/><ref name=LMS/>

==Algorithm details==

The 'split' variable controls the read operation, and the expansion operation.

Records in LH or LH* consists of a key and a content, the latter basically all the other attributes of the record.<ref name=WL80/><ref name=LMS/> They are stored in buckets. For example, in Ellis' implementation, a bucket is a linked list of records.<ref name=Ellis/> The file allows the key based CRUD operations create or insert, read, update, and delete as well as a scan operations that scans all records, for example to do a database select operation on a non-key attribute.<ref name=LMS/> Records are stored in buckets whose numbering starts with 0.<ref name=LMS/>

A read operation would use address(level,key) if address(level,key) is greater

than or equal to the 'split' variable. Otherwise, address(level+1,key) is used.

The key distinction from schemes such as Fagin's extendible hashing is that as the file expands due to insertions, only one bucket is split at a time, and the order in which buckets are split is already predetermined.<ref name="Fagin">

A linear hashing table expansion operation would consist of rehashing the

{{Citation |last1=Fagin |first1=Ronald |title=Extendible Hashing - A Fast Access Method for Dynamic Files |date=Sep 1979 |url=http://dl.acm.org/citation.cfm?doid=320083.320092 |journal=ACM Transactions on Database Systems |volume=4 |number=2 |pages=315–344 |doi=10.1145/320083.320092 |s2cid=2723596 |last2=Nievergelt |first2=Jurg |last3=Pippenger |first3=Nicholas |last4=Strong |first4=Raymond}}

entries at slot ___location indicated by the 'split' variable to the target slot ___location

</ref>

of address(level+1,key). The 'split' variable is incremented by 1 at the end of

the expansion operation. If the 'split' variable reaches N * 2^level, then the 'level'

variable is incremented by 1, and the 'split' variable is reset to 0.

===Hash functions===

Thus the hash buckets are expanded round robin, and seem unrelated to where buckets overflow at the time of expansion.

The hash function <math>h_i(c)</math> returns the 0-based index of the bucket that contains the record with key <math>c</math>. When a bucket which uses the hash function <math>h_i</math> is split into two new buckets, the hash function <math>h_i</math> is replaced with <math>h_{i+1}</math> for both of those new buckets. At any time, at most two hash functions <math>h_l</math> and <math>h_{l+1}</math> are used; such that <math>l</math> corresponds to the current '''level'''. The family of hash functions <math> h_i(c)</math> is also referred to as the dynamic hash function.

Overflow buckets are used at the sites of bucket overflow, but these are eventually reabsorbed when the round robin

comes to the bucket with the overflow bucket, and the contents of that bucket and the overflow bucket are redistributed

by the future hash function ( hash(key) mod N * (2^{level'''+1'''} ). The degenerate case, which is unlikely

with a randomized hash function, is that enough entries are hashed to the same bucket so that there is enough entries

to overflow more than one overflow bucket ( assuming overflow bucket size = normal bucket size), before being absorbed by

a delayed round robin split.

The point of the algorithm seems to be that overflow is preempted by gradually increasing the number of available buckets,

and overflow buckets are eventually reabsorbed during a later split, which must eventually happen because splitting occurs round robin.

Typically, the value of <math>i</math> in <math>h_i</math> corresponds to the number of rightmost binary digits of the key <math>c</math> that are used to segregate the buckets. This dynamic hash function can be expressed arithmetically as <math display="inline"> h_i(c) \mapsto (c \bmod 2^i) </math>. Note that when the total number of buckets is equal to one, <math>i=0</math>.

One problem is that after a bucket is split, it's old hash function has been replaced by the newer hash function ( hash(key) mod N * (2^{level'''+1'''} ).

To determine whether the old mod 2^level should be used, the old hash is applied first, and then if it is < split variable , then

it is recalculated with mod 2^{level +1} . Note this depends on the property that for any y = x mod M , y' = y or y + M if calculated as x mod M * 2 .

Therefore the new hash function will only hash to buckets created by previous splits.

Complete the calculations below to determine the correct hashing function for the given hashing key <math>c</math>.<ref name="LMS" /><syntaxhighlight lang="python">

and having the hash function conditional on the split variable, so any hash value greater than '''split''' is still used ( where hash is h(k) mod N * 2^level , as above ), but

# l represents the current level

any hash value <= split will be recalculated as h(k) mod N * 2^{level '''+ 1'''}

# s represents the split pointer index

a = h_l(c)

if (a < s): a = h_{l+1}(c)

</syntaxhighlight>

===Split control===

There is some flexibility in choosing how often the expansion operations are performed.

Linear hashing algorithms may use only controlled splits or both controlled and uncontrolled splits.

One obvious choice is to perform the expansion operation each time no more slots are available at the target slot ___location. Another choice

is to control the expansion with a programmer defined load factor.

'''Controlled splitting''' occurs if a split is performed whenever the [[load factor (computer science)|load factor]], which is monitored by the file, exceeds a predetermined threshold.<ref name="LMS" /> If the hash index uses controlled splitting, the buckets are allowed to overflow by using linked overflow blocks. When the ''load factor'' surpasses a set threshold, the ''split pointer's'' designated bucket is split. Instead of using the load factor, this threshold can also be expressed as an occupancy percentage, in which case, the maximum number of records in the hash index equals (occupancy percentage)*(max records per non-overflowed bucket)*(number of buckets).<ref name=":0">{{Cite book |last1=Silberschatz |first1=Abraham |title=Database system concepts |last2=Korth |first2=Henry F. |last3=Sudarshan |first3=S. |date=2020 |publisher=McGraw-Hill Education |isbn=978-1-260-08450-4 |edition=Seventh |___location=New York, NY}}</ref>

The hash table array for linear hashing is usually implemented with a [[dynamic array]]

algorithm.

An '''uncontrolled split''' occurs when a split is performed whenever a bucket overflows, in which case that bucket would be split into two separate buckets.

'''File contraction''' occurs in some LH algorithm implementations if a controlled split causes the load factor to sink below a threshold. In this case, a merge operation would be triggered which would undo the last split, and reset the file state.<ref name="LMS" />

===Split pointer===

The index of the next bucket to be split is part of the file state and called the '''split pointer''' <math>s</math>. The split pointer corresponds to the first bucket that uses the hash function <math>h_l</math> instead of <math>h_{l+1}</math>.<ref name="LMS" />

For example, if numerical records are inserted into the hash index according to their farthest right binary digits, the bucket corresponding with the appended bucket will be split. Thus, if we have the buckets labelled as 000, 001, 10, 11, 100, 101, we would split the bucket 10 because we are appending and creating the next sequential bucket 110. This would give us the buckets 000, 001, 010, 11, 100, 101, 110.<ref name=":0" />

When a bucket is split, split pointer and possibly the level are updated according to the following, such that the level is 0 when the linear hashing index only has 1 bucket.<ref name="LMS" /><syntaxhighlight lang="python">

# l represents the current level

# s represents the split pointer index

s = s + 1

if (s >= 2^l):

l = l + 1

s = 0

</syntaxhighlight>

===LH*===

The main contribution of LH* is to allow a client of an LH* file to find the bucket where

the record resides even if the client does not know the file state. Clients in fact store

their version of the file state, which is initially just the knowledge of the first bucket, namely Bucket 0. Based on their file state, a client calculates the address of a

key and sends a request to that bucket. At the bucket, the request is checked and if

the record is not at the bucket, it is forwarded. In a reasonably stable system, that is,

if there is only one split or merge going on while the request is processed, it can

be shown that there are at most two forwards. After a forward, the final bucket sends an

Image Adjustment Message to the client whose state is now closer to the state of the distributed file.<ref name="LMS" /> While forwards are reasonably rare for active clients,

their number can be even further reduced by additional information exchange between

servers and clients <ref name="CS">

{{Citation |last1=Chabkinian |first1=Juan |title=Fast LH* |date=2016 |journal=International Journal of Parallel Programming |volume=44 |number=4 |pages=709–734 |doi=10.1007/s10766-015-0371-8 |s2cid=7448240 |last2=Schwarz |first2=Thomas}}

</ref>

== Other properties ==

===File state calculation===

The file state consists of split pointer <math>s</math> and level <math>l</math>. If the original file started with <math>N=1</math> buckets, then the number of buckets <math>n</math> and the file state are related via

<ref name="CS" />

<math>n = 2^l+s </math>.

==Adoption in language systems==

Griswold and Townsend <ref> {{Citation | title=The Design and Implementation of Dynamic Hashing for Sets and Tables in Icon | first1=William G. | last1=Griswold | author1-link = Bill Griswold | first2=Gregg M. | last2=Townsend | journal=Software -: Practice and Experience | volume=23 | issue=4 | date=April 1993 | pages=351–367 | doi=10.1002/spe.4380230402 | s2cid=11595927 | url=http://citeseer.ist.psu.edu/griswold93design.html}} </ref> discussed the adoption of linear hashing in the [[Icon language]]. They discussed the implementation alternatives of [[dynamic array]] algorithm used in linear hashing, and presented performance comparisons using a list of Icon benchmark applications.

==Adoption in database systems==

Linear hashing is used in the [[Berkeley DB|Berkeley database system (BDB)]], which in turn is used by many software systems, using a C implementation derived from the [[Communications of the ACM|CACM]] article and first published on the Usenet in 1988 by Esmond Pitt.

==References==

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==External links==

* [https://tommyds.sourceforge.net/ TommyDS, C implementation of a Linear Hashtable]

*[http://www.concentric.net/~Ttwang/tech/sorthash.htm Sorted Linear Hash Table]

* [http://hackthology.com/an-in-memory-go-implementation-of-linear-hashing.html An in Memory Go Implementation with Explanation]

* {{DADS|linear hashing|linearHashing}}

* [https://github.com/KevinStern/index-cpp/blob/master/src/linear_hashing_table.h A C++ Implementation of Linear Hashtable which Supports Both Filesystem and In-Memory storage]

==See also==

* [[Extendible hashing]]

* [[Consistent hashing]]

* [[Spiral Hashing]]

[[Category:Search algorithms]]