Linear hashing: Difference between revisions

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

title = Recursive linear hashing |

journal = ACM Transactions on DatabasesDatabase 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 pre-determinedpredetermined bucket into two and contractsshrinks 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/>

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>

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===Hash functions===

ToThe determinehash 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 appliedsplit oninto thetwo keynew <math>c</math>. At any timebuckets, at most twothe hash functionsfunction <math>h_i</math> andis replaced with <math>h_{i+1}</math> arefor usedboth of those new buckets. WhenAt aany buckettime, whichat usesmost thetwo hash functionfunctions <math>h_ih_l</math> is split into two new buckets, the hash functionand <math>h_ih_{l+1}</math> isare replacedused; withsuch that <math>h_{i+1}l</math> forcorresponds bothto ofthe thosecurrent new buckets'''level'''. The family of hash functions <math> h_i(c)</math> is also referred to as the dynamic hash function.

====Determining the hashing function of a bucket====

To determine the 0-based index of the bucket that contains the record with key <math>c</math>, a hash function <math>h_i</math> is applied on the key <math>c</math>. At any time, at most two hash functions <math>h_i</math> and <math>h_{i+1}</math> are used. 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. The family of hash functions <math> h_i(c)</math> is also referred to as the dynamic hash function.

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>.

====Typical hashing function====

Typically, the value of <math>i</math> in <math>h_i</math> corresponds to the number of rightmost binary digits that are used to segregate the buckets. This dynamic hash function can be expressed arithmetically as

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

<math> h_i(c) \mapsto c \pmod{2^i}</math>.

# l represents the current level

# s represents the split pointer index

Note that when the total number of buckets is equal to one, <math>i=0</math>.<ref name="LMS" />

<math>a := h_l(c)</math>

<math>\text{if } (a < s): \quad a := h_{l+1}(c)</math>.

====Determining the hashing function for a key====

</syntaxhighlight>

Given the hashing key <math>c</math>, complete the calculations below which use the level <math>l</math> and the split pointer <math>s</math> to determine the correct hashing function for the given hashing key.

<math>a := h_l(c)</math>

<math>\text{if } (a < s): \quad a := h_{l+1}(c)</math>.

===Split control===

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

'''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 |lastlast1=Silberschatz |firstfirst1=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>

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.

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===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>. WhenThe asplit bucketpointer iscorresponds split,to splitthe pointerfirst andbucket possiblythat uses the levelhash arefunction updated<math>h_l</math> accordinginstead toof <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" />

<math>s := s+1</math>

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">

<math>\text{if } (s \ge 2^l): \quad l:=l+1, \ s:=0</math>.<ref name="LMS" />

# l represents the current level

The# levels is 0 whenrepresents the linearsplit hashingpointer index only has 1 bucket.

s = s + 1

if (s >= 2^l):

The split pointer corresponds to the first bucket that uses the hash function <math>h_l</math> instead of <math>h_{l+1}</math>.

l = l + 1

s = 0

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" />

</syntaxhighlight>

===LH*===

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==External links==

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

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

* [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]