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BNDM algorithm's matching complexity is not O(n), it's the same as for the Boyer-Moore - Omega(n/m) at best, O(n*m) at worst. https://users.dcc.uchile.cl/~gnavarro/ps/cpm98.pdf |
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{{short description|
A basic example of string searching is when the pattern and the searched text are [[Array data structure|arrays]] of elements of an [[Alphabet (computer science)|alphabet]] ([[finite set]]) Σ. Σ may be a human language alphabet, for example, the letters ''A'' through ''Z'' and other applications may use a ''binary alphabet'' (Σ = {0,1}) or a ''DNA alphabet'' (Σ = {A,C,G,T}) in [[bioinformatics]].
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The most basic case of string searching involves one (often very long) string, sometimes called the ''haystack'', and one (often very short) string, sometimes called the ''needle''. The goal is to find one or more occurrences of the needle within the haystack. For example, one might search for ''to'' within:
One might request the first occurrence of "to", which is the fourth word; or all occurrences, of which there are 3; or the last, which is the fifth word from the end.
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Another more complex type of search is [[regular expression]] searching, where the user constructs a pattern of characters or other symbols, and any match to the pattern should fulfill the search. For example, to catch both the American English word "color" and the British equivalent "colour", instead of searching for two different literal strings, one might use a regular expression such as:
where the "?" conventionally makes the preceding character ("u") optional.
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This article mainly discusses algorithms for the simpler kinds of string searching.
A similar problem introduced in the field of bioinformatics and genomics is the maximal exact matching (MEM).<ref>{{Cite journal|last1=Kurtz|first1=Stefan|last2=Phillippy|first2=Adam|last3=Delcher|first3=Arthur L|last4=Smoot|first4=Michael|last5=Shumway|first5=Martin|last6=Antonescu|first6=Corina|last7=Salzberg|first7=Steven L|date=2004|title=Versatile and open software for comparing large genomes |journal=Genome Biology |volume=5|issue=2|pages=R12|doi=10.1186/gb-2004-5-2-r12|issn=1465-6906 |pmc= 395750|pmid=14759262 |doi-access=free }}</ref> Given two strings, MEMs are common substrings that cannot be extended left or right without causing a mismatch.<ref>{{Cite journal|last1=Khan|first1=Zia|last2=Bloom|first2=Joshua S.|last3=Kruglyak|first3=Leonid|last4=Singh|first4=Mona|date=2009-07-01|title=A practical algorithm for finding maximal exact matches in large sequence datasets using sparse suffix arrays|url= |journal=Bioinformatics |volume=25|issue=13|pages=1609–1616|doi=10.1093/bioinformatics/btp275 |pmc=2732316|pmid=19389736}}</ref>
== Examples of search algorithms ==
=== Naive string search ===
A simple and inefficient way to see where one string occurs inside another is to check at each index, one by one. First, we see if there
=== Finite-state-automaton-based search ===
[[Image:DFA search mommy.svg|200px|right]]
In this approach, backtracking is avoided by constructing a [[deterministic finite automaton]] (DFA) that recognizes a stored search string. These are expensive to construct—they are usually created using the [[powerset construction]]—but are very quick to use. For example, the [[deterministic finite automaton|DFA]] shown to the right recognizes the word "MOMMY". This approach is frequently generalized in practice to search for arbitrary [[regular expression]]s.
===Stubs===
[[Knuth–Morris–Pratt algorithm|Knuth–Morris–Pratt]] computes a [[deterministic finite automaton|DFA]] that recognizes inputs with the string to search for as a suffix, [[Boyer–Moore string
=== Index methods ===
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== Classification of search algorithms ==
=== Classification by a number of patterns ===
The various [[algorithm]]s can be classified by the number of patterns each uses.
==== Single-pattern algorithms ====
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! Naïve algorithm
| none
| Θ(n+m) in average,<br/> O(mn)
| none
|-
! Automaton-based matching
| Θ(km)
| Θ(n)
| Θ(km)
|-
! [[Rabin–Karp algorithm|Rabin–Karp]]
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! [[Boyer–Moore string-search algorithm|Boyer–Moore]]
| Θ(m + k)
|
| Θ(k)
|-
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| Θ(m)
| O(n)
| O(
|-
! Backward Non-Deterministic [[
| O(m)
| Ω(n/m) at best,<br/> O(mn) at worst
|
|-
! Backward Oracle Matching (BOM)<ref>{{cite
| O(m)
| O(mn)
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:1.{{note|Asymptotic times}}Asymptotic times are expressed using [[Big O notation|O, Ω, and Θ notation]].
:2.{{note|libc}}Used to implement the ''memmem'' and [[C string handling#Functions|''strstr'']] search functions in the [[glibc]]<ref>{{cite web |title=glibc/string/str-two-way.h |url=https://code.woboq.org/userspace/glibc/string/str-two-way.h.html |access-date=2022-03-22 |archive-date=2020-09-20 |archive-url=https://web.archive.org/web/20200920180414/https://code.woboq.org/userspace/glibc/string/str-two-way.h.html |url-status=live }}</ref> and [[musl]]<ref>{{cite web |title=musl/src/string/memmem.c |url=http://git.musl-libc.org/cgit/musl/tree/src/string/memmem.c |access-date=23 November 2019 |archive-date=1 October 2020 |archive-url=https://web.archive.org/web/20201001184748/http://git.musl-libc.org/cgit/musl/tree/src/string/memmem.c |url-status=live }}</ref> [[C standard libraries]].
:3.{{note|fuzzy+regexp}}Can be extended to handle [[approximate string matching]] and (potentially-infinite) sets of patterns represented as [[regular
The '''[[Boyer–Moore string-search algorithm]]''' has been the standard benchmark for the practical string-search literature.<ref name=":0">{{cite journal |last1=Hume |last2=Sunday |year=1991 |title=Fast String Searching |journal=Software: Practice and Experience |volume=21 |issue=11 |pages=1221–1248 |doi=10.1002/spe.4380211105 |s2cid=5902579
==== Algorithms using a finite set of patterns ====
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! Space
|-
! [[Aho–Corasick
| [[
| Θ(m)
| Θ(n + o)
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! Patterns preprocessed
| Constructed search engines
| Signature methods
|}
=== Classification by matching strategies ===
Another one classifies the algorithms by their matching strategy:<ref>{{citation|surname1=Gonzalo Navarro|surname2=Mathieu Raffinot|title=Flexible Pattern Matching Strings: Practical On-Line Search Algorithms for Texts and Biological Sequences |isbn=978-0-521-03993-2|date= 2008|publisher=Cambridge University Press }}</ref>
* Match the prefix first (Knuth–Morris–Pratt, Shift-And, Aho–Corasick)
* Match the suffix first (Boyer–Moore and variants, Commentz-Walter)
* Match the best factor first (BNDM, BOM, Set-BOM)
* Other strategy (Naïve, Rabin–Karp, Vectorized)
===Real-time string matching===
In real-time string matching, one requires the matcher to output a response after reading each character of the text, that indicates
whether this is the last character of a match. The response has to be given within constant time.
The requirement regarding preprocessing vary: O(''m'') preprocessing may be allowed after the pattern is read (but before the reading of
the text), or a stricter requirement may be posed according to which the matcher has to also pause for at most a constant time after reading any character of the pattern (including the last).
For the more lenient version, if one does not mind that the preprocessing time and memory requirement dependend on the size of the alphabet, a real-time solution is provided by
automaton matching.
[[Zvi Galil]] developed a method to turn certain algorithms into real-time algorithms, and applied it to produce a variant of the KMP
matcher that runs in real time under the strict requirement.<ref>{{Cite journal
| last = Galil
| first = Zvi
| title = String matching in real time
| journal = Journal of the ACM
| volume = 28
| issue = 1
| pages = 134–149
| year = 1981
| doi = 10.1145/322234.322244
| pmid =
| url =
}}</ref>
==String searching with don't cares==
In this version of the string searching problem, there is a special symbol, ø (read: don't care), which can match any other symbol (including another ø).
Don't care symbols can appear either in the pattern or in the text. In 2002, an algorithm for this problem that runs in <math>O(n\log m)</math> time has been
given by [[Richard Cole]] and [[Ramesh Hariharan]], improving on a solution from 1973 by [[Michael J. Fischer|Fischer]] and [[Michael S. Paterson|Paterson]]
that has complexity <math>O(n\log m\log k)</math>, where ''k'' is the size of the alphabet.<ref>{{Cite conference
| last1 = Cole
| first1 = Richard
| last2 = Hariharan
| first2 = Ramesh
| title = Verifying candidate matches in sparse and wildcard matching
| book-title = Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
| editor =
| publisher =
| ___location =
| pages = 592–601
| year = 2002
| url =
| doi =
}}</ref> Another algorithm, claimed simpler, has been proposed by [[Peter Clifford (mathematician)|Clifford]] and [[Raphaël Clifford|Clifford]].<ref>{{cite journal |last1=Clifford |first1=Peter |last2=Clifford |first2=Raphaël |title=Simple deterministic wildcard matching |journal=Information Processing Letters |date=January 2007 |volume=101 |issue=2 |pages=53–54 |doi=10.1016/j.ipl.2006.08.002}}</ref>
==See also==
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*[[Compressed pattern matching]]
*[[Matching wildcards]]
*[[Approximate string matching]]
*[[Full-text search]]
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*[http://stringsandchars.amygdalum.net/ StringsAndChars] – Implementations of many String-Matching-Algorithms (for single and multiple patterns) in Java
*[http://www-igm.univ-mlv.fr/~lecroq/string/index.html Exact String Matching Algorithms] — Animation in Java, Detailed description and C implementation of many algorithms.
*[http://www.cs.ucr.edu/~stelo/cpm/cpm04/35_Navarro.pdf (PDF) Improved Single and Multiple Approximate String Matching] {{Webarchive|url=https://web.archive.org/web/20170311221134/http://www.cs.ucr.edu/~stelo/cpm/cpm04/35_Navarro.pdf |date=2017-03-11 }}
*[https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2647288/ Kalign2: high-performance multiple alignment of protein and nucleotide sequences allowing external features]
*[https://www.codeproject.com/Articles/5282980/Fastest-Fulltext-Vector-Scalar-Exact-Searcher NyoTengu – high-performance pattern matching algorithm in C] – Implementations of Vector and Scalar String-Matching-Algorithms in C
*[https://arxiv.org/html/2502.20338v3 Nathaniel K. Brown, et al.: "KeBaB: k-mer based breaking for finding long MEMs", arXiv:2502.20338v3 (09 Jun 2025).]
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