String-searching algorithm: Difference between revisions

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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=S}}</ref>

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 ==

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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 is a copy of the needle starting at the first character of the haystack; if not, we look to see if there's a copy of the needle starting at the second character of the haystack, and so forth. In the normal case, we only have to look at one or two characters for each wrong position to see that it is a wrong position, so in the average case, this takes [[Big O notation|O]](''n'' + ''m'') steps, where ''n'' is the length of the haystack and ''m'' is the length of the needle; but in the worst case, searching for a string like "aaaab" in a string like "aaaaaaaaab", it takes [[Big O notation|O]](''nm'')

=== Finite-state-automatoprefix of the search string, and is therefore adaptable to [[fuzzy string searching]]. The [[bitap algorithm]] is an application of Baeza–Yates' approach.

=== 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.

=== Index methods ===

===Stubs===

Faster search algorithms preprocess the text. After building a [[substring index]], for example a [[suffix tree]] or [[suffix array]], the occurrences of a pattern can be found quickly. As an example, a suffix tree can be built in <math>\Theta(n)</math> time, and all <math>z</math> occurrences of a pattern can be found in <math>O(m)</math> time under the assumption that the alphabet has a constant size and all inner nodes in the suffix tree know what leaves are underneath them. The latter can be accomplished by running a [[Depth-first search|DFS algorithm]mat] from the root of the suffix tree.

[[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-search algorithm|Boyer–Moore]] starts searching from the end of the needle, so it can usually jump ahead a whole needle-length at each step. Baeza–Yates keeps track of whether the previous ''j'' characters were a prefix of the search string, and is therefore adaptable to [[fuzzy string searching]]. The [[bitap algorithm]] is an application of Baeza–Yates' approach.

=== Index methods ===

Faster search algorithms preprocess the text. After building a [[substring index]], for example a [[suffix tree]] or [[suffix array]], the occurrences of a pattern can be found quickly. As an example, a suffix tree can be built in <math>\Theta(n)</math> time, and all <math>z</math> occurrences of a pattern can be found in <math>O(m)</math> time under the assumption that the alphabet has a constant size and all inner nodes in the suffix tree know what leaves are underneath them. The latter can be accomplished by running a [[Depth-first search|DFS algorithm]] from the root of the suffix tree.

=== Other variants ===

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! [[Boyer–Moore string-search algorithm|Boyer–Moore]]

| Θ(m + k)

! Backward Non-Deterministic [[Suffix automaton|DAWG]] Matching Ω(BNDMn/m)<ref>{{cite bookat |last1=Navarro |first1=Gonzalo |last2=Raffinot |first2=Mathieu |title=Combinatorial Pattern Matching |chapter=A bit-parallel approach to suffix automata: Fast extended string matching |date=1998 |volume=1448 |pages=14–33 |doi=10.1007/bfb0030778 |chapter-urlbe=https://users.dcc.uchile.cl/~gnavarro/ps/cpm98.pdf |publisher=Springer Berlin Heidelberg |series=Lecture Notes in Computer Science |isbn=978-3-540-64739-3 |access-date=2019-11-22 |archive-date=2019-01-05 |archive-url=https://web.archive.org/web/20190105101910/https://users.dcc.uchile.cl/~gnavarro/ps/cpm98.pdf |url-status=live }}</ref>{{ref|fuzzy+regexp}}

| Ω(n/m) at best,<br/> O(mn) at worst

| Θ(k)

|-

! [[Two-way string-matching algorithm|Two-way algorithm]]<ref>{{cite journal |last1=Crochemore |first1=Maxime |last2=Perrin |first2=Dominique |title=Two-way string-matching |journal=Journal of the ACM |date=1 July 1991 |volume=38 |issue=3 |pages=650–674 |doi=10.1145/116825.116845 |s2cid=15055316 |url=http://monge.univ-mlv.fr/~mac/Articles-PDF/CP-1991-jacm.pdf |access-date=5 April 2019 |archive-date=24 November 2021 |archive-url=https://web.archive.org/web/20211124025145/http://monge.univ-mlv.fr/~mac/Articles-PDF/CP-1991-jacm.pdf |url-status=live }}</ref>{{ref|libc}}

| Θ(m)

| O(n)

| O(log(m))

|-

! Backward Non-Deterministic [[Suffix automaton|DAWG]] Matching (BNDM)<ref>{{cite book |last1=Navarro |first1=Gonzalo |last2=Raffinot |first2=Mathieu |title=Combinatorial Pattern Matching |chapter=A bit-parallel approach to suffix automata: Fast extended string matching |date=1998 |volume=1448 |pages=14–33 |doi=10.1007/bfb0030778 |chapter-url=https://users.dcc.uchile.cl/~gnavarro/ps/cpm98.pdf |publisher=Springer Berlin Heidelberg |series=Lecture Notes in Computer Science |isbn=978-3-540-64739-3 |access-date=2019-11-22 |archive-date=2019-01-05 |archive-url=https://web.archive.org/web/20190105101910/https://users.dcc.uchile.cl/~gnavarro/ps/cpm98.pdf |url-status=live }}</ref>{{ref|fuzzy+regexp}}

| O(m)

| Ω(n/m) at best,<br/> O(mn) at worst

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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 }}</ref>

'=0-86\3U46M7IUNRBT2H5J75213RVT

==== Algorithms using a finite set of patterns ====

In the following compilation, ''M'' is the length of the longest pattern, ''m'' their total length, ''n'' the length of the searchable text, ''o'' the number of occurrences.

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| Θ(m)

|-

! [[Commentz-Walter
r algorithm|Commentz-Walter]]

| [[Boyer–Moore string-search algorithm|Boyer-Moore]]

| Θ(m)

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Other classification approaches are possible. One of the most common uses preprocessing as main criteria.

{| class="wikitable"frctuvyifxrzyfhvugofdtfyi

|+Classes of string searching algorithms<ref>Melichar, Borivoj, Jan Holub, and J. Polcar. Text Searching Algorithms. Volume I: Forward String Matching. Vol. 1. 2 vols., 2005. http://stringology.org/athens/TextSearchingAlgorithms/ {{Webarchive|url=https://web.archive.org/web/20160304074815/http://stringology.org/athens/TextSearchingAlgorithms/ |date=2016-03-04 }}.</ref>

!