Lexicographically minimal string rotation: Difference between revisions

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==Algorithms==

===The Naivenaive Algorithmalgorithm===

The naive algorithm for finding the lexicographically minimal rotation of a string is to iterate through successive rotations while keeping track of the most lexicographically minimal rotation encountered. If the string is of length {{mvar|n}}, this algorithm runs in {{math|''O''(''n''²)}} time in the worst case.

===Booth's Algorithmalgorithm===

An efficient algorithm was proposed by Booth (1980).<ref>{{cite journal

| author = Kellogg S. Booth

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Of interest is that removing all lines of code which modify the value of {{mvar|k}} results in the original Knuth-Morris-Pratt preprocessing function, as {{mvar|k}} (representing the rotation) will remain zero. Booth's algorithm runs in {{tmath|O(n)}} time, where {{mvar|n}} is the length of the string. The algorithm performs at most {{tmath|3n}} comparisons in the worst case, and requires auxiliary memory of length {{mvar|n}} to hold the failure function table.

===Shiloach's Fastfast Canonizationcanonization Algorithmalgorithm===

Shiloach (1981)<ref>{{cite journal

| title = Fast canonization of circular strings

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The algorithm is divided into two phases. The first phase is a quick sieve which rules out indices that are obviously not starting locations for the lexicographically minimal rotation. The second phase then finds the lexicographically minimal rotation start index from the indices which remain.

===Duval's Lyndon Factorizationfactorization Algorithmalgorithm===

Duval (1983)<ref>{{cite journal

| title = Factorizing words over an ordered alphabet

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proposed an algorithm to efficiently compare two circular strings for equality without a normalization requirement. An additional application which arises from the algorithm is the fast generation of certain chemical structures without repetitions.

A variant for [[Quantumquantum computing]] was proposed by Wang & Ying (2024).<ref name="Wang">{{cite journal

| author = Wang, Q., Ying, M.

| title = Quantum Algorithm for Lexicographically Minimal String Rotation

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| year = 2024

| doi = 10.1007/s00224-023-10146-8

| arxiv = 2012.09376

}}

</ref> They show that the quantum algorithm outperforms any (classical) randomized algorithms in both worst and average cases.