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Low level algorithms, bit wizardry, combinatorial generation, fast transforms, fast arithmetics and algorithms for finite fields. A real masterpiece of computation !
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Some algorithms: * [Efficient computation of absent words in genomic sequences](https://pubmed.ncbi.nlm.nih.gov/18366790) paper by Julia Herold, Stefan Kurtz and Robert Giegerich. * [keeSeek algo](http://www.medcomp.medicina.unipd.it/main_site/doku.php?id=keeseek) from [keeSeek: searching distant non-existing words in genomes for PCR-based applications](https://academic.oup.com/bioinformatics/article/30/18/2662/2475409) paper by Marco Falda, Paolo Fontana, Luisa Barzon, Stefano Toppo and Enrico Lavezzo. Can these algorithms be improved ?
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A famous example of the periodicity forcing word equation is $xy = yx$. It means that if a word $xy$ equals to a word $yx$, obtained by a _cut-and-swap_ process, there exists a word $w$ and integers $i,j$ such that $x = w^i$ and $y = w^j$. The paper's Section 1 and 5 contain a brief yet very helpful overview of existing results about periodicity forcing. About ten years ago, Czeizler, Kari and Seki, in their work [On a special class of primitive words](https://www.csd.uwo.ca/~lkari/mfcs.pdf), have introduced a notion of a pseudo-repetition. It can be explained with the following example. When DNA molecules are regarded as words, the Watson-Crick correspondence $$G \sim C, A \sim T$$ between guanine ($G$) and cytosine ($C$), adenine ($A$) and thymine ($T$) allows us to say that a word $ATCGATTGAGCTCTAGCG$ contains the same information as the word $TAGCTAACTCGAGATCGC$. A pseudo-repetition is a repetition of a word or its Watson-Crick complement. So, for instance $ATC \sim TAG$, and the word $ATCTAGATCATCTAG$ is a pseudo-repetition of the word $ATC$. In an abstract algebraic setting (involving monoids, morphic permutations, equivalence classes and anticongruences), the author studies an extension to the notion of periodicity forcing, using the recently introduced concept of pseudo-repetitions and related ideas. Holub proves, for instance, that any periodicity forcing classical word equation also forces a pseudo-periodicity. He provides interesting new results and generalizations of already known results. From MathSciNet MR4074724.
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