Deciding the Confusability of Words under Tandem Repeats in Linear Time

Abstract

Tandem duplication in DNA is the process of inserting a copy of a segment of DNA adjacent to the original position. Motivated by applications that store data in living organisms, Jain et al. (2016) proposed the study of codes that correct tandem duplications to improve the reliability of data storage. We investigate algorithms associated with the study of these codes. Two words are said to be ⩽-confusable if there exists a sequence of tandem duplications for each word, where each duplication is of length at most k , such that the resulting two words after duplications are equal. For k =3, we demonstrate that the problem of deciding whether two words is ⩽3-confusable is linear-time solvable through a characterisation that can be checked efficiently. Combining with previous results, the decision problem is linear-time solvable for k ⩽ 3. We conjecture that this problem is undecidable for k > 3. Using insights gained from the algorithm, we study the size of tandem-duplication codes. We improve the previous known upper bound and then construct codes with larger sizes as compared to the previous constructions. We determine the sizes of optimal tandem-duplication codes for lengths up to 20, develop recursive methods to construct tandem-duplication codes for all word lengths, and compute explicit lower bounds for the size of optimal tandem-duplication codes for lengths from 21 to 30.