2016/11/17 by Noga Alon, Jehoshua Bruck, Alon, Noga +5
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Algorithms and Data Compression #Coding theory and cryptography #Discrete Mathematics (cs.DM) #FOS: Biological sciences #FOS: Computer and information sciences #Genomics (q-bio.GN) #Information Theory (cs.IT) #cs.DM #cs.IT #math.IT #q-bio.GN #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1611.05537
submitted to IEEE Transactions on Information Theory
arxiv created 2016/11/17 · openalex publication_date 2016/11/17 · arxiv updated 2016/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
We study the tandem duplication distance between binary sequences and their roots. In other words, the quantity of interest is the number of tandem duplication operations of the form \seq x = \seq a \seq b \seq c → \seq y = \seq a \seq b \seq b \seq c, where \seq x and \seq y are sequences and \seq a, \seq b, and \seq c are their substrings, needed to generate a binary sequence of length n starting from a square-free sequence from the set \0,1,01,10,010,101\. This problem is a restricted case of finding the duplication/deduplication distance between two sequences, defined as the minimum number of duplication and deduplication operations required to transform one sequence to the other. We consider both exact and approximate tandem duplications. For exact duplication, denoting the maximum distance to the root of a sequence of length n by f(n), we prove that f(n)=Θ(n). For the case of approximate duplication, where a β-fraction of symbols may be duplicated incorrectly, we show that the maximum distance has a sharp transition from linear in n to logarithmic at β=1/2. We also study the duplication distance to the root for sequences with a given root and for special classes of sequences, namely, the de Bruijn sequences, the Thue-Morse sequence, and the Fibbonaci words. The problem is motivated by genomic tandem duplication mutations and the smallest number of tandem duplication events required to generate a given biological sequence.