Duplication Distance to the Root for Binary Sequences

Noga Alon, Jehoshua Bruck, Farzad Farnoud Hassanzadeh, Siddharth Jain

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


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 x = a b c\to y = a b b c, where x and y are sequences and a, b, and 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 the set of sequences arising from a given root and for special classes of sequences, namely, the De Bruijn sequences, the Thue-Morse sequence, and the Fibonacci 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.

Original languageEnglish (US)
Article number7993073
Pages (from-to)7793-7803
Number of pages11
JournalIEEE Transactions on Information Theory
Issue number12
StatePublished - Dec 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Sequences
  • deduplication
  • root
  • substrings
  • tandem duplication


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