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# Duplication Distance to the Root for Binary Sequences

Alon, Noga and Bruck, Jehoshua and Farnoud (Hassanzadeh), Farzad and Jain, Siddharth (2017) Duplication Distance to the Root for Binary Sequences. IEEE Transactions on Information Theory, 63 (12). pp. 7793-7803. ISSN 0018-9448. doi:10.1109/TIT.2017.2730864. https://resolver.caltech.edu/CaltechAUTHORS:20170726-162754925

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## Abstract

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 = abc → y = abbc, 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.

Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/TIT.2017.2730864DOIArticle
https://arxiv.org/abs/1611.05537arXivDiscussion Paper
http://resolver.caltech.edu/CaltechAUTHORS:20161108-134615672Related ItemTechnical Report
ORCID:
AuthorORCID
Alon, Noga0000-0003-1332-4883
Bruck, Jehoshua0000-0001-8474-0812
Jain, Siddharth0000-0002-9164-6119
Additional Information:© 2017 IEEE. Manuscript received October 26, 2016; revised June 14, 2017; accepted July 9, 2017. Date of publication July 26, 2017; date of current version November 20, 2017. This work was supported in part by the NSF Expeditions in Computing Program (The Molecular Programming Project), in part by USA-Israeli BSF under Grant 2012/107, in part by ISF under Grant 620/13, and in part by the Israeli I-Core Program. This paper was presented at the 2016 IEEE International Symposium on Information Theory.
Funders:
Funding AgencyGrant Number
NSFUNSPECIFIED
Binational Science Foundation (USA-Israel)2012/107
Israel Science Foundation620/13
I-CORE Program of the Planning and Budgeting CommitteeUNSPECIFIED
Subject Keywords:Sequences, substrings, tandem duplication, root, deduplication
Issue or Number:12
DOI:10.1109/TIT.2017.2730864
Record Number:CaltechAUTHORS:20170726-162754925
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170726-162754925
Official Citation:N. Alon, J. Bruck, F. Farnoud Hassanzadeh and S. Jain, "Duplication Distance to the Root for Binary Sequences," in IEEE Transactions on Information Theory, vol. 63, no. 12, pp. 7793-7803, Dec. 2017. doi: 10.1109/TIT.2017.2730864.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:79466
Collection:CaltechAUTHORS
Deposited By: Kristin Buxton
Deposited On:27 Jul 2017 00:18