TY - JOUR

T1 - Duplication Distance to the Root for Binary Sequences

AU - Alon, Noga

AU - Bruck, Jehoshua

AU - Hassanzadeh, Farzad Farnoud

AU - Jain, Siddharth

N1 - Funding Information:
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.

PY - 2017/12

Y1 - 2017/12

N2 - 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.

AB - 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.

KW - Sequences

KW - deduplication

KW - root

KW - substrings

KW - tandem duplication

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U2 - 10.1109/TIT.2017.2730864

DO - 10.1109/TIT.2017.2730864

M3 - Article

AN - SCOPUS:85028930570

VL - 63

SP - 7793

EP - 7803

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 12

M1 - 7993073

ER -