Output-Sensitive Algorithms for Finding the Nested Common Intervals of Two General Sequences

The focus of this paper is the problem of finding all nested common intervals of two general sequences. Depending on the treatment one wants to apply to duplicate genes, Blin et al. introduced three models to define nested common intervals of two sequences: the uniqueness, the free-inclusion, and the bijection models. We consider all the three models. For the uniqueness and the bijection models, we give O(n + N<sub>out</sub>)-time algorithms, where N<sub>out</sub> denotes the size of the output. For the free-inclusion model, we give an O(n<sup>1+ε</sup> + N<sub>out</sub>)-time algorithm, where ε >; 0 is an arbitrarily small constant. We also present an upper bound on the size of the output for each model. For the uniqueness and the free-inclusion models, we show that N<sub>out</sub> = O(n<sup>2</sup>). Let C = Σ<sub>gϵΓ</sub> o<sub>1</sub>(g)o<sub>2</sub>(5), where Γ is the set of distinct genes, and o<sub>1</sub>(g) and o<sub>2</sub>(g) are, respectively, the numbers of copies of g in the two given sequences. For the bijection model, we show that N<sub>out</sub> = O(Cn). In this paper, we also study the problem of finding all approximate nested common intervals of two sequences on the bijection model. An O(δn + N<sub>out</sub>)-time algorithm is presented, where δ denotes the maximum number of allowed gaps. In addition, we show that for this problem N<sub>out</sub> is O(δn<sup>3</sup>).

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