Abstract
Recently, Pereira Zanetti et al. [9] have proposed a new definition of a distance for which the computational complexity of the median problem is currently unknown. In this formulation, each genome is represented as a permutation on n elements that is the product of disjoint cycles of length 1 (telomeres) and length 2 (adjacencies). The permutations are converted into their matrix representation, and the rank distance d is used to define the median.
In their paper, the authors provide an \(O(n^3)\) algorithm for determining three candidate medians, prove the tight approximation ratio \(\frac{4}{3}\), and provide a sufficient condition for their candidates to be true medians. They also conduct some experiments that suggest that their method is accurate on simulated and real data.
In this paper, we extend their results and provide the following:
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2 invariants characterizing the problem of finding the median of 3 matrices
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a strengthening of one of the results in the work of Pereira Zanetti et al.
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a sufficient condition for optimality that can be checked in O(n) time
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a faster, \(O(n^2)\) algorithm for determining the median under this condition
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a new heuristic algorithm for this problem based on compressed sensing.
J. Meidanis is on Leave from University of Campinas, Campinas, Brazil.
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Acknowledgments
The authors would like to thank Cedric Chauve and Pedro Feijão for helpful discussions. LC would like to acknowledge financial support from NSERC, CIHR, Genome Canada and the Sloan Foundation. JM would like to acknowledge financial support from NSERC.
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Chindelevitch, L., Meidanis, J. (2017). On the Rank-Distance Median of 3 Permutations. In: Meidanis, J., Nakhleh, L. (eds) Comparative Genomics. RECOMB-CG 2017. Lecture Notes in Computer Science(), vol 10562. Springer, Cham. https://doi.org/10.1007/978-3-319-67979-2_14
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