Median Approximations for Genomes Modeled as Matrices

Zanetti, João Paulo Pereira; Biller, Priscila; Meidânis, João

doi:10.1007/s11538-016-0162-4

Cited by 14 publications

(56 citation statements)

References 23 publications

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“…Proof By the calculations in [23] recapitulated in Corollary 3, the median M A achieves the lower bound β. Furthermore, by the calculations in the analysis of the system (8), we see that there exists a unique matrix M that simultaneously satisfies the necessary conditions for attaining the lower bound β.…”

Section: Proof Of Uniqueness For the Special Casementioning

confidence: 87%

“…The second invariant, which was already identified in [23] as playing an important role in the median problem, is the dimension of the "triple agreement" subspace V 1 , i.e. :…”

Section: The Second Invariantmentioning

confidence: 99%

“…Pereira Zanetti et al [23] showed how to decompose the space R n into a direct sum of five subspaces, and expressed their median candidates using the projection operators of these subspaces. We now show how to express the dimensions of these subspaces using the invariants α, β and δ, and deduce a sufficient condition for their median candidate to be a true median.…”

Section: Subspace Dimensions In Terms Of Invariantsmentioning

confidence: 99%

“…In addition, we can now combine this with the expression in [23] for the score of their median candidates M A , M B , M C (obtained by taking the common value on each vector in an "agreement subspace", V 1 through V 4 , and the value corresponding to multiplication by A, B or C, respectively, on the remaining subspace…”

Section: Corollary 3 δ(A B C) ≥ 0 For Permutation Matrices a B Cmentioning

confidence: 99%

“…To clarify the procedure, we now illustrate the running of our algorithm on ρ = (12) (34), σ = (13) (24), τ = (14) (23), with n = 4. First, we note that the product of any two of these equals the third, and they are each their own inverse.…”

Section: An Example Of the Algorithmmentioning

confidence: 99%

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On the Rank-Distance Median of 3 Permutations

Chindelevitch

Meidânis

2017

Comparative Genomics

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Background: Recently, Pereira Zanetti, Biller and Meidanis have proposed a new definition of a rearrangement distance between genomes. In this formulation, each genome is represented as a matrix, and the distance d is the rank distance between these matrices. Although defined in terms of matrices, the rank distance is equal to the minimum total weight of a series of weighted operations that leads from one genome to the other, including inversions, translocations, transpositions, and others. The computational complexity of the median-of-three problem according to this distance is currently unknown. The genome matrices are a special kind of permutation matrices, which we study in this paper. In their paper, the authors provide an O n 3 algorithm for determining three candidate medians, prove the tight approximation ratio 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. Results: In this paper, we extend their results and provide the following: • Three invariants characterizing the problem of finding the median of 3 matrices • A sufficient condition for uniqueness of medians that can be checked in O(n) • A faster, O n 2 algorithm for determining the median under this condition • A new heuristic algorithm for this problem based on compressed sensing • A O n 4 algorithm that exactly solves the problem when the inputs are orthogonal matrices, a class that includes both permutations and genomes as special cases. Conclusions: Our work provides the first proof that, with respect to the rank distance, the problem of finding the median of 3 genomes, as well as the median of 3 permutations, is exactly solvable in polynomial time, a result which should be contrasted with its NP-hardness for the DCJ (double cut-and-join) distance and most other families of genome rearrangement operations. This result, backed by our experimental tests, indicates that the rank distance is a viable alternative to the DCJ distance widely used in genome comparisons.

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Section: Proof Of Uniqueness For the Special Casementioning

confidence: 87%

“…The second invariant, which was already identified in [23] as playing an important role in the median problem, is the dimension of the "triple agreement" subspace V 1 , i.e. :…”

Section: The Second Invariantmentioning

confidence: 99%

Section: Subspace Dimensions In Terms Of Invariantsmentioning

confidence: 99%

Section: Corollary 3 δ(A B C) ≥ 0 For Permutation Matrices a B Cmentioning

confidence: 99%

Section: An Example Of the Algorithmmentioning

confidence: 99%

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On the Rank-Distance Median of 3 Permutations

Chindelevitch

Meidânis

2017

Comparative Genomics

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A Cubic Algorithm for the Generalized Rank Median of Three Genomes

Chindelevitch

Meidânis

2018

Comparative Genomics

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The area of genome rearrangements has given rise to a number of interesting biological, mathematical and algorithmic problems. Among these, one of the most intractable ones has been that of finding the median of three genomes, a special case of the ancestral reconstruction problem. In this work we reexamine our recently proposed way of measuring genome rearrangement distance, namely, the rank distance between the matrix representations of the corresponding genomes, and show that the median of three genomes can be computed exactly in polynomial time O(n ω), where ω ≤ 3, with respect to this distance, when the median is allowed to be an arbitrary orthogonal matrix. We define the five fundamental subspaces depending on three input genomes, and use their properties to show that a particular action on each of these subspaces produces a median. In the process we introduce the notion of M-stable subspaces. We also show that the median found by our algorithm is always orthogonal, symmetric, and conserves any adjacencies or telomeres present in at least 2 out of 3 input genomes. We test our method on both simulated and real data. We find that the majority of the realistic inputs result in genomic outputs, and for those that do not, our two heuristics perform well in terms of reconstructing a genomic matrix attaining a score close to the lower bound, while running in a reasonable amount of time. We conclude that the rank distance is not only theoretically intriguing, but also practically useful for median-finding, and potentially ancestral genome reconstruction.

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