A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1s on each row are consecutive. Algorithmic issues of the C1P are central in computational molecular biology,in particular for physical mapping and ancestral genome reconstruction. In 1972,Tucker gave a characterization of matrices that have the C1P by a set of forbidden submatrices,and a substantial amount of research has been devoted to the problem of efficiently finding such a minimum size forbidden submatrix. This paper presents a new O(Δ^3 m^2(mΔ+n 3)) time algorithm for this particular task for a m×n binary matrix with at most Δ 1-entries per row,thereby improving the O(Δ^3 m^2(mn+n 3)) time algorithm of M. Dom,J. Guo and R. Niedermeier [Approximation and fixed-parameter algorithms for consecutive ones submatrix problems,Journal of Computer and System Sciences,76(3–4): 204–221,2010]. Moreover,this approach can be used—with a much heavier machinery—to address harder problems related to Minimal Conflicting Set [G. Blin,R. Rizzi,and S. Vialette. A Polynomial-Time Algorithm for Finding Minimal Conflicting Sets,Proc. 6th International Computer Science Symposium in Russia (CSR),[2011]].
A Faster Algorithm for Finding Minimum Tucker Submatrices
RIZZI, ROMEO;
2012-01-01
Abstract
A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1s on each row are consecutive. Algorithmic issues of the C1P are central in computational molecular biology,in particular for physical mapping and ancestral genome reconstruction. In 1972,Tucker gave a characterization of matrices that have the C1P by a set of forbidden submatrices,and a substantial amount of research has been devoted to the problem of efficiently finding such a minimum size forbidden submatrix. This paper presents a new O(Δ^3 m^2(mΔ+n 3)) time algorithm for this particular task for a m×n binary matrix with at most Δ 1-entries per row,thereby improving the O(Δ^3 m^2(mn+n 3)) time algorithm of M. Dom,J. Guo and R. Niedermeier [Approximation and fixed-parameter algorithms for consecutive ones submatrix problems,Journal of Computer and System Sciences,76(3–4): 204–221,2010]. Moreover,this approach can be used—with a much heavier machinery—to address harder problems related to Minimal Conflicting Set [G. Blin,R. Rizzi,and S. Vialette. A Polynomial-Time Algorithm for Finding Minimal Conflicting Sets,Proc. 6th International Computer Science Symposium in Russia (CSR),[2011]].
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