Matrix Tightness: A Linear-Algebraic Framework for Sorting by Transpositions

  • Tzvika Hartman
  • Elad Verbin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4209)


We study the problems of sorting signed permutations by reversals (SBR) and sorting unsigned permutations by transpositions (SBT), which are central problems in computational molecular biology. While a polynomial-time solution for SBR is known, the computational complexity of SBT has been open for more than a decade and is considered a major open problem.

In the first efficient solution of SBR, Hannenhalli and Pevzner [HP99] used a graph-theoretic model for representing permutations, called the interleaving graph. This model was crucial to their solution. Here, we define a new model for SBT, which is analogous to the interleaving graph. Our model has some desirable properties that were lacking in earlier models for SBT. These properties make it extremely useful for studying SBT.

Using this model, we give a linear-algebraic framework in which SBT can be studied. Specifically, for matrices over any algebraic ring, we define a class of matrices called tight matrices. We show that an efficient algorithm which recognizes tight matrices over a certain ring, \(\mathbb{M}\), implies an efficient algorithm that solves SBT on an important class of permutations, called simple permutations. Such an algorithm is likely to lead to an efficient algorithm for SBT that works on all permutations.

The problem of recognizing tight matrices is also a generalization of SBR and of a large class of other “sorting by rearrangements” problems, and seems interesting in its own right as. We give an efficient algorithm for recognizing tight symmetric matrices over any field of characteristic 2. We leave as an open problem to find an efficient algorithm for recognizing tight matrices over the ring \(\mathbb{M}\).


Genome Rearrangement Identity Permutation Signed Permutation Black Edge Breakpoint Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Tzvika Hartman
    • 1
  • Elad Verbin
    • 2
  1. 1.Dept. of Computer ScienceBar-Ilan UniversityRamat-GanIsrael
  2. 2.School of Computer ScienceTel-Aviv UniversityTel-AvivIsrael

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