Sorting a Permutation by Best Short Swaps

A permutation is happy, if it can be transformed into the identity permutation using as many short swaps as one third times the number of inversions in the permutation. The complexity of the decision version of sorting a permutation by short swaps, is still open. We present an O(n) time algorithm to decide whether it is true for a permutation to be happy, where n is the number of elements in the permutation. If a permutation is happy, we give an $$O(n^2)$$ time algorithm to find a sequence of as many short swaps as one third times the number of its inversions, to transform it into the identity permutation. A permutation is lucky, if it can be transformed into the identity permutation using as many short swaps as one fourth times the length sum of the permutation’s element vectors. We present an O(n) time algorithm to decide whether it is true for a permutation to be lucky, where n is the number of elements in the permutation. If a permutation is lucky, we give an $$O(n^2)$$ time algorithm to find a sequence of as many short swaps as one fourth times the length sum of its element vectors to transform it into the identity permutation. This improves upon the $$O(n^2)$$ time algorithm proposed by Heath and Vergara to decide whether a permutation is lucky. We show that there are at least $$2^{\lceil \frac{n}{2}\rceil -2}$$ happy permutations as well as $$2^{n-4}$$ lucky permutations of n elements.

[1]  Ivan Hal Sudborough,et al.  Improved upper bound for sorting by short swaps , 2004, 7th International Symposium on Parallel Architectures, Algorithms and Networks, 2004. Proceedings..

[2]  T. Dobzhansky,et al.  Inversions in the Chromosomes of Drosophila Pseudoobscura. , 1938, Genetics.

[3]  D. Sankoff,et al.  Gene order comparisons for phylogenetic inference: evolution of the mitochondrial genome. , 1992, Proceedings of the National Academy of Sciences of the United States of America.

[4]  Lenwood S. Heath,et al.  Sorting by Short Swaps , 2003, J. Comput. Biol..

[5]  Binhai Zhu,et al.  A (1+ε)-approximation algorithm for sorting by short block-moves , 2012, Theor. Comput. Sci..

[6]  S Heath Lenwood,et al.  Sorting by Short Block-Moves , 1998 .

[7]  Marek Karpinski,et al.  1.375-Approximation Algorithm for Sorting by Reversals , 2002, ESA.

[8]  Haim Kaplan,et al.  A Faster and Simpler Algorithm for Sorting Signed Permutations by Reversals , 1999, SIAM J. Comput..

[9]  David Sankoff,et al.  Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement , 1995, Algorithmica.

[10]  Alfredo De Santis,et al.  A simple algorithm for the constrained sequence problems , 2004, Information Processing Letters.

[11]  Zanoni Dias,et al.  Approximation algorithms for sorting by signed short reversals , 2014, BCB.

[12]  Pavel A. Pevzner,et al.  Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals , 1999, J. ACM.

[13]  P. Pevzner,et al.  Genome rearrangements in mammalian evolution: lessons from human and mouse genomes. , 2003, Genome research.

[14]  Jeffrey D. Palmer,et al.  Plant mitochondrial DNA evolved rapidly in structure, but slowly in sequence , 2005, Journal of Molecular Evolution.

[15]  Daming Zhu,et al.  A 14/11-approximation algorithm for sorting by short block-moves , 2010, Science China Information Sciences.

[16]  W. Ewens,et al.  The chromosome inversion problem , 1982 .

[17]  Mark Jerrum,et al.  The Complexity of Finding Minimum-Length Generator Sequences , 1985, Theor. Comput. Sci..

[18]  Jingjing Ma,et al.  Can a permutation be sorted by best short swaps? , 2018, CPM.

[19]  T. F. Rinke de Wit,et al.  Comparison of Perceived and Technical Healthcare Quality in Primary Health Facilities: Implications for a Sustainable National Health Insurance Scheme in Ghana , 2015, PloS one.

[20]  Zanoni Dias,et al.  Sorting signed permutations by short operations , 2015, Algorithms for Molecular Biology.

[21]  P. Pevzner,et al.  Genome-scale evolution: reconstructing gene orders in the ancestral species. , 2002, Genome research.

[22]  Vineet Bafna,et al.  Genome Rearrangements and Sorting by Reversals , 1996, SIAM J. Comput..

[23]  P. V. Vara Prasad,et al.  Evaluation of Wheat Chromosome Translocation Lines for High Temperature Stress Tolerance at Grain Filling Stage , 2015, PloS one.

[24]  Alberto Caprara,et al.  Sorting Permutations by Reversals and Eulerian Cycle Decompositions , 1999, SIAM J. Discret. Math..

[25]  Daming Zhu,et al.  An 5/4-Approximation Algorithm for Sorting Permutations by Short Block Moves , 2014, ISAAC.