skip to main content
research-article

An Audit Tool for Genome Rearrangement Algorithms

Published: 07 January 2015 Publication History

Abstract

We consider the combinatorial problem of sorting a permutation using a minimum number of rearrangement events, which finds application in the estimation of evolutionary distance between species. Many variants of this problem, which we generically refer to as the rearrangement sorting problem, have been tackled in the literature, and for most of them, the best known algorithms are approximations or heuristics. In this article, we present a tool, called GRAAu, to aid in the evaluation of the results produced by these algorithms. To illustrate its application, we use GRAAu to evaluate the results of four approximation algorithms regarding two variants of the rearrangement sorting problem: the problem of sorting by prefix reversals and the problem of sorting by prefix transpositions. As a result, we show that the approximation ratios of three algorithms are tight and conjecture that the approximation ratio of the remaining one is also tight.

References

[1]
D. Bader, B. Moret, and M. Yan. 2001. A linear-time algorithm for computing inversion distance between signed permutations with an experimental study. Journal of Computational Biology 8, 5, 483--491.
[2]
V. Bafna and P. A. Pevzner. 1996. Genome rearrangements and sorting by reversals. SIAM Journal on Computing 25, 2, 272--289.
[3]
V. Bafna and P. A. Pevzner. 1998. Sorting by transpositions. SIAM Journal on Discrete Mathematics 11, 2, 224--240.
[4]
M. Benoît-Gagné and S. Hamel. 2007. A new and faster method of sorting by transpositions. In Combinatorial Pattern Matching. Lecture Notes in Computer Science, Vol. 4580. Springer, 131--141.
[5]
P. Berman, S. Hannenhalli, and M. Karpinski. 2002. 1.375-approximation algorithm for sorting by reversals. In Algorithms—ESA 2002. Lecture Notes in Computer Science, Vol. 2461. Springer, 200--210.
[6]
L. Bulteau, G. Fertin, and I. Rusu. 2012a. Pancake flipping is hard. In Mathematical Foundations of Computer Science 2012. Lecture Notes in Computer Science, Vol. 7464. Springer, 247--258.
[7]
L. Bulteau, G. Fertin, and I. Rusu. 2012b. Sorting by transpositions is difficult. SIAM Journal on Discrete Mathematics 26, 3, 1148--1180.
[8]
A. Caprara. 1999. Sorting permutations by reversals and Eulerian cycle decompositions. SIAM Journal on Discrete Mathematics 12, 1, 91--110.
[9]
B. Chitturi, W. Fahle, Z. Meng, L. Morales, C. Shields, I. H. Sudborough, and W. Voit. 2009. An (18/11)n upper bound for sorting by prefix reversals. Theoretical Computer Science 410, 36, 3372--3390.
[10]
B. Chitturi and I. H. Sudborough. 2012. Bounding prefix transposition distance for strings and permutations. Theoretical Computer Science 421, 15--24.
[11]
D. A. Christie. 1998. Genome Rearrangement Problems. Ph.D. Dissertation. University of Glasgow, Glasgow, Scotland.
[12]
D. S. Cohen and M. Blum. 1995. On the problem of sorting burnt pancakes. Discrete Applied Mathematics 61, 2, 105--120.
[13]
U. Dias and Z. Dias. 2010. An improved 1.375-approximation algorithm for the transposition distance problem. In Proceedings of the 1st ACM International Conference on Bioinformatics and Computational Biology (BCB’10). ACM, New York, NY, 334--337.
[14]
U. Dias and Z. Dias. 2013. Heuristics for the transposition distance problem. Journal of Bioinformatics and Computational Biology 11, 1350013.
[15]
Z. Dias and J. Meidanis. 2002. Sorting by prefix transpositions. In String Processing and Information Retrieval. Lecture Notes in Computer Science, Vol. 2476. Springer, 65--76.
[16]
H. Dweighter. 1975. Elementary problems and solutions, problem E2569. American Mathematical Monthly 82, 1010.
[17]
I. Elias and T. Hartman. 2006. A 1.375-approximation algorithm for sorting by transpositions. IEEE/ACM Transactions on Computational Biology and Bioinformatics 3, 4, 369--379.
[18]
H. Eriksson, K. Eriksson, J. Karlander, L. Svensson, and J. Wastlund. 2001. Sorting a bridge hand. Discrete Mathematics 241, 1--3, 289--300.
[19]
G. Fertin, A. Labarre, I. Rusu, E. Tannier, and S. Vialette. 2009. Combinatorics of Genome Rearrangements. MIT Press, Cambridge, MA.
[20]
J. Fischer and S. W. Ginzinger. 2005. A 2-approximation algorithm for sorting by prefix reversals. In Algorithms—ESA 2005. Lecture Notes in Computer Science, Vol. 3669. Springer, 415--425.
[21]
S. Grusea and A. Labarre. 2013. The distribution of cycles in breakpoint graphs of signed permutations. Discrete Applied Mathematics 161, 10--11, 1448--1466.
[22]
Q. Gu, S. Peng, and I. H. Sudborough. 1999. A 2-approximation algorithm for genome rearrangements by reversals and transpositions. Theoritical Computer Science 210, 2, 327--339.
[23]
S. Hannenhalli and P. A. Pevzner. 1995. Transforming men into mice (polynomial algorithm for genomic distance problem). In Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS’95). IEEE, Los Alamitos, CA, 581--592.
[24]
T. Hartman and R. Sharan. 2005. A 1.5-approximation algorithm for sorting by transpositions and transreversals. Journal of Computer and System Sciences 70, 3, 300--320.
[25]
M. H. Heydari and I. H. Sudborough. 1997. On the diameter of the pancake network. Journal of Algorithms 25, 1, 67--94.
[26]
J. D. Kececioglu and D. Sankoff. 1995. Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement. Algorithmica 13, 1--2, 80--110.
[27]
A. Labarre. 2008. Combinatorial Aspects of Genome Rearrangements and Haplotype Networks. Ph.D. Dissertation. Université Libre de Bruxelles, Brussels, Belgium.
[28]
A. Labarre. 2013. Lower bounding edit distances between permutations. SIAM Journal on Discrete Mathematics 27, 1410--1428.
[29]
L. Lu and Y. Yang. 2010. A lower bound on the transposition diameter. SIAM Journal on Discrete Mathematics 24, 4, 1242--1249.
[30]
J. Meidanis, M. M. Walter, and Z. Dias. 2002. A lower bound on the reversal and transposition diameter. Journal of Computational Biology 9, 5, 743--745.
[31]
J. I. Munro, R. Raman, V. Raman, and S. S. Rao. 2012. Succinct representations of permutations and functions. Theoretical Computer Science 22, 74--88.
[32]
W. Myrvold and F. Ruskey. 2001. Ranking and unranking permutations in linear time. Information Processing Letters 79, 6, 281--284.
[33]
A. Rahman, S. Shatabda, and M. Hasan. 2008. An approximation algorithm for sorting by reversals and transpositions. Journal of Discrete Algorithms 6, 3, 449--457.
[34]
M. Sharmin, R. Yeasmin, M. Hasan, A. Rahman, and M. S. Rahman. 2010. Pancake flipping with two spatulas. Electronic Notes in Discrete Mathematics 36, 231--238.
[35]
E. Tannier, A. Bergeron, and M. F. Sagot. 2007. Advances on sorting by reversals. Discrete Applied Mathematics 155, 6--7, 881--888.
[36]
G. Tesler. 2002. GRIMM: Genome rearrangements Web server. Bioinformatics 18, 492--493.
[37]
J. P. C. Vergara. 1998. Sorting by Bounded Permutations. Ph.D. Dissertation. Virginia Polytechnic Institute and State University, Blacksburg, VA.
[38]
M. E. M. T. Walter, Z. Dias, and J. Meidanis. 1998. Reversal and transposition distance of linear chromosomes. In Proceedings of the 5th International Symposium on String Processing and Information Retrieval (SPIRE’98). IEEE, Los Alamitos, CA, 96--102.
[39]
M. E. M. T. Walter, Z. Dias, and J. Meidanis. 2000. A new approach for approximating the transposition distance. In Proceedings of the 7th International Symposium on String Processing Information Retrieval (SPIRE’00). IEEE, Los Alamitos, CA, 199--208.
[40]
M. E. M. T. Walter, M. C. Sobrinho, E. T. G. Oliveira, L. S. Soares, A. G. Oliveira, T. E. S. Martins, and T. M. Fonseca. 2005. Improving the algorithm of Bafna and Pevzner for the problem of sorting by transpositions: A practical approach. Journal of Discrete Algorithms 3, 2--4, 342--361.
[41]
G. A. Watterson, W. J. Ewens, T. E. Hall, and A. Morgan. 1982. The chromosome inversion problem. Journal of Theoretical Biology 99, 1, 1--7.
[42]
A. Williams. 2010. O(1)-time unsorting by prefix-reversals in a boustrophedon linked list. In Fun With Algorithms. Lecture Notes in Computer Science, Vol. 6099. Springer, 368--379.

Cited By

View all
  • (2022)A hapless mathematical contribution to biologyHistory and Philosophy of the Life Sciences10.1007/s40656-022-00514-x44:3Online publication date: 2-Aug-2022

Index Terms

  1. An Audit Tool for Genome Rearrangement Algorithms

    Recommendations

    Comments

    Information & Contributors

    Information

    Published In

    cover image ACM Journal of Experimental Algorithmics
    ACM Journal of Experimental Algorithmics  Volume 19, Issue
    2014
    402 pages
    ISSN:1084-6654
    EISSN:1084-6654
    DOI:10.1145/2627368
    Issue’s Table of Contents
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

    Publisher

    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 07 January 2015
    Accepted: 01 August 2014
    Revised: 01 June 2014
    Received: 01 January 2014
    Published in JEA Volume 19

    Author Tags

    1. Genome rearrangements
    2. ranking and unranking permutations
    3. sorting by prefix reversals
    4. sorting by prefix transpositions

    Qualifiers

    • Research-article
    • Research
    • Refereed

    Funding Sources

    Contributors

    Other Metrics

    Bibliometrics & Citations

    Bibliometrics

    Article Metrics

    • Downloads (Last 12 months)3
    • Downloads (Last 6 weeks)1
    Reflects downloads up to 05 Mar 2025

    Other Metrics

    Citations

    Cited By

    View all
    • (2022)A hapless mathematical contribution to biologyHistory and Philosophy of the Life Sciences10.1007/s40656-022-00514-x44:3Online publication date: 2-Aug-2022

    View Options

    Login options

    Full Access

    View options

    PDF

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    Figures

    Tables

    Media

    Share

    Share

    Share this Publication link

    Share on social media