Skip to main content
Springer Nature Link
Log in

Approximating Maximum Agreement Forest on Multiple Binary Trees

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Given a collection of phylogenetic trees on the same leaf label-set, the Maximum Agreement Forest problem (Maf) asks for a largest common subforest of these trees. The Maf problem on two binary phylogenetic trees has been studied extensively. In this paper, we are focused on the Maf problem on multiple (i.e., two or more) binary phylogenetic trees and present two polynomial-time approximation algorithms, one for the Maf problem on multiple rooted trees, and the other for the Maf problem on multiple unrooted trees. The ratio of our algorithm for the Maf problem on multiple rooted trees is 3, which is an improvement over the previous best ratio 8 for the problem. Our approximation algorithm of ratio 4 for the Maf problem on multiple unrooted trees is the first constant ratio approximation algorithm for the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Some definitions in the study of maximum agreement forests have been somewhat confusing and misleading. If size denotes the number of edges in a forest, then the size of a forest is equal to the number of vertices minus its order. Thus, when the number of vertices is fixed, a forest of large size implies a small order. The terminology of “maximum agreement forest” means an agreement forest of the maximum size. However, as it has been studied in the literature, the maximum agreement forest problem is indeed a minimization problem, with the objective of minimizing the order of an agreement forest.

  2. The indices used here are slightly different from that used in the algorithm Apx-MAF: in the algorithm Apx-MAF, step 2 operates on \(F_1\) and \(F_{i+1}\) for \(1 \le i \le m-1\), which simplifies the notations in the proof of Theorem 1; while in this section, we let step 2 of the algorithm operate on \(F_1\) and \(F_i\) for \(2 \le i \le m\) to simplify the descriptions of our meta-steps.

  3. During the preparation of the final version of this manuscript, the authors were informed by an anonymous referee that Mukhopadhyay and Bhabak had announced an \(O(kn^5)\)-time approximation algorithm of ratio 3 for the Maf problem on k rooted binary phylogenetic trees [16].

References

  1. Aho, A., Hopcroft, J., Ullman, J.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)

    MATH  Google Scholar 

  2. Allen, B., Steel, M.: Subtree transfer operations and their induced metrics on evolutionary trees. Ann. Comb. 5(1), 1–15 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beiko, R.G., Hamilton, N.: Phylogenetic identification of lateral genetic transfer events. BMC Evolut. Biol. 6(1), 15 (2006)

    Article  Google Scholar 

  4. Bonet, M., John, K.S., Mahindru, R., Amenta, N.: Approximating subtree distances between phylogenies. J. Comput. Biol. 13(8), 1419–1434 (2006)

    Article  MathSciNet  Google Scholar 

  5. Bordewich, M., McCartin, C., Semple, C.: A 3-approximation algorithm for the subtree distance between phylogenies. J. Discrete Algorithms 6(3), 458–471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bordewich, M., Semple, C.: On the computational complexity of the rooted subtree prune and regraft distance. Ann. Comb. 8(4), 409–423 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buneman, P.: The recovery of trees from measures of dissimilarity. In: Kendall, D., Tauta, P. (eds.) Mathematics in the Archaeological and Historical Sciences, pp. 387–395. Edinburgh University Press, Edinburgh (1971)

    Google Scholar 

  8. Chataigner, F.: Approximating the maximum agreement forest on \(k\) trees. Inf. Process. Lett. 93, 239–244 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, J., Fan, J.-H., Sze, S.-H.: Parameterized and approximation algorithms for maximum agreement forest in multifurcating trees. Theor. Comput. Sci. 562, 496–512 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, Z., Wang, L.: Algorithms for reticulate networks of multiple phylogenetic trees. IEEE/ACM Trans. Comput. Biol. Bioinform. 9, 372–384 (2012)

    Article  Google Scholar 

  11. Diestel, R.: Graph Theory, 4th edition. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2010)

  12. Dudas, G., Bedford, T., Lycett, S., Rambaut, A.: Reassortment between influenza B lineages and the emergence of a coadapted PB1-PB2-HA gene complex. Mol. Biol. Evol. 32(1), 162–172 (2014). (supplemental information)

    Article  Google Scholar 

  13. Hallett, M., McCartin, C.: A faster FPT algorithm for the maximum agreement forest problem. Theory Comput. Syst. 41(3), 539–550 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hein, J., Jiang, T., Wang, L., Zhang, K.: On the complexity of comparing evolutionary trees. Discrete Appl. Math. 71, 153–169 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, M., Tromp, J., Zhang, L.: On the nearest neighbour interchange distance between evolutionary trees. J. Theor. Biol. 182(4), 463–467 (1996)

    Article  Google Scholar 

  16. Mukhopadhyay, A., Bhabak, P.: A 3-factor approximation algorithm for a minimum acyclic agreement forest on \(k\) rooted, binary phylogenetic trees. CoRR abs/1407.7125 (2014)

  17. Robinson, D., Foulds, L.: Comparison of phylogenetic trees. Math. Biosci. 53(1–2), 131–147 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rodrigues, M., Sagot, M., Wakabayashi, Y.: Some approximation results for the maximum agreement forest problem. In: Proceedigs of the RANDOM-APPROX 2001, Lecture Notes in Computer Science, vol. 2129, pp. 159–169 (2001)

  19. Rodrigues, E., Sagot, M., Wakabayashi, Y.: The maximum agreement forest problem: approximation algorithms and computational experiments. Theor. Comput. Sci. 374, 91–110 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shi, F., Wang, J., Chen, J., Feng, Q., Guo, J.: Algorithms for parameterized maximum agreement forest problem on multiple trees. Theor. Comput. Sci. 554, 207–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Shi, F., Feng, Q., You, J., Wang, J.: Improved approximation algorithm for maximum agreement forest of two rooted binary phylogenetic trees. J. Comb. Optim. (2015a). doi:10.1007/s10878-015-9921-7

    MathSciNet  MATH  Google Scholar 

  22. Shi, F., Wang, J., Yang, Y., Feng, Q., Li, W., Chen, J.: A fixed-parameter algorithm for the maximum agreement forset problem on multifurcating trees. Sci. China Inf. Sci. (2015b). doi:10.1007/s11432-015-5355-1

    Google Scholar 

  23. Swofford, D., Olsen, G., Waddell, P., Hillis, D.: Phylogenetic inference. In: Hillis, D., Moritz, D., Mable, B. (eds.) Molecular Systematics, 2nd edn, pp. 407–514. Sinauer Associates, Sunderiand (1996)

    Google Scholar 

  24. Whidden, C., Zeh, N.: A unifying view on approximation and FPT of agreement forests. In: Proceedings of the WABI 2009, Lecture Notes in Computer Science, vol. 5724, pp. 390–401 (2009)

  25. Whidden, C., Beiko, R.G., Zeh, N.: Fixed-parameter algorithms for maximum agreement forests. SIAM J. Comput. 42(4), 1431–1466 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Whidden, C., Zeh, N., Beiko, R.G.: Supertrees based on the subtree prune-and-regraft distance. Syst. Biol. 63(4), 566–581 (2014)

    Article  Google Scholar 

  27. Whidden, C., Matsen IV, F.A.: Quantifying MCMC exploration of phylogenetic tree space. Syst. Biol. 64(3), 472 (2015)

    Article  Google Scholar 

  28. Wu, Y.: Close lower and upper bounds for the minimum reticulate network of multiple phylogenetic trees. Bioinformatics 26(12), i140–i148 (2010)

    Article  Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous referees, whose comments and suggestions have greatly improved the presentation of this paper. In particular, a referee provided further pointers to applications of algorithms for maximum agreement forests on multiple trees, and another referee updated us of the status of approximation algorithms for maximum agreement forests on multiple rooted trees.

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Jianer Chen, Feng Shi & Jianxin Wang

  2. Department of Computer Science, Texas A&M University, College Stataion, TX, 77843-3112, USA

    Jianer Chen

Authors
  1. Jianer Chen
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Feng Shi
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Jianxin Wang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jianxin Wang.

Additional information

A preliminary version of this work was reported in the Proceedings of the 20th International Computing and Combinatorics Conference, Lecture Notes in Computer Science, vol. 8591, pp. 381–392, 2014. This work is supported by the National Natural Science Foundation of China under Grants (61232001, 61472449, 61370172, 61420106009), the Major Science and Technology Research Program for Strategic Emerging Industry of Hunan (Grant No. 2012GK4054), and the Research Fund for the Doctoral Program of Higher Education of China (NO. 20130162130001).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, J., Shi, F. & Wang, J. Approximating Maximum Agreement Forest on Multiple Binary Trees. Algorithmica 76, 867–889 (2016). https://doi.org/10.1007/s00453-015-0087-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-015-0087-6

Keywords