We present efficient fixed-parameter and approximation algorithms for the NP-hard problem of computing a maximum agreement forest (MAF) of a pair of multifurcating (nonbinary) rooted trees. Multifurcating trees arise naturally as a result of statistical uncertainty in current tree construction methods. The size of an MAF corresponds to the subtree prune-and-regraft distance of the two trees and is intimately connected to their hybridization number. These distance measures are essential tools for understanding reticulate evolution, such as lateral gene transfer, recombination, and hybridization. Our algorithms nearly match the running times of the currently best algorithms for the binary case. This is achieved using a combination of efficient branching rules (similar to but more complex than in the binary case) and a novel edge protection scheme that further reduces the size of the search space the algorithms need to explore.
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This is similar to the generalization of the hybridization number used by Linz and Semple .
We excluded this case from the statement of the lemma, in order to keep the cases covered by the different lemmas disjoint, but the lemma also holds for \(s = 1\). A similar comment applies to Lemma 12.
This is a fairly loose bound on \(I(k,t)\), but it is easy to manipulate.
Albrecht, B., Scornavacca, C., Cenci, A., Huson, D.H.: Fast computation of minimum hybridization networks. Bioinformatics 28(2), 191–197 (2012)
Allen, B.L., Steel, M.: Subtree transfer operations and their induced metrics on evolutionary trees. Ann. Comb. 5(1), 1–15 (2001)
Baroni, M., Grünewald, S., Moulton, V., Semple, C.: Bounding the number of hybridisation events for a consistent evolutionary history. J. Math. Biol. 51(2), 171–182 (2005)
Beiko, R.G., Hamilton, N.: Phylogenetic identification of lateral genetic transfer events. BMC Evol. Biol. 6(1), 15 (2006)
Bonet, M.L., St. John, K., Mahindru, R., Amenta, N.: Approximating subtree distances between phylogenies. J. Comput. Biol. 13(8), 1419–1434 (2006)
Bordewich, M., McCartin, C., Semple, C.: A 3-approximation algorithm for the subtree distance between phylogenies. J. Discret. Algorithms 6(3), 458–471 (2008)
Bordewich, M., Semple, C.: On the computational complexity of the rooted subtree prune and regraft distance. Ann. Combin. 8(4), 409–423 (2005)
Bordewich, M., Semple, C.: Computing the minimum number of hybridization events for a consistent evolutionary history. Discret. Appl. Math. 155(8), 914–928 (2007)
Chen, Z.-Z., Wang, L.: Hybridnet: a tool for constructing hybridization networks. Bioinformatics 26(22), 2912–2913 (2010)
Chen, Z.-Z., Wang, L.: Algorithms for reticulate networks of multiple phylogenetic trees. IEEE/ACM Trans. Comput. Biol. Bioinform. 9, 372–384 (2012)
Chen, Z.-Z., Wang, L.: Faster exact computation of rSPR distance. In: Frontiers in Algorithmics and Algorithmic Aspects in Information and Management, pp. 36–47. Springer (2013)
Chen, Z.-Z., Wang, L.: An ultrafast tool for minimum reticulate networks. J. Comput. Biol. 20(1), 38–41 (2013)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill, New York (2001)
Hein, J., Jiang, T., Wang, L., Zhang, K.: On the complexity of comparing evolutionary trees. Discrete Appl. Math. 71(1–3), 153–169 (1996)
Hickey, G., Dehne, F., Rau-Chaplin, A., Blouin, C.: SPR distance computation for unrooted trees. Evolut. Bioinform. 4, 17–27 (2008)
Hillis, D.M., Moritz, C., Mable, B.K. (eds.): Molecular Systematics. Sinauer Associates, Sunderland (1996)
van Iersel, L., Kelk, S., Lekić, N., Stougie, L.: Computing nonbinary agreement forests. SIAM J. Discret. Math. 28(1), 49–66 (2014)
Linz, S., Semple, C.: Hybridization in nonbinary trees. IEEE/ACM Trans. Comput. Biol. Bioinform. 6, 30–45 (2009)
Maddison, W.: Reconstructing character evolution on polytomous cladograms. Cladistics 5(4), 365–377 (1989)
Rodrigues, E.M., Sagot, M.-F., Wakabayashi, Y.: The maximum agreement forest problem: approximation algorithms and computational experiments. Theor. Comput. Sci. 374(1–3), 91–110 (2007)
Rosas-Magallanes, V., Deschavanne, P., Quintana-Murci, L., Brosch, R., Gicquel, B., Neyrolles, O.: Horizontal transfer of a virulence operon to the ancestor of mycobacterium tuberculosis. Mol. Biol. Evol. 23(6), 1129–1135 (2006)
Whidden, C.: RSPR software. http://kiwi.cs.dal.ca/Software/RSPR
Whidden, C.: SPRSupertrees software. http://kiwi.cs.dal.ca/Software/SPRSupertrees
Whidden, C., Beiko, R.G., Zeh, N.: Fast FPT algorithms for computing rooted agreement forests: theory and experiments. In: Proceedings of the 9th International Symposium on Experimental Algorithms, SEA 2010, vol. 6049 of Lecture Notes in Computer Science, pp. 141–153. Springer (2010)
Whidden, C., Beiko, R.G., Zeh, N.: Fixed-parameter algorithms for maximum agreement forests. SIAM J. Comput. 42(4), 1431–1466 (2013)
Whidden, C., Zeh, N.: A unifying view on approximation and FPT of agreement forests. In: Proceedings of the 9th International Workshop, WABI 2009, volume 5724 of Lecture Notes in Bioinformatics, pp. 390–401. Springer (2009)
Whidden, C., Zeh, N., Beiko, R.G.: Supertrees based on the subtree prune-and-regraft distance. Syst. Biol. 63(4), 566–581 (2014)
This work was done while Chris Whidden was a PhD student at Dalhousie University. Chris Whidden was supported by a Killam predoctoral scholarship and the Tula Foundation, Robert G. Beiko is a Canada Research Chair and was supported by NSERC, Genome Atlantic, and the Canada Foundation for Innovation, and Norbert Zeh is a Canada Research Chair and was supported by NSERC and the Canada Foundation for Innovation.
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Whidden, C., Beiko, R.G. & Zeh, N. Fixed-Parameter and Approximation Algorithms for Maximum Agreement Forests of Multifurcating Trees. Algorithmica 74, 1019–1054 (2016). https://doi.org/10.1007/s00453-015-9983-z
- Fixed-parameter tractability
- Approximation algorithms