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Linear-Time Tree Containment in Phylogenetic Networks

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Comparative Genomics (RECOMB-CG 2018)

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Abstract

We consider the NP-hard Tree Containment problem that has important applications in phylogenetics. The problem asks if a given single-rooted leaf-labeled network (“phylogenetic network”) N “contains” a given leaf-labeled tree (“phylogenetic tree”) T. We develop a fast algorithm for the case that N is a phylogenetic tree in which multiple leaves might share a label. Generalizing a previously known decomposition scheme lets us leverage this algorithm, yielding linear-time algorithms for so-called “reticulation visible” networks and“nearly stable” networks. While these are special classes of networks, they rank among the most general of the previously considered cases. We also present a dynamic programming algorithm that solves the general problem in \(O(3^{t^*}\cdot |N|\cdot |T|)\) time, where the parameter \(t^*\) is the maximum number of “tree components with unstable roots” in any block of the input network. Notably, \(t^*\) is stronger (that is, smaller on all networks) than the previously considered parameter “number of reticulations” and even the popular parameter “level” of the input network.

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Notes

  1. 1.

    Herein, is the maximum out-degree in T and is the maximum in-degree in the result of contracting all arcs between reticulations in N.

  2. 2.

    u is stable on \(\ell \) if all root-\(\ell \)-paths contain u. The notion of stability is equivalent to the notion of “dominators” in directed graphs [1, 24].

  3. 3.

    A biconnected component (or “block”) of a network is a subdigraph induced by the vertices of a biconnected component of its underlying undirected graph, that is, a connected component in the result of removing all bridges.

  4. 4.

    The level of a phylogenetic network is the largest number of reticulations in any biconnected component (of its underlying undirected graph).

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Acknowledgement

Thanks to Celine Scornavacca for her thorough proof-reading.

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Appendix

Appendix

Proof

(Proof of correctness of Rule 2). Let \(S^v\) be a subdivision of \(T_v\) in P and let \((N',T')\) be the result of applying Rule 2 to (NT).

\(\Leftarrow \)”: Let \(N'\) contain a subdivision \(S'\) of \(T'\). It suffices to show that the result S of replacing \(\rho \left( P\right) \) with \(S^v\) in \(S'\) is contained in N since S is clearly a subdivision of T. Since \(S^v\) is contained in P, it suffices to show that \(S'\) and \(S^v\) are vertex disjoint (except for \(\rho \left( P\right) \)). Towards a contradiction, assume that \(S'\) and \(S^v\) both contain a vertex \(u\ne \rho \left( P\right) \) of P. Since \(\mathcal {L}(S')\) and \(\mathcal {L}(S^v)\) are disjoint, u is ancestor to at least two different leaves in N. Thus, u is in the tip of P, contradicting that u is in \(N'\).

\(\Rightarrow \)”: Let N contain a subdivision S of T and let . Since \(\rho \left( P\right) \) is stable on c and \(c\in \mathcal {L}(T_v)\), we have \(u\le _N \rho \left( P\right) \), implying \(\mathcal {L}(S_{\rho \left( P\right) })\supseteq \mathcal {L}(T_v)\). Further, maximality of v implies \(\mathcal {L}(S_{\rho \left( P\right) })\subseteq \mathcal {L}(T_v)\). Let \(S'\) result from S by contracting \(S_{\rho \left( P\right) }\) into a single vertex and labeling this vertex \(\lambda \). Since \(\mathcal {L}(S_{\rho \left( P\right) })=\mathcal {L}(T_v)\), we know that \(S'\) is a subdivision of \(T'\) and it suffices to show that \(N'\) contains \(S'\). To do this, we show that all vertices of \(S'\) are in \(N'\). Assume towards a contradiction that \(S'\) contains a vertex w that is not in \(N'\). Then, w is in the tip of P, implying \(\mathcal {L}(S_w)\subseteq \mathcal {L}(S_{\rho \left( P\right) })\). Thus, w is a vertex of \(S_{\rho \left( P\right) }\) contradicting w being in \(S'\).

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Weller, M. (2018). Linear-Time Tree Containment in Phylogenetic Networks. In: Blanchette, M., Ouangraoua, A. (eds) Comparative Genomics. RECOMB-CG 2018. Lecture Notes in Computer Science(), vol 11183. Springer, Cham. https://doi.org/10.1007/978-3-030-00834-5_18

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  • DOI: https://doi.org/10.1007/978-3-030-00834-5_18

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