Reconstructing Evolution of Natural Languages: Complexity and Parameterized Algorithms

Abstract

Abstract. In a recent article, Nakhleh, Ringe and Warnow introduced perfect phylogenetic networks-a model of language evolution where languages do not evolve via clean speciation-and formulated a set of problems for their accurate reconstruction. Their new methodology assumes networks, rather than trees, as the correct model to capture the evolutionary history of natural languages. They proved the NP-hardness of the problem of testing whether a network is a perfect phylogenetic one for characters exhibiting at least three states, leaving open the case of binary characters, and gave a straightforward brute-force parameterized algorithm for the problem of running time O(3 k n), where k is the number of bidirectional edges in the network and n is its size. In this paper, we first establish the NP-hardness of the binary case of the problem. Then we provide a more efficient parameterized algorithm for this case running in time O(2 k n 2 ). The presented algorithm is very simple, and utilizes some structural results and elegant operations developed in this paper that can be useful on their own in the design of heuristic algorithms for the problem. The analysis phase of the algorithm is very elegant using amortized techniques to show that the upper bound on the running time of the algorithm is much tighter than the upper bound obtained under a conservative worst-case scenario assumption. Our results bear significant impact on reconstructing evolutionary histories of languagesparticularly from phonological and morphological character data, most of which exhibit at most two states (i.e., are binary), as well as on the design and analysis of parameterized algorithms.

Seeing the trees and their branches in the network is hard

“…Kanj et al [7] proved the NP-completeness for the problem of character compatibility on phylogenetic networks when the network edges are bi-directional. We modify their proof to make it work for the ISPN problem and present the theorem.…”

“…This improvement is very significant in practice. [12] Kanj et al [7] considered the problem of character compatibility on a different model of phylogenetic networks that is used in historical linguistics. Whereas the NP-hardness result from that work is modified and used here, that is not the case, however, for the new parameterized algorithm that we present here.…”

In the light of deep coalescence: revisiting trees within networks

“…This is why the set U(ψ) has played a fundamental role in most results established for phylogenetic networks. Examples of the prominent use of U(ψ) include: (1) Parsimonious phylogenetic networks that fit the evolution of a sequence of sequences under the infinite sites model [7,8,9,10,11,12,13,14]; (2) extending the maximum parsimony and maximum likelihood criteria from trees to networks [15,16,17,18,19,20]; (3) inferring minimal networks from sets of gene trees [21,22,23,24]; (4) establishing identifiability results related to networks [25]; (5) establishing complexity results related to networks [26,27,28,29,30,31]; and (6) identifying special trees within the network [32,33,34,35]. One of the evolutionary phenomena that has been extensively documented in recent analyses and targeted for computational developments is deep coalescence, or incomplete lineage sorting [36].…”