A Rearrangement Distance for Fully-Labelled Trees

The problem of comparing trees representing the evolutionary histories of cancerous tumors has turned out to be crucial, since there is a variety of different methods which typically infer multiple possible trees. A departure from the widely studied setting of classical phylogenetics, where trees are leaf-labelled, tumoral trees are fully labelled, i.e., every vertex has a label. In this paper we provide a rearrangement distance measure between two fully-labelled trees. This notion originates from two operations: one which modifies the topology of the tree, the other which permutes the labels of the vertices, hence leaving the topology unaffected. While we show that the distance between two trees in terms of each such operation alone can be decided in polynomial time, the more general notion of distance when both operations are allowed is NP-hard to decide. Despite this result, we show that it is fixed-parameter tractable, and we give a 4-approximation algorithm when one of the trees is binary

On Two Measures of Distance Between Fully-Labelled Trees

The last decade brought a significant increase in the amount of data and a variety of new inference methods for reconstructing the detailed evolutionary history of various cancers. This brings the need of designing efficient procedures for comparing rooted trees representing the evolution of mutations in tumor phylogenies. Bernardini et al. [CPM 2019] recently introduced a notion of the rearrangement distance for fully-labelled trees motivated by this necessity. This notion originates from two operations: one that permutes the labels of the nodes, the other that affects the topology of the tree. Each operation alone defines a distance that can be computed in polynomial time, while the actual rearrangement distance, that combines the two, was proven to be NP-hard. We answer two open question left unanswered by the previous work. First, what is the complexity of computing the permutation distance? Second, is there a constant-factor approximation algorithm for estimating the rearrangement distance between two arbitrary trees? We answer the first one by showing, via a two-way reduction, that calculating the permutation distance between two trees on n nodes is equivalent, up to polylogarithmic factors, to finding the largest cardinality matching in a sparse bipartite graph. In particular, by plugging in the algorithm of Liu and Sidford [ArXiv 2020], we obtain an ??(n^{4/3+o(1}) time algorithm for computing the permutation distance between two trees on n nodes. Then we answer the second question positively, and design a linear-time constant-factor approximation algorithm that does not need any assumption on the trees

Regenerative tree growth: structural results and convergence

We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n>=1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as non-exchangeable models such as the alpha-theta model, the alpha-gamma model and all restricted exchangeable models previously studied. Our main structural result is a representation of the growth rule by a sigma-finite dislocation measure kappa on the set of partitions of the natural numbers extending Bertoin's notion of exchangeable dislocation measures from the setting of homogeneous fragmentations. We use this representation to establish necessary and sufficient conditions on the growth rule under which we can apply results by Haas and Miermont for unlabelled and not necessarily consistent trees to establish self-similar random trees and residual mass processes as scaling limits. While previous studies exploited some form of exchangeability, our scaling limit results here only require a regularity condition on the convergence of asymptotic frequencies under kappa, in addition to a regular variation condition.Comment: 23 pages, new title, restructured, presentation improve

Synchronisation Properties of Trees in the Kuramoto Model

We consider the Kuramoto model of coupled oscillators, specifically the case of tree networks, for which we prove a simple closed-form expression for the critical coupling. For several classes of tree, and for both uniform and Gaussian vertex frequency distributions, we provide tight closed form bounds and empirical expressions for the expected value of the critical coupling. We also provide several bounds on the expected value of the critical coupling for all trees. Finally, we show that for a given set of vertex frequencies, there is a rearrangement of oscillator frequencies for which the critical coupling is bounded by the spread of frequencies.Comment: 21 pages, 19 Figure

Using F-structures in machine translation evaluation

Despite a growing interest in automatic evaluation methods for Machine Translation (MT) quality, most existing automatic metrics are still limited to surface comparison of translation and reference strings. In this paper we show how Lexical-Functional Grammar (LFG) labelled dependencies obtained from an automatic parse can be used to assess the quality of MT on a deeper linguistic level, giving as a result higher correlations with human judgements

Limited Lifespan of Fragile Regions in Mammalian Evolution

An important question in genome evolution is whether there exist fragile regions (rearrangement hotspots) where chromosomal rearrangements are happening over and over again. Although nearly all recent studies supported the existence of fragile regions in mammalian genomes, the most comprehensive phylogenomic study of mammals (Ma et al. (2006) Genome Research 16, 1557-1565) raised some doubts about their existence. We demonstrate that fragile regions are subject to a "birth and death" process, implying that fragility has limited evolutionary lifespan. This finding implies that fragile regions migrate to different locations in different mammals, explaining why there exist only a few chromosomal breakpoints shared between different lineages. The birth and death of fragile regions phenomenon reinforces the hypothesis that rearrangements are promoted by matching segmental duplications and suggests putative locations of the currently active fragile regions in the human genome