Homeomorphic Alignment of Weighted Trees

International audienceMotion capture, a currently active research area, needs estimation of the pose of the subject. For this purpose, we match the tree representation of the skeleton of the 3D shape to a pre-specified tree model. Unfortunately, the tree representation can contain vertices that split limbs in multiple parts, which do not allow a good match by usual methods. To solve this problem, we propose a new alignment, taking into account the homeomorphism between trees, rather than the isomorphism, as in prior works. Then, we develop several computationally efficient algorithms for reaching real-time motion capture

Mirroring co-evolving trees in the light of their topologies

Determining the interaction partners among protein/domain families poses hard computational problems, in particular in the presence of paralogous proteins. Available approaches aim to identify interaction partners among protein/domain families through maximizing the similarity between trimmed versions of their phylogenetic trees. Since maximization of any natural similarity score is computationally difficult, many approaches employ heuristics to maximize the distance matrices corresponding to the tree topologies in question. In this paper we devise an efficient deterministic algorithm which directly maximizes the similarity between two leaf labeled trees with edge lengths, obtaining a score-optimal alignment of the two trees in question. Our algorithm is significantly faster than those methods based on distance matrix comparison: 1 minute on a single processor vs. 730 hours on a supercomputer. Furthermore we have advantages over the current state-of-the-art heuristic search approach in terms of precision as well as a recently suggested overall performance measure for mirrortree approaches, while incurring only acceptable losses in recall. A C implementation of the method demonstrated in this paper is available at http://compbio.cs.sfu.ca/mirrort.htmComment: 13 pages, 2 figures, Iman Hajirasouliha and Alexander Sch\"onhuth are joint first author

Crinoid phylogeny: a preliminary analysis (Echinodermata: Crinoidea)

We describe the first molecular and morphological analysis of extant crinoid high-level inter-relationships. Nuclear and mitochondrial gene sequences and a cladistically coded matrix of 30 morphological characters are presented, and analysed by phylogenetic methods. The molecular data were compiled from concatenated nuclear-encoded 18S rDNA, internal transcribed spacer 1, 5.8S rDNA, and internal transcribed spacer 2, together with part of mitochondrial 16S rDNA, and comprised 3,593 sites, of which 313 were parsimony-informative. The molecular and morphological analyses include data from the bourgueticrinid Bathycrinus; the antedonid comatulids Dorometra and Florometra; the cyrtocrinids Cyathidium, Gymnocrinus, and Holopus; the isocrinids Endoxocrinus, and two species of Metacrinus; as well as from Guillecrinus and Caledonicrinus, whose ordinal relationships are uncertain, together with morphological data from Proisocrinus. Because the molecular data include indel-rich regions, special attention was given to alignment procedure, and it was found that relatively low, gene-specific, gap penalties gave alignments from which congruent phylogenetic information was obtained from both well-aligned, indel-poor and potentially misaligned, indel-rich regions. The different sequence data partitions also gave essentially congruent results. The overall direction of evolution in the gene trees remains uncertain: an asteroid outgroup places the root on the branch adjacent to the slowly evolving isocrinids (consistent with palaeontological order of first appearances), but maximum likelihood analysis with a molecular clock places it elsewhere. Despite lineage-specific rate differences, the clock model was not excluded by a likelihood ratio test. Morphological analyses were unrooted. All analyses identified three clades, two of them generally well-supported. One well-supported clade (BCG) unites Bathycrinus and Guillecrinus with the representative (chimaeric) comatulid in a derived position, suggesting that comatulids originated from a sessile, stalked ancestor. In this connection it is noted that because the comatulid centrodorsal ossicle originates ontogenetically from the column, it is not strictly correct to describe comatulids as unstalked crinoids. A second, uniformly well-supported clade contains members of the Isocrinida, while the third clade contains Gymnocrinus, a well-established member of the Cyrtocrinida, together with the problematic taxon Caledonicrinus, currently classified as a bourgueticrinid. Another cyrtocrinid, Holopus, joins this clade with only weak molecular, but strong morphological support. In one morphological analysis Proisocrinus is weakly attached to the isocrinid clade. Only an unusual, divergent 18S rDNA sequence was obtained from the morphologically strange cyrtocrinid Cyathidium. Although not analysed in detail, features of this sequence suggested that it may be a PCR artefact, so that the apparently basal position of this taxon requires confirmation. If not an artefact, Cyathidium either diverged from the crinoid stem much earlier than has been recognised hitherto (i.e., it may be a Palaeozoic relic), or it has an atypically high rate of molecular evolution

FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees

The Gromov-Hausdorff distance is a natural way to measure the distortion between two metric spaces. However, there has been only limited algorithmic development to compute or approximate this distance. We focus on computing the Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric tree is a metric space that can be realized by the shortest path metric on a tree. Any finite tree with positive edge weight can be viewed as a metric tree where the weight is treated as edge length and the metric is the induced shortest path metric in the tree. Previously, Agarwal et al. showed that even for trees with unit edge length, it is NP-hard to approximate the Gromov-Hausdorff distance between them within a factor of 3. In this paper, we present a fixed-parameter tractable (FPT) algorithm that can approximate the Gromov-Hausdorff distance between two general metric trees within a multiplicative factor of 14. Interestingly, the development of our algorithm is made possible by a connection between the Gromov-Hausdorff distance for metric trees and the interleaving distance for the so-called merge trees. The merge trees arise in practice naturally as a simple yet meaningful topological summary (it is a variant of the Reeb graphs and contour trees), and are of independent interest. It turns out that an exact or approximation algorithm for the interleaving distance leads to an approximation algorithm for the Gromov-Hausdorff distance. One of the key contributions of our work is that we re-define the interleaving distance in a way that makes it easier to develop dynamic programming approaches to compute it. We then present a fixed-parameter tractable algorithm to compute the interleaving distance between two merge trees exactly, which ultimately leads to an FPT-algorithm to approximate the Gromov-Hausdorff distance between two metric trees. This exact FPT-algorithm to compute the interleaving distance between merge trees is of interest itself, as it is known that it is NP-hard to approximate it within a factor of 3, and previously the best known algorithm has an approximation factor of O(sqrt{n}) even for trees with unit edge length