Triplet Distance Research Articles

Building a small and informative phylogenetic supertree

We combine two fundamental optimization problems related to the construction of phylogenetic trees called maximum rooted triplets consistency and minimally resolved supertree into a new problem, which we call q-maximum rooted triplets consistency (q-MAXRTC). It takes as input a set R of rooted, binary phylogenetic trees with three leaves each and asks for a phylogenetic tree with exactly q internal nodes that contains the largest possible number of trees from R. We prove that q-MAXRTC is NP-hard to approximate within a constant, develop polynomial-time approximation algorithms for different values of q, and show experimentally that representing a phylogenetic tree by one having much fewer nodes typically does not destroy too much branching information. To demonstrate the algorithmic advantage of using trees with few internal nodes, we also propose a new algorithm for computing the rooted triplet distance that is faster than the existing algorithms when restricted to such trees.

Trying out a million genes to find the perfect pair with RTIST.

Consensus methods can be used for reconstructing a species tree from several gene trees, which exhibit incompatible topologies due to incomplete lineage sorting. Motivated by the fact that there are no anomalous rooted gene trees with three taxa and no anomalous unrooted gene trees with four taxa in the multispecies coalescent model, several contemporary methods form the gene tree consensus by finding the median tree with respect to the triplet or quartet distance-i.e. estimate the species tree as the tree which minimizes the sum of triplet or quartet distances to the input gene trees. These methods reformulate the solution to the consensus problem as the solution to a recursively solved dynamic programming (DP) problem. We present an iterative, easily parallelizable approach to finding the exact median triplet tree and implement it as an open source software package that can also find suboptimal consensus trees within a specified triplet distance to the gene trees. The most time-consuming step for methods of this type is the creation of a weights array for all possible subtree bipartitions. By grouping the relevant calculations and array update operations of different bipartitions of the same subtree together, this implementation finds the exact median tree of many gene trees faster than comparable methods, has better scaling properties with respect to the number of gene trees and has a smaller memory footprint. RTIST (Rooted Triple Inference of Species Trees) finds the exact median triplet tree of a set of gene trees. Its runtime and memory footprints scale better than existing algorithms. RTIST can resolve all the non-unique median trees, as well as sub-optimal consensus trees within a user-specified triplet distance to the median. Although it is limited in the number of taxa (≤20), its runtime changes little when the number of gene trees is changed by several orders of magnitude. RTIST is written in C and Python. It is freely available at https://github.com/glebzhelezov/rtist.

Automatic expression recognition of face image sequence based on key-frame generation and differential emotion feature

Using two key-frames with the neutral expression and maximum expression intensity respectively in face image sequence to alleviate inter-subject variations for Facial Expression Recognition (FER) has become the research focus of computer vision. To determine the two key-frames, an automatic Generation Difference Convolutional Neural Network (GDCNN) framework was presented to reduce the influences of high inter-subject variations caused by individual differences. Firstly, for any given input expression, a trained conditional Generative Adversarial Network (cGAN) is utilized to generate the frame with neutral expression while keeping the identity-related information. Secondly, the fine-tuned triplet distance model is used to detect the frame with the maximum expression intensity. Finally, an optimized two-stream CNN model is designed to reduce the influences of inter-subject variations and trained to extract the differential emotion features for FER. Extensive comparisons were performed on the CK+, MMI, and Beihang University databases and the results demonstrated the superiority of the proposed GDCNN framework over the state-of-the-art methods.

Cache Oblivious Algorithms for Computing the Triplet Distance between Trees

We consider the problem of computing the triplet distance between two rooted unordered trees with n labeled leaves. Introduced by Dobson in 1975, the triplet distance is the number of leaf triples that induce different topologies in the two trees. The current theoretically fastest algorithm is an O( n log n ) algorithm by Brodal et al. (SODA 2013). Recently, Jansson and Rajaby proposed a new algorithm that, while slower in theory, requiring O( n log 3 n ) time, in practice it outperforms the theoretically faster O( n log n ) algorithm. Both algorithms do not scale to external memory. We present two cache oblivious algorithms that combine the best of both worlds. The first algorithm is for the case when the two input trees are binary trees, and the second is a generalized algorithm for two input trees of arbitrary degree. Analyzed in the RAM model, both algorithms require O( n log n ) time, and in the cache oblivious model O( n / B log 2 n / M ) I/Os. Their relative simplicity and the fact that they scale to external memory makes them achieve the best practical performance. We note that these are the first algorithms that scale to external memory, both in theory and in practice, for this problem.

Computing the Rooted Triplet Distance Between Phylogenetic Networks

The rooted triplet distance measures the structural dissimilarity of two phylogenetic trees or phylogenetic networks by counting the number of rooted phylogenetic trees with exactly three leaf labels (called rooted triplets, or triplets for short) that occur as embedded subtrees in one, but not both, of them. Suppose that N_1 = (V_1, E_1) and N_2 = (V_2, E_2) are phylogenetic networks over a common leaf label set of size n, that N_i has level k_i and maximum in-degree d_i for i in {1,2}, and that the networks’ out-degrees are unbounded. Write N = max (|V_1|, |V_2|), M = max (|E_1|, |E_2|), k = max (k_1, k_2), and d = max (d_1, d_2). Previous work has shown how to compute the rooted triplet distance between N_1 and N_2 in mathrm {O}(n log n) time in the special case k le 1. For k > 1, no efficient algorithms are known; applying a classic method from 1980 by Fortune et al. in a direct way leads to a running time of {Omega }(N^{6} n^{3}) and the only existing non-trivial algorithm imposes restrictions on the networks’ in- and out-degrees (in particular, it does not work when non-binary vertices are allowed). In this article, we develop two new algorithms with no such restrictions. Their running times are mathrm {O}(N^{2} M + n^{3}) and mathrm {O}(M + N k^{2} d^{2} + n^{3}), respectively. We also provide implementations of our algorithms, evaluate their performance on simulated and real datasets, and make some observations on the limitations of the current definition of the rooted triplet distance in practice. Our prototype implementations have been packaged into the first publicly available software for computing the rooted triplet distance between unrestricted networks of arbitrary levels.

Attributed network representation learning via improved graph attention with robust negative sampling

Attributed network representation learning is to embed graphs in low dimensional vector space such that the embedded vectors follow the differences and similarities of the source graphs. To capture structural features and node attributes of attributed network, we propose a novel graph auto-encoder method which is stacked encoder-decoder layers based on graph attention with robust negative sampling. Here, minimize the negative log-likelihood, triplet distance, and weighted neighborhood attributes are proposed as the loss function. To alleviate the over-fitting on reconstruct graph structural features or node attributes, a trade off algorithm between reconstruction loss of node attributes and reconstruction loss of structural features is proposed. Furthermore, to alleviate the impact of random sampling, we propose additional constraints on negative sampling based on node degree. Experimental results on several benchmark datasets for transductive and inductive learning tasks show that the proposed model is competitive against well-known methods in node classification and link prediction.

An Efficient Algorithm for the Rooted Triplet Distance Between Galled Trees.

The previous fastest algorithm for computing the rooted triplet distance between two input galled trees (i.e., phylogenetic networks whose cycles are vertex-disjoint) runs in [Formula: see text] time, where n is the cardinality of the leaf label set. In this article, we present an [Formula: see text]-time solution. Our strategy is to transform the input so that the answer can be obtained by applying an existing [Formula: see text]-time algorithm for the simpler case of two phylogenetic trees a constant number of times. The new algorithm has been implemented, and applying it to pairs of randomly generated galled trees with up to [Formula: see text] leaves confirms that it is fast in practice.

A More Practical Algorithm for the Rooted Triplet Distance

The rooted triplet distance is a measure of the dissimilarity of two phylogenetic trees with identical leaf label sets. An algorithm by Brodal et al. that computes it in [Formula: see text] time and [Formula: see text] space, where n is the number of leaf labels, has recently been implemented in the software package tqDist. In this article, we show that replacing the hierarchical decomposition tree used in Brodal et al.'s algorithm by a centroid paths-based data structure yields an [Formula: see text]-time and [Formula: see text]-space algorithm that, although slower in theory, is faster in practice as well as less memory consuming. Simulations for values of n up to 4,000,000 support our claims experimentally.

TqDist: a library for computing the quartet and triplet distances between binary or general trees

tqDist is a software package for computing the triplet and quartet distances between general rooted or unrooted trees, respectively. The program is based on algorithms with running time [Formula: see text] for the triplet distance calculation and [Formula: see text] for the quartet distance calculation, where n is the number of leaves in the trees and d is the degree of the tree with minimum degree. These are currently the fastest algorithms both in theory and in practice. tqDist can be installed on Windows, Linux and Mac OS X. Doing this will install a set of command-line tools together with a Python module and an R package for scripting in Python or R. The software package is freely available under the GNU LGPL licence at http://birc.au.dk/software/tqDist.

Computing the rooted triplet distance between galled trees by counting triangles

We consider a generalization of the rooted triplet distance between two phylogenetic trees to two phylogenetic networks. We show that if each of the two given phylogenetic networks is a so-called galled tree with n leaves then the rooted triplet distance can be computed in o(n2.687) time. Our upper bound is obtained by reducing the problem of computing the rooted triplet distance between two galled trees to that of counting monochromatic and almost-monochromatic triangles in an undirected, edge-colored graph. To count different types of colored triangles in a graph efficiently, we extend an existing technique based on matrix multiplication and obtain several new algorithmic results that may be of independent interest: (i) the number of triangles in a connected, undirected, uncolored graph with m edges can be computed in o(m1.408) time; (ii) if G is a connected, undirected, edge-colored graph with n vertices and C is a subset of the set of edge colors then the number of monochromatic triangles of G with colors in C can be computed in o(n2.687) time; and (iii) if G is a connected, undirected, edge-colored graph with n vertices and R is a binary relation on the colors that is computable in O(1) time then the number of R-chromatic triangles in G can be computed in o(n2.687) time.

A practical O(n log2 n) time algorithm for computing the triplet distance on binary trees

The triplet distance is a distance measure that compares two rooted trees on the same set of leaves by enumerating all sub-sets of three leaves and counting how often the induced topologies of the tree are equal or different. We present an algorithm that computes the triplet distance between two rooted binary trees in time O (n log2 n). The algorithm is related to an algorithm for computing the quartet distance between two unrooted binary trees in time O (n log n). While the quartet distance algorithm has a very severe overhead in the asymptotic time complexity that makes it impractical compared to O (n2) time algorithms, we show through experiments that the triplet distance algorithm can be implemented to give a competitive wall-time running time.