A Fast Quartet Tree Heuristic for Hierarchical Clustering

The Minimum Quartet Tree Cost problem is to construct an optimal weight tree from the $3{n \choose 4}$ weighted quartet topologies on $n$ objects, where optimality means that the summed weight of the embedded quartet topologies is optimal (so it can be the case that the optimal tree embeds all quartets as nonoptimal topologies). We present a Monte Carlo heuristic, based on randomized hill climbing, for approximating the optimal weight tree, given the quartet topology weights. The method repeatedly transforms a dendrogram, with all objects involved as leaves, achieving a monotonic approximation to the exact single globally optimal tree. The problem and the solution heuristic has been extensively used for general hierarchical clustering of nontree-like (non-phylogeny) data in various domains and across domains with heterogeneous data. We also present a greatly improved heuristic, reducing the running time by a factor of order a thousand to ten thousand. All this is implemented and available, as part of the CompLearn package. We compare performance and running time of the original and improved versions with those of UPGMA, BioNJ, and NJ, as implemented in the SplitsTree package on genomic data for which the latter are optimized. Keywords: Data and knowledge visualization, Pattern matching--Clustering--Algorithms/Similarity measures, Hierarchical clustering, Global optimization, Quartet tree, Randomized hill-climbing,Comment: LaTeX, 40 pages, 11 figures; this paper has substantial overlap with arXiv:cs/0606048 in cs.D

A New Quartet Tree Heuristic for Hierarchical Clustering

We consider the problem of constructing an an optimal-weight tree from the 3*(n choose 4) weighted quartet topologies on n objects, where optimality means that the summed weight of the embedded quartet topologiesis optimal (so it can be the case that the optimal tree embeds all quartets as non-optimal topologies). We present a heuristic for reconstructing the optimal-weight tree, and a canonical manner to derive the quartet-topology weights from a given distance matrix. The method repeatedly transforms a bifurcating tree, with all objects involved as leaves, achieving a monotonic approximation to the exact single globally optimal tree. This contrasts to other heuristic search methods from biological phylogeny, like DNAML or quartet puzzling, which, repeatedly, incrementally construct a solution from a random order of objects, and subsequently add agreement values.Comment: 22 pages, 14 figure

Reconstructing phylogenies from noisy quartets in polynomial time with a high success probability

<p>Abstract</p> <p>Background</p> <p>In recent years, quartet-based phylogeny reconstruction methods have received considerable attentions in the computational biology community. Traditionally, the accuracy of a phylogeny reconstruction method is measured by simulations on synthetic datasets with known "true" phylogenies, while little theoretical analysis has been done. In this paper, we present a new model-based approach to measuring the accuracy of a quartet-based phylogeny reconstruction method. Under this model, we propose three efficient algorithms to reconstruct the "true" phylogeny with a high success probability.</p> <p>Results</p> <p>The first algorithm can reconstruct the "true" phylogeny from the input quartet topology set without quartet errors in <it>O</it>(<it>n</it>²) time by querying at most (<it>n </it>- 4) log(<it>n </it>- 1) quartet topologies, where <it>n </it>is the number of the taxa. When the input quartet topology set contains errors, the second algorithm can reconstruct the "true" phylogeny with a probability approximately 1 - <it>p </it>in <it>O</it>(<it>n</it>⁴log <it>n</it>) time, where <it>p </it>is the probability for a quartet topology being an error. This probability is improved by the third algorithm to approximately <inline-formula><m:math name="1748-7188-3-1-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:semantics><m:mrow><m:mfrac><m:mn>1</m:mn><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:msup><m:mi>q</m:mi><m:mn>2</m:mn></m:msup><m:mo>+</m:mo><m:mfrac><m:mn>1</m:mn><m:mn>2</m:mn></m:mfrac><m:msup><m:mi>q</m:mi><m:mn>4</m:mn></m:msup><m:mo>+</m:mo><m:mfrac><m:mn>1</m:mn><m:mrow><m:mn>16</m:mn></m:mrow></m:mfrac><m:msup><m:mi>q</m:mi><m:mn>5</m:mn></m:msup></m:mrow></m:mfrac></m:mrow><m:annotation encoding="MathType-MTEF"> MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqaIXaqmaeaacqaIXaqmcqGHRaWkcqWGXbqCdaahaaqabeaacqaIYaGmaaGaey4kaSYaaSaaaeaacqaIXaqmaeaacqaIYaGmaaGaemyCae3aaWbaaeqabaGaeGinaqdaaiabgUcaRmaalaaabaGaeGymaedabaGaeGymaeJaeGOnaydaaiabdghaXnaaCaaabeqaaiabiwda1aaaaaaaaa@3D5A@</m:annotation></m:semantics></m:math></inline-formula>, where <inline-formula><m:math name="1748-7188-3-1-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:semantics><m:mrow><m:mi>q</m:mi><m:mo>=</m:mo><m:mfrac><m:mi>p</m:mi><m:mrow><m:mn>1</m:mn><m:mo>−</m:mo><m:mi>p</m:mi></m:mrow></m:mfrac></m:mrow><m:annotation encoding="MathType-MTEF"> MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGacaGaaiaabeqaaeqabiWaaaGcbaGaemyCaeNaeyypa0tcfa4aaSaaaeaacqWGWbaCaeaacqaIXaqmcqGHsislcqWGWbaCaaaaaa@3391@</m:annotation></m:semantics></m:math></inline-formula>, with running time of <it>O</it>(<it>n</it>⁵), which is at least 0.984 when <it>p </it>< 0.05.</p> <p>Conclusion</p> <p>The three proposed algorithms are mathematically guaranteed to reconstruct the "true" phylogeny with a high success probability. The experimental results showed that the third algorithm produced phylogenies with a higher probability than its aforementioned theoretical lower bound and outperformed some existing phylogeny reconstruction methods in both speed and accuracy.</p

Constructing level-2 phylogenetic networks from triplets

Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of species), it is possible to determine in polynomial time whether there exists a level-1 network consistent with T, and if so to construct such a network. They also showed that, unlike in the case of trees (i.e. level-0 networks), the problem becomes NP-hard when the input is non-dense. Here we further extend this work by showing that, when the set of input triplets is dense, the problem is even polynomial-time solvable for the construction of level-2 networks. This shows that, assuming density, it is tractable to construct plausible evolutionary histories from input triplets even when such histories are heavily non-tree like. This further strengthens the case for the use of triplet-based methods in the construction of phylogenetic networks. We also show that, in the non-dense case, the level-2 problem remains NP-hard

Festparameter-Algorithmen fuer die Konsens-Analyse Genomischer Daten

Fixed-parameter algorithms offer a constructive and powerful approach to efficiently obtain solutions for NP-hard problems combining two important goals: Fixed-parameter algorithms compute optimal solutions within provable time bounds despite the (almost inevitable) computational intractability of NP-hard problems. The essential idea is to identify one or more aspects of the input to a problem as the parameters, and to confine the combinatorial explosion of computational difficulty to a function of the parameters such that the costs are polynomial in the non-parameterized part of the input. This makes especially sense for parameters which have small values in applications. Fixed-parameter algorithms have become an established algorithmic tool in a variety of application areas, among them computational biology where small values for problem parameters are often observed. A number of design techniques for fixed-parameter algorithms have been proposed and bounded search trees are one of them. In computational biology, however, examples of bounded search tree algorithms have been, so far, rare. This thesis investigates the use of bounded search tree algorithms for consensus problems in the analysis of DNA and RNA data. More precisely, we investigate consensus problems in the contexts of sequence analysis, of quartet methods for phylogenetic reconstruction, of gene order analysis, and of RNA secondary structure comparison. In all cases, we present new efficient algorithms that incorporate the bounded search tree paradigm in novel ways. On our way, we also obtain results of parameterized hardness, showing that the respective problems are unlikely to allow for a fixed-parameter algorithm, and we introduce integer linear programs (ILP's) as a tool for classifying problems as fixed-parameter tractable, i.e., as having fixed-parameter algorithms. Most of our algorithms were implemented and tested on practical data.Festparameter-Algorithmen bieten einen konstruktiven Ansatz zur Loesung von kombinatorisch schwierigen, in der Regel NP-harten Problemen, der zwei Ziele beruecksichtigt: innerhalb von beweisbaren Laufzeitschranken werden optimale Ergebnisse berechnet. Die entscheidende Idee ist dabei, einen oder mehrere Aspekte der Problemeingabe als Parameter der Problems aufzufassen und die kombinatorische Explosion der algorithmischen Schwierigkeit auf diese Parameter zu beschraenken, so dass die Laufzeitkosten polynomiell in Bezug auf den nicht-parametrisierten Teil der Eingabe sind. Gibt es einen Festparameter-Algorithmus fuer ein kombinatorisches Problem, nennt man das Problem festparameter-handhabbar. Die Entwicklung von Festparameter-Algorithmen macht vor allem dann Sinn, wenn die betrachteten Parameter im Anwendungsfall nur kleine Werte annehmen. Festparameter-Algorithmen sind zu einem algorithmischen Standardwerkzeug in vielen Anwendungsbereichen geworden, unter anderem in der algorithmischen Biologie, wo in vielen Anwendungen kleine Parameterwerte beobachtet werden koennen. Zu den bekannten Techniken fuer den Entwurf von Festparameter-Algorithmen gehoeren unter anderem groessenbeschraenkte Suchbaeume. In der algorithmischen Biologie gibt es bislang nur wenige Beispiele fuer die Anwendung von groessenbeschraenkten Suchbaeumen. Diese Arbeit untersucht den Einsatz groessenbeschraenkter Suchbaeume fuer NP-harte Konsens-Probleme in der Analyse von DNS- und RNS-Daten. Wir betrachten Konsens-Probleme in der Analyse von DNS-Sequenzdaten, in der Analyse von sogenannten Quartettdaten zur Erstellung von phylogenetischen Hypothesen, in der Analyse von Daten ueber die Anordnung von Genen und beim Vergleich von RNS-Strukturdaten. In allen Faellen stellen wir neue effiziente Algorithmen vor, in denen das Paradigma der groessenbeschraenkten Suchbaeume auf neuartige Weise realisiert wird. Auf diesem Weg zeigen wir auch Ergebnisse parametrisierter Haerte, die zeigen, dass fuer die dabei betrachteten Probleme ein Festparameter-Algorithmus unwahrscheinlich ist. Ausserdem fuehren wir ganzzahliges lineares Programmieren als eine neue Technik ein, um die Festparameter-Handhabbarkeit eines Problems zu zeigen. Die Mehrzahl der hier vorgestellten Algorithmen wurde implementiert und auf Anwendungsdaten getestet

A list of parameterized problems in bioinformatics

In this report we present a list of problems that originated in bionformatics. Our aim is to collect information on such problems that have been analyzed from the point of view of Parameterized Complexity. For every problem we give its definition and biological motivation together with known complexity results.Postprint (published version