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An Even Faster and More Unifying Algorithm for Comparing Trees via Unbalanced Bipartite Matchings

TL;DR: In this article, the authors presented an algorithm for comparing trees that are labeled in an arbitrary manner, which is faster than the previous algorithms and is at the core of their maximum agreement subtree algorithm.
Abstract: A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled by the same symbol) and evolutionary trees (i.e., leaf-labeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms. Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are node-unbalanced or weight-unbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm.
Citations
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Book ChapterDOI
29 Oct 2012
TL;DR: A label-based closest-neighbor trimming method to trim a phylogenetic tree built from nucleotide sequences according to the labels of leaves and a trim distance is formulated as the LCA-preserving distance between the trimmed phylogenetic trees according to nucleotides occurring in the positions.
Abstract: In this paper, first we introduce a label-based closest-neighbor trimming method to trim a phylogenetic tree built from nucleotide sequences according to the labels of leaves. Next, by replacing the indices of nucleotide sequences as labels of leaves with the nucleotides occurring in a position, we formulate a trim distance between two positions in nucleotide sequences as the LCA-preserving distance between the trimmed phylogenetic trees according to nucleotides occurring in the positions. Finally, we apply the trim distance to compare pandemic influenza A (H1N1) viruses with non-pandemic ones.

5 citations

Dissertation
01 Jan 2008
TL;DR: Cette these etudie d'un point de vue algorithmique diverses methodes de consensus portant sur des collections d'objets etiquetes implique des problemes etudies impliquent des objects peuvent etre des arbres enracines ou des sequences, avec des applications a la bioinformatique.
Abstract: Cette these etudie d'un point de vue algorithmique diverses methodes de consensus portant sur des collections d'objets etiquetes. Les problemes etudies impliquent des objets etiquetes sans repetition d'etiquettes ; ces objets peuvent etre des arbres enracines ou des sequences, avec des applications a la bioinformatique. Ainsi, les problemes sur les arbres consideres dans cette these peuvent trouver des applications pour l'estimation de congruence entre phylogenies, pour la construction de superarbres, et pour l'identification de transferts horizontaux de genes. Pour leur part, les problemes sur les sequences consideres dans cette these ont des applications potentielles pour le calcul de distance genomique base sur les ordres de genes. De maniere generale, ce travail met a profit les theories de la complexite parametrique et de l'approximabilite pour obtenir des algorithmes et des resultats de difficulte pour les problemes etudies.

4 citations

Journal ArticleDOI
TL;DR: A linear-time approximation algorithm to solve the complement version of MAST, namely identifying the smallest set of leaves to remove from input trees to obtain isomorphic trees, achieves significantly lower running times than previously known algorithms.
Abstract: Given a set of leaf-labeled trees with identical leaf sets, the well-known Maximum Agreement SubTree (MAST) problem consists in finding a subtree homeomorphically included in all input trees and with the largest number of leaves. MAST and its variant called Maximum Compatible Tree (MCT) are of particular interest in computational biology. This article presents a linear-time approximation algorithm to solve the complement version of MAST, namely identifying the smallest set of leaves to remove from input trees to obtain isomorphic trees. We also present an O(n2 + kn) algorithm to solve the complement version of MCT. For both problems, we thus achieve significantly lower running times than previously known algorithms. Fast running times are especially important in phylogenetics where large collections of trees are routinely produced by resampling procedures, such as the nonparametric bootstrap or Bayesian MCMC methods.

4 citations

Journal ArticleDOI
TL;DR: Given a set of k phylogenetic trees whose leaves are drawn from {1,2,...,n} and the leaves for two arbitrary trees are not necessary the same, a linear-time algorithm is presented to final all the maximal leaf-agreement descendent subtrees.

4 citations

Book ChapterDOI
27 Oct 2013
TL;DR: By applying the agreement subtree mapping kernel to trimmed phylogenetic trees obtained from all the positions in nucleotide sequences for A (H1N1) influenza viruses, pandemic viruses from non-pandemic viruses and viruses in one region from viruses in the other regions are classified.
Abstract: In this paper, we introduce an agreement subtree mapping kernel counting all of the agreement subtree mappings and design the algorithm to compute it for phylogenetic trees, which are unordered leaf-labeled full binary trees, in quadratic time. Then, by applying the agreement subtree mapping kernel to trimmed phylogenetic trees obtained from all the positions in nucleotide sequences for A (H1N1) influenza viruses, we classify pandemic viruses from non-pandemic viruses and viruses in one region from viruses in the other regions. On the other hand, for leaf-labeled trees, we show that the problem of counting all of the agreement subtree mappings is #P-complete.

3 citations

References
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Book
01 Jan 1990
TL;DR: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.
Abstract: From the Publisher: The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Like the first edition,this text can also be used for self-study by technical professionals since it discusses engineering issues in algorithm design as well as the mathematical aspects. In its new edition,Introduction to Algorithms continues to provide a comprehensive introduction to the modern study of algorithms. The revision has been updated to reflect changes in the years since the book's original publication. New chapters on the role of algorithms in computing and on probabilistic analysis and randomized algorithms have been included. Sections throughout the book have been rewritten for increased clarity,and material has been added wherever a fuller explanation has seemed useful or new information warrants expanded coverage. As in the classic first edition,this new edition of Introduction to Algorithms presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers. Further,the algorithms are presented in pseudocode to make the book easily accessible to students from all programming language backgrounds. Each chapter presents an algorithm,a design technique,an application area,or a related topic. The chapters are not dependent on one another,so the instructor can organize his or her use of the book in the way that best suits the course's needs. Additionally,the new edition offers a 25% increase over the first edition in the number of problems,giving the book 155 problems and over 900 exercises thatreinforcethe concepts the students are learning.

21,651 citations

Journal ArticleDOI
TL;DR: This paper presents algorithms for the assignment problem, the transportation problem, and the minimum- cost flow problem of operations research that find a minimum-cost solution, yet run in time close to the best-known bounds for the corresponding problems without costs.
Abstract: This paper presents algorithms for the assignment problem, the transportation problem, and the minimum-cost flow problem of operations research. The algorithms find a minimum-cost solution, yet run in time close to the best-known bounds for the corresponding problems without costs. For example, the assignment problem (equivalently, minimum-cost matching in a bipartite graph) can be solved in $O(\sqrt {nm} \log (nN))$ time, where $n,m$, and N denote the number of vertices, number of edges, and largest magnitude of a cost; costs are assumed to be integral. The algorithms work by scaling. As in the work of Goldberg and Tarjan, in each scaled problem an approximate optimum solution is found, rather than an exact optimum.

457 citations

Journal ArticleDOI
TL;DR: This paper presents another approach to the problem of comparing many secondary structures by utilizing a very efficient tree-matching algorithm that will compare two trees in O([T1] X [T2] X L1 X L2) in the worst case and very close to O[T1?] for average trees representing secondary structures.
Abstract: In a previous paper, an algorithm was presented for analyzing multiple RNA secondary structures utilizing a multiple string alignment algorithm. In this paper we present another approach to the problem of comparing many secondary structures by utilizing a very efficient tree-matching algorithm that will compare two trees in O([T1] X [T2] X L1 X L2) in the worst case and very close to O([T1] X [T2]) for average trees representing secondary structures. The result of the pairwise comparison algorithm is then used with a cluster algorithm to produce a multiple structure clustering which can be displayed in a taxonomy tree to show related structures.

346 citations

Journal ArticleDOI
TL;DR: The tree obtained by regrafting branches on to a largest common pruned tree is shown to contain all the classes present in the strict consensus tree.
Abstract: Given two or more dendrograms (rooted tree diagrams) based on the same set of objects, ways are presented of defining and obtaining common pruned trees. Bounds on the size of a largest common pruned tree are introduced, as is a categorization of objects according to whether they belong to all, some, or no largest common pruned trees. Also described is a procedure for regrafting pruned branches, yielding trees for which one can assess the reliability of the depicted relationships. The tree obtained by regrafting branches on to a largest common pruned tree is shown to contain all the classes present in the strict consensus tree. The theory is illustrated by application to two classifications of a set of forty-nine stratigraphical pollen spectra.

221 citations