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Extending Partial Representations of Interval Graphs

algorithmICA 2017年第3期78卷 945-967页

作者： Klavik, Pavel Kratochvil, Jan Otachi, Yota Saitoh, Toshiki Vyskoil, Tomas Charles Univ Prague Fac Math & Phys Comp Sci Inst Malostranske Namesti 25 Prague 11800 Czech Republic Charles Univ Prague Fac Math & Phys Dept Appl Math Malostranske Namesti 25 Prague 11800 Czech Republic Japan Adv Inst Sci & Technol Sch Informat Sci Asahidai 1-1 Nomi Ishikawa 9231292 Japan Kobe Univ Grad Sch Engn Nada Ku Rokkodai 1-1 Kobe Hyogo 6578501 Japan Rutgers State Univ NewJersey Dept Comp Sci 110 Frelinghuysen Rd Piscataway NJ 08854 USA

Interval graphs are intersection graphs of closed intervals of the real-line. The well-known computational problem, called recognition, asks whether an input graph G can be represented by closed intervals, i.e., whether G is an interval graph. There are several linear-time algorithms known for recognizing interval graphs, the oldest one is by Booth and Lueker (J Comput Syst Sci 13: 335-379, 1976) based on PQ-trees. In this paper, we study a generalization of recognition, called partial representation extension. The input of this problem consists of a graph G with a partial representation R' fixing the positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation R of the entire graph G extending R'. We generalize the characterization of interval graphs by Fulkerson and Gross (Pac J Math 15: 835-855, 1965) to extendible partial representations. Using it, we give a linear-time algorithm for partial representation extension based on a reordering problem of PQ-trees.

关键词： Interval graphs Partial representation extension PQ-trees linear-time algorithm

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Rphylopars: fast multivariate phylogenetic comparative methods for missing data and within-species variation

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METHODS IN ECOLOGY AND EVOLUTION 2017年第1期8卷 22-27页

作者： Goolsby, Eric W. Bruggeman, Jorn Ane, Cecile Harvard Univ Arnold Arboretum Boston MA 02131 USA Univ Georgia Interdisciplinary Toxicol Program Dept Plant Biol Athens GA 30602 USA Plymouth Marine Lab Prospect Pl Plymouth PL1 3DH Devon England Univ Wisconsin Madison Dept Stat 1300 Univ Ave Madison WI 53706 USA Univ Wisconsin Madison Dept Bot Madison WI 53706 USA

Over the past several years, phylogenetic comparative studies have increasingly approached trait evolution in a multivariate context, with a number of taxa that continues to rise dramatically. Recent methods for phylogenetic comparative studies have provided ways to incorporate measurement error and to address computational challenges. However, missing data remain a particularly common problem, in which data are unavailable for some but not all traits of interest for a given species (or individual), leaving researchers with the choice between omitting observations or utilizing imputation-based approaches. Here, we introduce an r implementation of PhyloPars, a tool for phylogenetic imputation of missing data and estimation of trait covariance across species (phylogenetic covariance) and within species (phenotypic covariance). Rphylopars provides expanded capabilities over the original PhyloPars interface including a fast linear-time algorithm, thus allowing for extremely large data sets (which were previously computationally infeasible) to be analysed in seconds or minutes rather than hours. In addition to providing fast and computationally efficient implementations, we introduce in Rphylopars methods to estimate macroevolutionary parameters under alternative evolutionary models (e.g. Early-Burst, multivariate Ornstein-Uhlenbeck). By providing fast and computationally efficient methods with flexible options for various phylogenetic comparative approaches, Rphylopars expands the possibilities for researchers to analyse large and complex data with missing observations, within-species variation and deviations from Brownian motion.

关键词： fast methods linear-time algorithm missing data multivariate Ornstein-Uhlenbeck phylogenetic comparative method phylogenetic generalized least squares phylogenetic imputation

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Rapid maximum likelihood ancestral state reconstruction of continuous characters: A rerooting-free algorithm

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ECOLOGY AND EVOLUTION 2017年第8期7卷 2791-2797页

作者： Goolsby, Eric W. Brown Univ Dept Ecol & Evolutionary Biol Providence RI 02912 USA

Ancestral state reconstruction is a method used to study the evolutionary trajectories of quantitative characters on phylogenies. Although efficient methods for univariate ancestral state reconstruction under a Brownian motion model have been described for at least 25years, to date no generalization has been described to allow more complex evolutionary models, such as multivariate trait evolution, non-Brownian models, missing data, and within-species variation. Furthermore, even for simple univariate Brownian motion models, most phylogenetic comparative R packages compute ancestral states via inefficient tree rerooting and full tree traversals at each tree node, making ancestral state reconstruction extremely time-consuming for large phylogenies. Here, a computationally efficient method for fast maximum likelihood ancestral state reconstruction of continuous characters is described. The algorithm has linear complexity relative to the number of species and outperforms the fastest existing R implementations by several orders of magnitude. The described algorithm is capable of performing ancestral state reconstruction on a 1,000,000-species phylogeny in fewer than 2s using a standard laptop, whereas the next fastest R implementation would take several days to complete. The method is generalizable to more complex evolutionary models, such as phylogenetic regression, within-species variation, non-Brownian evolutionary models, and multivariate trait evolution. Because this method enables fast repeated computations on phylogenies of virtually any size, implementation of the described algorithm can drastically alleviate the computational burden of many otherwise prohibitively time-consuming tasks requiring reconstruction of ancestral states, such as phylogenetic imputation of missing data, bootstrapping procedures, Expectation-Maximization algorithms, and Bayesian estimation. The described ancestral state reconstruction algorithm is implemented in the Rphylopars functions a

关键词： ancestral state reconstruction fast methods linear-time algorithm maximum likelihood phylogenetic comparative method phylogenetic generalized least squares

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linear-time version of Holub's algorithm for morphic imprimitivity testing

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THEORETICAL COMPUTER SCIENCE 2015年 602卷 7-21页

作者： Kociumaka, T. Radoszewski, J. Rytter, W. Walen, T. Univ Warsaw Fac Math Informat & Mech PL-02097 Warsaw Poland

Stepan Holub (Discr. Math., 2009) gave the first polynomial-time algorithm deciding whether a given word is a nontrivial fixed point of a morphism. His algorithm works in quadratic time for large alphabets. We improve the algorithm to work in linear time. Our improvement starts with a careful choice of a subset of rules used in Holub's algorithm that is necessary to grant correctness of the algorithm. Afterwards we show how to choose the order of applying the rules that allows to avoid unnecessary operations on sets. We obtain linear time using efficient data structures for implementation of the rules. Holub's algorithm maintains connected components of a graph corresponding to specially marked positions in a word. This graph is of quadratic size for large alphabet. In our algorithm only a linear number of edges of this conceptual graph is processed. A preliminary version of this paper appeared at LATA 2013 conference [11]. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Word morphism Morphical primitivity linear-time algorithm

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Induced disjoint paths in circular-arc graphs in linear time

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THEORETICAL COMPUTER SCIENCE 2016年 640卷 70-83页

作者： Golovach, Petr A. Paulusma, Daniel van Leeuwen, Erik Jan Univ Bergen Dept Informat N-5020 Bergen Norway Univ Durham Sch Engn & Comp Sci Durham DH1 3HP England Max Planck Inst Informat Saarbrucken Germany

The INDUCED DISJOINT PATHS problem is to test whether an graph G on n vertices with k distinct pairs of vertices (S-i, t(i)) contains paths P-1, . . . , P-k such that P-i connects s(i) and t(i) for i = 1, . . . , k, and P-i and P-j have neither common vertices nor adjacent vertices (except perhaps their ends) for 1 <= i <= j <= k. We present a linear-time algorithm that solves INDUCED DISJOINT PATHS and finds the corresponding paths (if they exist) on circular arc graphs. For interval graphs, we exhibit a linear-time algorithm for the generalization of INDUCED DISJOINT PATHS where the pairs (s(i), t(i)) are not necessarily distinct. In both cases, if a representation of the graph is given, then the algorithms run in O(n + k) time. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Induced disjoint paths Circular-arc graphs linear-time algorithm

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linear-time Fitting of a k-Step Function 2nd

Linear-Time Fitting of a <i>k</i>-Step Function

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2nd Conference on algorithms and Discrete Applied Mathematics (CALDAM)

作者： Bhattacharya, Binay Das, Sandip Kameda, Tsunehiko Simon Fraser Univ Sch Comp Sci Burnaby BC V5A 1S6 Canada Indian Stat Inst Kolkata India

ISBN: (纸本)9783319292212;9783319292205

Given a set of n weighted points on the x-y plane, we want to find a step function consisting of k horizontal steps such that the maximum vertical weighted distance from any point to a step is minimized. We solve this problem in O(n) time when k is a constant. Our approach relies on the prune-and-search technique, and can be adapted to design similar linear time algorithms to solve the line-constrained k-center problem and the size-k histogram construction problem as well.

关键词： linear-time algorithm Step function fitting Weighted points Prune-and-search Anchored step function

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Extending Convex Partial Drawings of Graphs

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algorithmICA 2016年第1期76卷 47-67页

作者： Mchedlidze, Tamara Noellenburg, Martin Rutter, Ignaz KIT Karlsruhe Germany

Given a plane graph G (i.e., a planar graph with a fixed planar embedding and outer face) and a biconnected subgraph with a fixed planar straight-line convex drawing, we consider the question whether this drawing can be extended to a planar straight-line drawing of G. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, one can be computed in the same running time.

关键词： Extension of a partial drawing Fixed cycle Fixed inner face Convex shape Straight-line drawing linear-time algorithm

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Remarks on Privileged Words

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INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE 2016年第4期27卷 431-442页

作者： Forsyth, Michael Jayakumar, Amlesh Peltomaki, Jarkko Shallit, Jeffrey Univ Waterloo Sch Comp Sci Waterloo ON N2L 3G1 Canada Turku Ctr Comp Sci TUCS Turku 20520 Finland Univ Turku Dept Math & Stat Turku 20014 Finland

We discuss the notion of privileged word, recently introduced by Kellendonk, Lenz and Savinien. A word w is privileged if it is of length = 1, then w(j) is privileged for all j >= 0;(2) the language of privileged w... 详细信息

We discuss the notion of privileged word, recently introduced by Kellendonk, Lenz and Savinien. A word w is privileged if it is of length <= 1, or has a privileged border that occurs exactly twice in w. We prove the following results: (1) if w(k) is privileged for some k >= 1, then w(j) is privileged for all j >= 0;(2) the language of privileged words is not context-free;(3) there is a linear-time algorithm to check if a given word is privileged;and (4) there are at least 2(n-4)/n(2) privileged binary words of length n

关键词： Privileged word regular language linear-time algorithm context-free language enumeration

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linear-time graph distance and diameter approximation

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INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH 2016年第5期23卷 843-851页

作者： Machado, Raphael C. S. Figueiredo, Celina M. H. de Univ Fed Rio de Janeiro COPPE Rio de Janeiro Brazil

In this study, we consider the problem of estimating the diameter of a graph, that is, the maximum distance between any two vertices, in linear time. We address a question posed in the literature-whether there exists an interesting graph class with arbitrarily large cycles for which breadth first search (BFS) would always return high-eccentricity vertices. We answer this question positively, defining a class of graphs that generalizes AT-free graphs and has no bound on the size of induced cycles, yet BFS always returns a vertex whose eccentricity is within a constant difference from the diameter. In addition, we consider the question-also explicitly stated in the literature-whether some variant of the so-called multisweep algorithm would always return a high-eccentricity vertex. We show that the answer is negative by describing a family of graphs for which no variant of multisweep BFS can return a vertex of eccentricity higher than half of the diameter plus a constant.

关键词： graph diameter distance in graphs breadth first search linear-time algorithm

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Computing the directed Cartesian-product decomposition of a directed graph from its undirected decomposition in linear time

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DISCRETE MATHEMATICS 2015年第12期338卷 2393-2407页

作者： Crespelle, Christophe Thierry, Eric Univ Lyon 1 ENS Lyon DANTE INRIA LIP UMR CNRS 5668 F-69622 Villeurbanne France Univ Lyon ENS Lyon LIP UMR CNRS 5668 Lyon France

In this paper, we design an algorithm that, given a directed graph G and the Cartesian-product decomposition of its underlying undirected graph (G) over tilde, produces the directed Cartesian-product decomposition of G in linear time. This is the first time that the linear complexity is achieved for this problem, which has two major consequences. Firstly, it shows that the directed and undirected versions of the Cartesian-product decomposition of graphs are linear-time equivalent problems. And secondly, as there already exists a linear-time algorithm for solving the undirected version of the problem, combined together, it provides the first linear-time algorithm for computing the directed Cartesian-product decomposition of a directed graph. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Cartesian product Directed graphs linear-time algorithm Graph decompositions

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