Optimal algorithms for comparing trees with labeled leaves

LetR _n denote the set of rooted trees withn leaves in which: the leaves are labeled by the integers in {1, ...,n}; and among interior vertices only the root may have degree two. Associated with each interior vertexv in such a tree is the subset, orcluster, of leaf labels in the subtree rooted atv. Cluster {1, ...,n} is calledtrivial. Clusters are used in quantitative measures of similarity, dissimilarity and consensus among trees. For anyk trees inR _n , thestrict consensus tree C(T ₁, ...,T _k ) is that tree inR _n containing exactly those clusters common to every one of thek trees. Similarity between treesT ₁ andT ₂ inR _n is measured by the numberS(T ₁,T ₂) of nontrivial clusters in bothT ₁ andT ₂; dissimilarity, by the numberD(T ₁,T ₂) of clusters inT ₁ orT ₂ but not in both. Algorithms are known to computeC(T ₁, ...,T _k ) inO(kn ²) time, andS(T ₁,T ₂) andD(T ₁,T ₂) inO(n ²) time. I propose a special representation of the clusters of any treeT R _n , one that permits testing in constant time whether a given cluster exists inT. I describe algorithms that exploit this representation to computeC(T ₁, ...,T _k ) inO(kn) time, andS(T ₁,T ₂) andD(T ₁,T ₂) inO(_n) time. These algorithms are optimal in a technical sense. They enable well-known indices of consensus between two trees to be computed inO(n) time. All these results apply as well to comparable problems involving unrooted trees with labeled leaves.The Natural Sciences and Engineering Research Council of Canada partially supported this work with grant A-4142.     

Two Earth-sized planets orbiting Kepler-20

Since the discovery of the first extrasolar giant planets around Sun-like stars, evolving observational capabilities have brought us closer to the detection of true Earth analogues. The size of an exoplanet can be determined when it periodically passes in front of (transits) its parent star, causing a decrease in starlight proportional to its radius. The smallest exoplanet hitherto discovered has a radius 1.42 times that of the Earth's radius (R(⊕)), and hence has 2.9 times its volume. Here we report the discovery of two planets, one Earth-sized (1.03R(⊕)) and the other smaller than the Earth (0.87R(⊕)), orbiting the star Kepler-20, which is already known to host three other, larger, transiting planets. The gravitational pull of the new planets on the parent star is too small to measure with current instrumentation. We apply a statistical method to show that the likelihood of the planetary interpretation of the transit signals is more than three orders of magnitude larger than that of the alternative hypothesis that the signals result from an eclipsing binary star. Theoretical considerations imply that these planets are rocky, with a composition of iron and silicate. The outer planet could have developed a thick water vapour atmosphere.     

Transiting circumbinary planets Kepler-34 b and Kepler-35 b

Most Sun-like stars in the Galaxy reside in gravitationally bound pairs of stars (binaries). Although long anticipated, the existence of a 'circumbinary planet' orbiting such a pair of normal stars was not definitively established until the discovery of the planet transiting (that is, passing in front of) Kepler-16. Questions remained, however, about the prevalence of circumbinary planets and their range of orbital and physical properties. Here we report two additional transiting circumbinary planets: Kepler-34 (AB)b and Kepler-35 (AB)b, referred to here as Kepler-34 b and Kepler-35 b, respectively. Each is a low-density gas-giant planet on an orbit closely aligned with that of its parent stars. Kepler-34 b orbits two Sun-like stars every 289?days, whereas Kepler-35 b orbits a pair of smaller stars (89% and 81% of the Sun's mass) every 131?days. The planets experience large multi-periodic variations in incident stellar radiation arising from the orbital motion of the stars. The observed rate of circumbinary planets in our sample implies that more than ～1% of close binary stars have giant planets in nearly coplanar orbits, yielding a Galactic population of at least several million.     

Guest Editorial:Special Issue on Visualization and Computer Graphics

In the decade of data explosion, a significant challenge is how to transform data into understandings and insights that are useful to people. Visualization,the use of computer-supported, interactive visual     

Broad phylogenomic sampling improves resolution of the animal tree of life

Long-held ideas regarding the evolutionary relationships among animals have recently been upended by sometimes controversial hypotheses based largely on insights from molecular data. These new hypotheses include a clade of moulting animals (Ecdysozoa) and the close relationship of the lophophorates to molluscs and annelids (Lophotrochozoa). Many relationships remain disputed, including those that are required to polarize key features of character evolution, and support for deep nodes is often low. Phylogenomic approaches, which use data from many genes, have shown promise for resolving deep animal relationships, but are hindered by a lack of data from many important groups. Here we report a total of 39.9 Mb of expressed sequence tags from 29 animals belonging to 21 phyla, including 11 phyla previously lacking genomic or expressed-sequence-tag data. Analysed in combination with existing sequences, our data reinforce several previously identified clades that split deeply in the animal tree (including Protostomia, Ecdysozoa and Lophotrochozoa), unambiguously resolve multiple long-standing issues for which there was strong conflicting support in earlier studies with less data (such as velvet worms rather than tardigrades as the sister group of arthropods), and provide molecular support for the monophyly of molluscs, a group long recognized by morphologists. In addition, we find strong support for several new hypotheses. These include a clade that unites annelids (including sipunculans and echiurans) with nemerteans, phoronids and brachiopods, molluscs as sister to that assemblage, and the placement of ctenophores as the earliest diverging extant multicellular animals. A single origin of spiral cleavage (with subsequent losses) is inferred from well-supported nodes. Many relationships between a stable subset of taxa find strong support, and a diminishing number of lineages remain recalcitrant to placement on the tree.     

膨胀的多重宇宙

安东尼·阿圭雷博士(基础问题研究所科学副总监)在过去的十年中,宇宙学领域内发生了两件令人震惊的大事:一是宇宙学家们建立了一个真实的宇宙学"标准模型".通过结合一系列相对简单的对早期宇宙的猜想和已经证明的引力、核子、等离子和天体物理学知识,该理论能够适用于几乎所有现存的观测数据,并可重复测试和相互校验该模型的基本构成.二是科学家们意识到,通过宇宙膨胀--这一理论原本是为了解释构建"标准模型"所需的宇宙早期情况而提出的--目前所观测到的宇宙可能仅仅是一个巨大、复杂、多元的多重宇宙的"沧海一瞥",毫不足奇.多重宇宙的结构如何?我们所处的宇宙与其他宇宙关系如何?如果根据该理论推导出来诸多属性不一的宇宙,我们将使用哪些属性来与所能观测到的宇宙相互印证,以证明该理论的真伪?这些就是本文中两个主题人物所提出的一部分问题.亚历山大·维兰金(Alexander Vilenkin)是膨胀理论的先驱之一,亦是多重宇宙研究领域的顶尖科学家.理查·伊瑟(Richard Easther)则是一位致力于研究多重宇宙理论中一个最为微妙的方面的年轻科学家.他的研究方向是:如何在清点多重宇宙中的宇宙?