Abstract Evolutionary trees describing the relationship for a set of species are central in evolutionarybiology, and quantifying differences between evolutionary trees is therefore an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook,McMorris and Meacham. The quartet distance between two unrooted evolutionary trees is the number of quartet topology differences between the two trees, where a quartet topologyis the topological subtree induced by four species. In this paper, we present an algorithm for computing the quartet distance between two unrooted evolutionary trees of n species, whereall internal nodes have degree three, in time O(n log n). The previous best algorithm for theproblem uses time O(n2).

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Reconstructing phylogenetic trees efficiently and accurately from distance estimates is an ongoing challenge in computational biology from both practical and theoretical considerations. We study algorithms which are based on a characterization of edge-weighted trees by distances to LCAs (Least Common Ancestors). This characterization enables a direct application of ultrametric reconstruction techniques to trees which are not necessarily ultrametric. A simple and natural neighbor joining criterion based on this observation is used to provide a family of efficient neighbor-joining algorithms. These algorithms are shown to reconstruct a refinement of the Buneman tree, which implies optimal robustness to noise under criteria defined by Atteson. In this sense, they outperform many popular algorithms such as Saitou&Nei’s NJ. One member of this family is used to provide a new simple version of the 3-approximation algorithm for the closest additive metric under the l ∞ norm. A byproduct of our work is a novel technique 1 which yields a time optimal O(n 2) implementation of common clustering algorithms such as UPGMA. 1

Culberson and Rudnicki [1] gave an algorithm that reconstructs a degree d restricted tree from its distance matrix. According to their analysis, it runs in time O(dn log d n) for topological trees. However, this turns out to be false; we show that the algorithm takes Θ(n 3/2 √ d) time in the topological case, giving tight examples. Key words: Analysis of algorithms, graph algorithms 1

In this report, we explore various aspects of query learning. We focus on learning hidden structures given various queries. In Chapter 1, we consider learning evolutionary trees given distance queries. In Chapter 2 we focus on learning and verifying general graph structures with various queries. In Chapter 3 we are interested in learning circuits with value-injection queries. Chapter 1 is based on a paper coauthored with Nikhil Srivastava, entitled “On the Longest Path Algorithm for Reconstructing Trees from Distance Matrices.” This

www.elsevier.com/locate/ipl On the longest path algorithm for reconstructing trees from distance matrices

From molecular arrangements to biological organisms, our world is composed of systems of small components interacting with and affecting each other. Scientists often learn the structure of such systems by tampering with them and making observations. In this thesis, we develop methods for automating this process from an active learning perspective, a setting where the learner is not restricted to making passive observations, but can choose to query the data. First, we consider the setting of learning hidden graphs with queries. Each query type is motivated by a real-world problem, from genome sequencing to evolutionary tree reconstruction. We give new algorithms for learning graphs and also consider the problem of verifying the results of the learning task. Next, we turn to value injection queries, which model experiments used to identify gene regulatory networks. We analyze the complexity of learning large alphabet and analog circuits with value injection queries. We then apply this model to social networks, allowing the learner to activate and suppress agents in the network, and we give an optimal algorithm and matching lower bound for this problem. Finally, we examine the passive learner, who watches the output of agents in a social network and must deduce the most likely underlying network. Last, we consider a classical problem in query learning: learning finite automata, which themselves are networks of connected states. We introduce label queries as a generalization of the well studied membership queries. We give algorithms for learning automata using label queries and analyze other models for learning automata.