Above are papers related to the following paper: [Paper_ID] Paper #1135750. 	[Title] TREE CONTRACTIONS AND EVOLUTIONARY TREES [Abstract] An evolutionary tree is a rooted tree where each internal vertex has at least two children and where the leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary history of species. An agreement subtree of two evolutionary trees is an evolutionary tree which is also a topological subtree of the two given trees. We give an algorithm to determine the largest possible number of leaves in any agreement subtree of two trees T 1 and T 2 with n leaves each. If the maximum degree d of these trees is bounded by a constant, the time complexity is O(n log 2 n) and is within a log n factor of optimal. For general d, this algorithm runs in O(nd 2 log d log 2 n) time or alternately in O(nd p d log 3 n) time. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1138755. 	[Title] Design of Optimization Criteria for Multiple Sequence Alignment [Abstract] DIMACS Technical Report 96-53 January 1997 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1129608. 	[Title] A New Look at Tree Models for Multiple Sequence Alignment [Abstract] Evolutionary trees are frequently used as the underlying model in the design of algorithms, optimization criteria and software packages for multiple sequence alignment (MSA). In this paper, we reexamine the suitability of trees as a universal model for MSA in light of the broad range of biological questions that MSA's are used to address. A tree model consists of a tree topology and a model of accepted mutations along the branches. After surveying the major applications of MSA, examples from the molecular biology literature are used to illustrate situations in which this tree model fails. This occurs when the relationship between residues in a column cannot be described by a tree; for example, in some structural and functional applications of MSA. It also occurs in situations, such as lateral gene transfer, where an entire gene cannot be modeled by a unique tree. In cases of nonparsimonous data or convergent evolution, it may be difficult to find a consistent mutational model. We hope that this survey will promote dialogue between biologists and computer scientists, leading to more biologically realistic research on MSA. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1110256. 	[Title] Learning Distributions from Random Walks [Abstract] We introduce a new model of distributions generated by random walks on graphs. This model suggests a variety of learning problems, using the definitions and models of distribution learning defined in [6]. Our framework is general enough to model previously studied distribution learning problems, as well as to suggest new applications. We describe special cases of the general problem, and investigate their relative difficulty. We present algorithms to solve the learning problem under various conditions. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #573553. 	[Title] On the Approximability of Numerical Taxonomy (Fitting Distances by Tree Metrics) [Abstract] We consider the problem of fitting an n fi n distance matrix D by a tree metric T . Let " be the distance to the closest tree metric, that is, " = min T fk T; D k 1 g. First we present an O(n 2 ) algorithm for finding an additive tree T such that k T; D k 1 3", giving the first algorithm for this problem with a performance guarantee. Second we show that it is N P-hard to find a tree T such that k T; D k 1 &lt; 9 Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #171954. 	[Title] Efficient Algorithms for Inverting Evolution [Abstract] Evolution is a stochastic process which operates on the DNA of species. The evolutionary process leaves tell-tale signs in the DNA which can be used to construct phylogenies, or evolutionary trees, for a set of species. Maximum Likelihood Estimations (MLE) methods seek the evolutionary tree which is most likely to have produced the DNA under consideration. While these methods are widely accepted and intellectually satisfying, they have been computationally intractable. In this paper, we address the intractability of MLE methods as follows. We introduce a metric on stochastic process models of evolution. We show that this metric is meaningful by proving that in order for any algorithm to distinguish between two stochatic models that are close according to this metric, it needs to be given a lot of observations. We complement this result with a simple and efficient algorithm for inverting the stochastic process of evolution, that is, for building the tree from observations on the DNA of the species. Our result can be viewed as a result on the PAC-learnability of the class of distributions produced by tree-like processes. Though there have been many heuristics suggested for this problem, our algorithm is the first one with a guaranteed convergence rate, and further, this rate is within a polynomial of the lower-bound rate we establish. Ours is also the the first polynomial-time algorithm which is guaranteed to converge at all to the correct tree. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #693143. 	[Title] A Fast Algorithm for the Computation and Enumeration of Perfect Phylogenies [Abstract] The Perfect Phylogeny Problem is a classical problem in computational evolutionary biology, in which a set of species/taxa is described by a set of qualitative characters. In recent years, the problem has been shown to be NP-Complete in general, while the different fixed parameter versions can each be solved in polynomial time. In particular, Agarwala and Fernandez-Baca have developed an O(2 3r (nk 3 +k 4 )) algorithm for the perfect phylogeny problem for n species defined by k r-state characters. Since commonly the character data is drawn from alignments of molecular sequences, k is the length of the sequences and can thus be very large (in the hundreds or thousands). Thus, it is imperative to develop algorithms which run efficiently for large values of k. In this paper we make additional observations about the structure of the problem and produce an algorithm for the problem that runs in time O(2 2r k 2 n). We also show how it is possible to efficiently build a structure that implicitly represents the set of all perfect phylogenies, and to randomly sample from that set. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #695284. 	[Title] The Hardness of Problems on Thin Colored Graphs [Abstract] In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded treewidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in "yes"-instances. For all of these problems with the exceptions of feasible register assignment and module allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W [t] for all t 2 Z + . We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #820662. 	[Title] Constructing Big Trees from Short Sequences [Abstract] The construction of evolutionary trees is a fundamental problem in biology, and yet methods for reconstructing evolutionary trees are not reliable when it comes to inferring accurate topologies of large divergent evolutionary trees from realistic length sequences. We address this problem and present a new polynomial time algorithm for reconstructing evolutionary trees called the Short Quartets Method which is consistent and which has greater statistical power than other polynomial time methods, such as Neighbor-Joining and the 3-approximation algorithm by Agarwala et al. (and the "Double Pivot" variant of the Agarwala et al. algorithm by Cohen and Farach) for the L 1 -nearest tree problem. Our study indicates that our method will produce the correct topology from shorter sequences than can be guaranteed using these other methods. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #67415. 	[Title] On the Learnability of Discrete Distributions (extended abstract) [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1138970. 	[Title] The Disk-Covering Method for Tree Reconstruction [Abstract] Evolutionary tree reconstruction is a very important step in many biological research problems, and yet is extremely difficult for a variety of computational, statistical, and scientific reasons. In particular, the reconstruction of very large trees containing significant amounts of divergence is especially challenging. We present in this paper a new tree reconstruction method, which we call the Disk-Covering Method, which can be used to recover accurate estimations of the evolutionary tree for otherwise intractable datasets. DCM obtains a decomposition of the input dataset into small overlapping sets of closely related taxa, reconstructs trees on these subsets (using a "base" phylogenetic method of choice), and then combines the subtrees into one tree on the entire set of taxa. Because the subproblems analyzed by DCM are smaller, com-putationally expensive methods such as maximum likelihood estimation can be used without incurring too much cost. At the same time, because the taxa within each subset are closely related, even very simple methods (such as neighbor-joining) are much more likely to be highly accurate. The result is that DCM-boosted methods are typically faster and more accurate as compared to "naive" use of the same method. In this paper we describe the basic ideas and techniques in DCM, and demonstrate the advantages of DCM experimentally by simulating sequence evolution on a variety of trees. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1138043. 	[Title] Inserting the best known bounds for weighted bipar tite matching [11], with 1=2 p polynomial-time [Abstract] we apply the reduction to their two core children, the total sum of their matching weights becomes O(n), and if for each comparison of a spine node and a critical node we apply the reduction to the core child of the spine node, the total sum of their matching weights becomes O(n). With regards to the O( 2 ) comparisons of two critical nodes, their sum cannot exceed O( 2 n) in total weight. Thus, since we have a total of O(n) edges involved in the matchings, in time O(n), we can reduce the total sum of the matching weights to O( 2 n). Theorem 6.6 Let M : IR 1 ! IR be a monotone function bounding the time complexity UWBM. Moreover, let M satisfy that M (x) = x 1+" f(x), where " 0 is a constant, f (x) = O(x o(1) ), f is monotone, and for some constants b 1 ; b 2 , 8x; y b 1 : f(xy) b 2 f (x)f (y). Then, with = " p in time O(n 1+o(1) + M (n)). Proof: We spend O(n polylog n + time(UWBM(k 2 n))) on the matchings. So, by Theorem 5.10, we have that Comp-Core-Trees can be computed in time O(n polylog n + time(UWBM(k 2 n))). Applying Theo 4 = 16, we get Corollary 6.7 MAST is computable in time O(n 1:5 log n). [4] W.H.E. Day. Computational complexity of inferring phylogenies from dissimilarity matrices. Bulletin of Mathematical Biology, 49(4):461-467, 1987. [5] M. Farach, S. Kannan, and T. Warnow. A robust model for finding optimal evolutionary trees. Al-gorithmica, 1994. In press. See also STOC '93. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #573535. 	[Title] Fast and Simple Algorithms for Perfect Phylogeny and Triangulating Colored Graphs [Abstract] This paper presents an O((r n=m) m rnm) algorithm for determining whether a set of n species has a perfect phylogeny, where m is the number of characters used to describe a species and r is the maximum number of states that a character can be in. The perfect phylogeny algorithm leads to an O((2e=k) k e 2 k) algorithm for triangulating a k-colored graph having e edges. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #66751. 	[Title] A Faster Algorithm for the Perfect Phylogeny Problem when the number of Characters is Fixed TR94-05 [Abstract]  Please summarise all the given information.

Paper #1135750 has the title `TREE CONTRACTIONS AND EVOLUTIONARY TREES  
` and the abstract `An evolutionary tree is a rooted tree where each internal vertex has at least two children and where the leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary history of species. An agreement subtree of two evolutionary trees is an evolutionary tree which is also a topological subtree of the two given trees. We give an algorithm to determine the largest possible number of leaves in any agreement subtree of two trees T 1 and T 2 with n leaves each. If the maximum degree d of these trees is bounded by a constant, the time complexity is O(n log 2 n) and is within a log n factor of optimal. For general d, this algorithm runs in O(nd 2 log d log 2 n) time or alternately in O(nd p d log 3 n) time. 
`.
Question: Which category should Paper #1135750 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #1135750 has the title `TREE CONTRACTIONS AND EVOLUTIONARY TREES  
` and the abstract `An evolutionary tree is a rooted tree where each internal vertex has at least two children and where the leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary history of species. An agreement subtree of two evolutionary trees is an evolutionary tree which is also a topological subtree of the two given trees. We give an algorithm to determine the largest possible number of leaves in any agreement subtree of two trees T 1 and T 2 with n leaves each. If the maximum degree d of these trees is bounded by a constant, the time complexity is O(n log 2 n) and is within a log n factor of optimal. For general d, this algorithm runs in O(nd 2 log d log 2 n) time or alternately in O(nd p d log 3 n) time. 
`.
Question: Which category should Paper #1135750 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Neural_Networks