Showing papers by "Martin Farach published in 1999"

On the Approximability of Numerical Taxonomy (Fitting Distances by Tree Metrics)

Abstract: We consider the problem of fitting an n × n distance matrix D by a tree metric T. Let $\varepsilon$ be the distance to the closest tree metric under the $L_{\infty}$ norm; that is, $\varepsilon=\min_T\{\parallel T-D\parallel{\infty}\}$. First we present an O(n2) algorithm for finding a tree metric T such that $\parallel T-D\parallel{\infty}\leq 3\varepsilon$. Second we show that it is ${\cal NP}$-hard to find a tree metric T such that $\parallel T-D\parallel{\infty}<\frac{9}{8}\varepsilon$. This paper presents the first algorithm for this problem with a performance guarantee.

Efficient algorithms for inverting evolution

Abstract: Evolution can be mathematically modelled by a stochastic process that operates on the DNA of species. Such models are based on the established theory that the DNA sequences, or genomes, of all extant species have been derived from the genome of the common ancestor of all species by a process of random mutation and natural selection.A stochastic model of evolution can be used to construct phylogenies, or evolutionary trees, for a set of species. Maximum Likelihood Estimation (MLE) methods seek the evolutionary tree which is most likely to have produced the DNA under consideration. While these methods are intellectually satisfying, they have not been widely accepted because of their computational intractability.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 stochastic models that are close according to this metric, it needs to be given many observations. We complement this result with a simple and efficient algorithm for inverting the stochastic process of evolution, that is, for building a tree from observations on two-state characters. (We will use the same techniques in a subsequent paper to solve the problem for multistate characters, and hence for building a tree from DNA sequence data.) The tree we build is provably close, in our metric, to the tree generating the data and gets closer as more observations become available.Though there have been many heuristics suggested for the problem of finding good approximations to the most likely tree, 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 first polynomial-time algorithm that is proven to converge at all to the correct tree.