![]() ![]() Therefore, for two parameters, our recursive approach to solving the parametric problem yields an O( Kmn) algorithm, matching the running time of xparal and the conjecture of Waterman et al. This procedure is a primitive of the divide-and-conquer approach to convex hull computation, and there is a well known O( K) algorithm for solving it (ref. However, this running time can be improved by observing that the convex hull computations that need to be carried out have a very special form namely, in each step of the algorithm, we need to compute the convex hull of two superimposed convex polygons. Therefore, each convex hull computation requires at most O( Klog( K)) operations, thus giving an O( nmKlog( K)) algorithm for solving the parametric alignment problem. The number of points in each computation is bounded by the total number of points in the final convex hull (or equivalently, the number, K, of explanations). The polytope propagation algorithm has the same running time as xparal: for two sequences of length n and m, the method requires O( nm) two-dimensional convex hull computations. xparal will return four cones however, a computation of the Newton polytope reveals seven vertices (three of which correspond to positive mis or gap values). Consider the example of the following two sequences: σ 1 = AGGACCGATTACAGTTCAA and σ 2 = TTCCTAGGTTAAACCTCATGCA. 14, section 6.3).Įfficient software for parametrically aligning the sequences with two free parameters exists ( xparal ref. ![]() These coefficients are known as Delannoy numbers in combinatorics (ref. This book is the first of its kind to provide a large collection of bioinformatics problems with. Formally, given two sequences \(\begin\) of all alignments can be computed as the coefficient of x my n in the generating function 1/(1 - x - y - xy). Problems and Solutions in Biological Sequence Analysis. The sequence alignment problem is concerned with finding the best alignment between two sequences that have evolved from a common ancestor by means of a series of mutations, insertions, and deletions. Although it is exponential in the number of parameters, it is polynomial in the size of the graphical model. The algorithm is a geometric version of the sum-product algorithm, which is the standard tool for inference with graphical models. In certain cases, it is not much slower than solving problem 2 for fixed parameters. In thesis iii, we claim that the polytope propagation algorithm is efficient for solving the parametric inference problem for a small number of parameters. Our aim is to develop this approach for arbitrary graphical models. This approach has been applied successfully to the problem of pairwise sequence alignment in which parameter choices are known to be crucial in obtaining good alignments ( 4- 6). In this way, we can decide whether a solution obtained for particular parameters is an artifact or if it is largely independent of the chosen parameters. By parametric inference, we mean the solution of problem 2 for all model parameters simultaneously. Thesis ii suggests that a parametric solution to the inference problem can help in ascertaining the reliability, robustness, and biological meaning of an inference result.
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