Estimating the Maximum Hidden Vertex Set in Polygons

It is known that the MAXIMUM HIDDEN VERTEX SET problem on a given simple polygon is NP-hard [11], therefore we focused on the development of approximation algorithms to tackle it. We propose four strategies to solve this problem, the first two (based on greedy constructive search) are designed specifically to solve it, and the other two are based on the general metaheuristics Simulated Annealing and Genetic Algorithms. We conclude, through experimentation, that our best approximate algorithm is the one based on the Simulated Annealing metaheuristic. The solutions obtained with it are very satisfactory in the sense that they are always close to optimal (with an approximation ratio of 1.7, for arbitrary polygons; and with an approximation ratio of 1.5, for orthogonal polygons). We, also, conclude, that on average the maximum number of hidden vertices in a simple polygon (arbitrary or orthogonal) with n vertices is n4

Parametric inference of recombination in HIV genomes

Recombination is an important event in the evolution of HIV. It affects the global spread of the pandemic as well as evolutionary escape from host immune response and from drug therapy within single patients. Comprehensive computational methods are needed for detecting recombinant sequences in large databases, and for inferring the parental sequences. We present a hidden Markov model to annotate a query sequence as a recombinant of a given set of aligned sequences. Parametric inference is used to determine all optimal annotations for all parameters of the model. We show that the inferred annotations recover most features of established hand-curated annotations. Thus, parametric analysis of the hidden Markov model is feasible for HIV full-length genomes, and it improves the detection and annotation of recombinant forms. All computational results, reference alignments, and C++ source code are available at http://bio.math.berkeley.edu/recombination/.Comment: 20 pages, 5 figure

Minimum Vertex Guard problem for orthogonal polygons: a genetic approach

The problem of minimizing the number of guards placed on vertices needed to guard a given simple polygon (MINIMUM VERTEX GUARD problem) is NP-hard. This computational complexity opens two lines of investigation: the development of algorithms that determine approximate solutions and the determination of optimal solutions for special classes of simple polygons. In this paper we follow the first line of investigation proposing an approximation algorithm based on the general met heuristic Genetic Algorithms to solve the MINIMUM VERTEXGUARD problem

Optimizing the Minimum Vertex Guard Set on Simple Polygons via a Genetic Algorithm

The problem of minimizing the number of vertex-guards necessary to cover a given simple polygon (MINIMUM VERTEX GUARD (MVG) problem) is NP-hard. This computational complexity opens two lines of investigation: the development of algorithms that establish approximate solutions and the determination of optimal solutions for special classes of simple polygons. In this paper we follow the first line of investigation and propose an approximation algorithm based on general metaheuristic genetic algorithms to solve the MVG problem. Based on our algorithm, we conclude that on average the minimum number of vertex-guards needed to cover an arbitrary and an orthogonal polygon with n vertices is n / 6.38 and n / 6.40 , respectively. We also conclude that this result is very satisfactory in the sense that it is always close to optimal (with an approximation ratio of 2, for arbitrary polygons; and with an approximation ratio of 1.9, for orthogonal polygons)

Parametric Inference for Biological Sequence Analysis

One of the major successes in computational biology has been the unification, using the graphical model formalism, of a multitude of algorithms for annotating and comparing biological sequences. Graphical models that have been applied towards these problems include hidden Markov models for annotation, tree models for phylogenetics, and pair hidden Markov models for alignment. A single algorithm, the sum-product algorithm, solves many of the inference problems associated with different statistical models. This paper introduces the \emph{polytope propagation algorithm} for computing the Newton polytope of an observation from a graphical model. This algorithm is a geometric version of the sum-product algorithm and is used to analyze the parametric behavior of maximum a posteriori inference calculations for graphical models.Comment: 15 pages, 4 figures. See also companion paper "Tropical Geometry of Statistical Models" (q-bio.QM/0311009

Parametric Alignment of Drosophila Genomes

The classic algorithms of Needleman--Wunsch and Smith--Waterman find a maximum a posteriori probability alignment for a pair hidden Markov model (PHMM). In order to process large genomes that have undergone complex genome rearrangements, almost all existing whole genome alignment methods apply fast heuristics to divide genomes into small pieces which are suitable for Needleman--Wunsch alignment. In these alignment methods, it is standard practice to fix the parameters and to produce a single alignment for subsequent analysis by biologists. Our main result is the construction of a whole genome parametric alignment of Drosophila melanogaster and Drosophila pseudoobscura. Parametric alignment resolves the issue of robustness to changes in parameters by finding all optimal alignments for all possible parameters in a PHMM. Our alignment draws on existing heuristics for dividing whole genomes into small pieces for alignment, and it relies on advances we have made in computing convex polytopes that allow us to parametrically align non-coding regions using biologically realistic models. We demonstrate the utility of our parametric alignment for biological inference by showing that cis-regulatory elements are more conserved between Drosophila melanogaster and Drosophila pseudoobscura than previously thought. We also show how whole genome parametric alignment can be used to quantitatively assess the dependence of branch length estimates on alignment parameters. The alignment polytopes, software, and supplementary material can be downloaded at http://bio.math.berkeley.edu/parametric/.Comment: 19 pages, 3 figure

Escondiendo puntos en espirales e histogramas

El problema de maximizar el número de vértices que no son visibles dos a dos en un polígono simple P, (MAXIMUN HIDDEN VERTEX SET) es un problema NP-duro [6]. En este trabajo se resuelve el problema para dos tipos de polígonos: espirales e histogramas. Para los primeros se obtiene un algoritmo lineal que resuelve el problema MHVS y cotas para el máximo número h de vértices ocultos, [r2 ]+ 1 ≤ h ≤ r + 1, siendo r el número de vértices cóncavos del polígono espiral. Para polígonos histograma se demuestra que h = r − (p − 1), siendo p el número de lados fondo

Graph coloring with no large monochromatic components

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed family of graphs. We show that \mcc_2(G) = O(n^{2/3}) for any n-vertex graph G \in F. This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such F and every fixed t we show that mcc_t(G)=O(n^{2/(t+1)}). On the other hand we have examples of graphs G with no K_{t+3} minor and with mcc_t(G)=\Omega(n^{2/(2t-1)}). It is also interesting to consider graphs of bounded degrees. Haxell, Szabo, and Tardos proved \mcc_2(G) \leq 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with \mcc_2(G)=\Omega(n), and more sharply, for every \epsilon>0 there exists c_\epsilon>0 and n-vertex graphs of maximum degree 7, average degree at most 6+\epsilon for all subgraphs, and with mcc_2(G)\ge c_\eps n. For 6-regular graphs it is known only that the maximum order of magnitude of \mcc_2 is between \sqrt n and n. We also offer a Ramsey-theoretic perspective of the quantity \mcc_t(G).Comment: 13 pages, 2 figure

Design of a VLSI scan conversion processor for high performance 3-D graphics systems

Scan conversion processing is the bottleneck in the image generation process. To solve the problem of smooth shading and hidden surface elimination, a new processor architecture has been invented which has been labeled as a scan conversion processor architecture (SCP). The SCP is designed to perform hidden surface elimination and scan conversion for 64 pixels. The color intensities are dual-buffered so that when one buffer is being updated the other can be scanned out. Z-depth is used to perform the hidden surface elimination. The key operation performed by the SCP is the evaluation of linear functions of a form like F(X,Y) = A * X + B * Y + C. The computation is further simplified by using incremental addition. The z-depth buffer and the color buffers are incorporated onto the same chip. The SCP receives from its preprocessor the information for the definition of polygons and the computation of z-depth and RGB color intensities;Many copies of this processor will be used in a high performance graphics system. The SCP processes one polygon at a time. Many polygons can be processed at the same time when several Bounds-Checking Processors are added to the system. Each Bounds-Checking Processor handles a specific area of the display screen. If one polygon has intersection with a Bounds-Checking Processor\u27s controlled area, the related information will be rebroadcasted to SCPs in that area. The SCP chip uses about 26,000 transistors. 16 SCPs can be put on one chip if the 1 [mu]m CMOS technology is used