Department of Computer Science Activity 1998-2004

This report summarizes much of the research and teaching activity of the Department of Computer Science at Dartmouth College between late 1998 and late 2004. The material for this report was collected as part of the final report for NSF Institutional Infrastructure award EIA-9802068, which funded equipment and technical staff during that six-year period. This equipment and staff supported essentially all of the department\u27s research activity during that period

Closest and Farthest-Line Voronoi Diagrams in the Plane

Voronoi diagrams are a geometric structure containing proximity information useful in efficiently answering a number of common geometric problems associated with a set of points in the plane.. They have applications in fields ranging from crystallography to biology. Diagrams of sites other than points and with different distance metrics have been studied. This paper examines the Voronoi diagram of a set of lines, which has escaped study in the computational geometry literature. The combinatorial and topological properties of the closest and farthest Voronoi diagrams are analyzed and O(n^2) and O(n log n) algorithms are presented for their computation respectively

GEOMETRIC OPTIMIZATION IN SOME PROXIMITY AND BIOINFORMATICS PROBLEMS

The theme of this dissertation is geometric optimization and its applications. We study geometric proximity problems and several bioinformatics problems with a geometric content, requiring the use of geometric optimization tools. We have investigated the following type of proximity problems. Given a point set in a plane with n distinct points, for each point in the set find a pair of points from the remaining points in the set such that the three points either maximize or minimize some geometric measure defined on these. The measures include (a) sum and product; (b) difference; (c) line–distance; (d) triangle area; (e) triangle perimeter; (f) circumcircle–radius; and (g) triangle–distance in three dimensions. We have also studied the application of a linear time incremental geometric algorithm to test the linear separability of a set of blue points from a set of red points, in two and three–dimensional Euclidean spaces. We have used this geometric separability tool on 4 different gene expression data–sets, enumerating gene–pairs and gene–triplets that are linearly separable. Pushing on further, we have exploited this novel tool to identify some bio–marker genes for a classifier. The gene selection method proposed in the dissertation exhibits good classification accuracy as compared to other known feature (or gene) selection methods such as t–values, FCS (Fisher Criterion Score) and SAM (Significance Analysis of Microarrays). Continuing this line of investigation further, we have also designed an efficient algorithm to find the minimum number of outliers when the red and blue point sets are not fully linearly separable. We have also explored the applicability of geometric optimization techniques to the problem of protein structure similarity. We have come up with two new algorithms, EDAlignres and EDAlignsse, for pairwise protein structure alignment. EDAlignres identifies the best structural alignment of two equal length proteins by refining the correspondence obtained from eigendecomposition and to maximize the similarity measure for the refined correspondence. EDAlignsse, on the other hand, does not require the input proteins to be of equal length. These have been fully implemented and tested against well-established protein alignment program

Модели и методы решения непрерывных задач оптимального разбиения множеств: линейные, нелинейные, динамические задачи

Излагается математическая теория непрерывных задач оптимального разбиения множеств n-мерного евклидова пространства, являющихся неклассическими задачами бесконечномерного математического программирования. Особое внимание уделяется наиболее сложным задачам, характеризующимся нелинейностью критериев оптимальности разбиения, наличием дополнительных ограничений, динамическим характером параметров. Рассматриваются также модели и методы решения непрерывных задач оптимального разбиения множеств, возникающих при управлении распределенными системами, в частности, системами параболического типа. Приводится широкий спектр практических приложений непрерывных задач оптимального разбиения множеств и родственных с ними задач оптимального покрытия, геометрического проектирования. Описываются некоторые направления дальнейшего развития теории и методов решения непрерывных задач оптимального разбиения множеств. Для специалистов в области вычислительной и прикладной математики, научных работников, аспирантов и студентов, интересующихся современными проблемами теории оптимизации, в том числе недифференцируемой, математическим моделированием, проблемами территориального планирования, оптимального размещения объектов различной природы в заданной области, других задач, сводящихся к моделям оптимального разбиения множеств.Викладається математична теорія неперервних задач оптимального розбиття множин n-вимірного евклідового простору, які є некласичними задачами нескінченновимірного математичного програмування. Особлива увага приділяється найбільш складним задачам, що характеризуються нелінійністю критеріїв оптимальності розбиття, наявністю додаткових обмежень, динамічним характером параметрів. Розглядаються також моделі і методи розв'язання неперервних задач оптимального розбиття множин, що виникають при керуванні розподіленими системами, зокрема, системами параболічного типу. Наводиться широкий спектр практичних застосувань неперервних задач оптимального розбиття множин і споріднених з ними задач оптимального покриття, геометричного проектування. Описуються деякі напрямки подальшого розвитку теорії і методів розв'язання неперервних задач оптимального розбиття множин. Для фахівців у галузі обчислювальної та прикладної математики, науковців, аспірантів і студентів, які цікавляться сучасними проблемами теорії оптимізації, у тому числі недиференційовної, математичним моделюванням, проблемами територіального планування, оптимального розміщення об'єктів різної природи в заданій області, інших задач, що зводяться до моделей оптимального розбиття множин.E.M. Kiseleva, L.S. Korіashkina. Models and Methods for Solving Continuous Problems of Optimal Set Partitioning: Linear, Nonlinear, and Dynamic Problems: a monograph [in Russian]. – K .: Naukova Dumka, 2013. – 606 p. We present the mathematical theory of continuous problems of optimal partitioning of sets from an n-dimensional Euclidean space, which are non-classical problems of infinite-dimensional mathematical programming. Particular attention is given to the most complex problems characterized by non-linearity of partition optimality criteria, additional restrictions, dynamic nature of the parameters. We also consider the models and methods for solving of continuous problems of optimal partitioning of sets arising in the control of distributed systems, in particular parabolic type systems. We present a wide range of practical applications of continuous problems of optimal partitioning of sets, optimal covering and geometric design. We describe some areas for further development of theory and methods for solving continuous problems of optimal partitioning of sets. For experts in the field of computational and applied mathematics, researchers and students interested in modern problems of the optimization theory, including non-differentiable, mathematical modeling, the problems of spatial planning, optimal location of different nature objects in a given area, the other problems that reduce to models of optimal sets partitioning

2-Point Site Voronoi Diagrams

. In this paper we investigate a new type of Voronoi diagrams in which every region is defined by a pair of point sites and some distance function from a point to two points. We analyze the complexity of the respective nearest- and furthest-neighbor diagrams of several such distance functions, and show how to compute the diagrams efficiently. 1 Introduction The standard Voronoi Diagram of a set of n given points (called sites) is a subdivision of the plane into n regions, one associated with each site. Each site's region consists of all points in the plane closer to it than to any of the other sites. One application is what Knuth called the "post office" problem. Given a letter to be delivered, the nearest post office to the destination can be found by locating the destination point in the Voronoi diagram of the post office sites. This is called a "locus approach" to solving the problem---points in the plane are broken into sets by the answer to a query (in this case, "Which post offi..