Showing papers by "Herbert Edelsbrunner published in 1984"

Efficient algorithms for agglomerative hierarchical clustering methods

Abstract: Whenevern objects are characterized by a matrix of pairwise dissimilarities, they may be clustered by any of a number of sequential, agglomerative, hierarchical, nonoverlapping (SAHN) clustering methods. These SAHN clustering methods are defined by a paradigmatic algorithm that usually requires 0(n 3) time, in the worst case, to cluster the objects. An improved algorithm (Anderberg 1973), while still requiring 0(n 3) worst-case time, can reasonably be expected to exhibit 0(n 2) expected behavior. By contrast, we describe a SAHN clustering algorithm that requires 0(n 2 logn) time in the worst case. When SAHN clustering methods exhibit reasonable space distortion properties, further improvements are possible. We adapt a SAHN clustering algorithm, based on the efficient construction of nearest neighbor chains, to obtain a reasonably general SAHN clustering algorithm that requires in the worst case 0(n 2) time and space. Whenevern objects are characterized byk-tuples of real numbers, they may be clustered by any of a family of centroid SAHN clustering methods. These methods are based on a geometric model in which clusters are represented by points ink-dimensional real space and points being agglomerated are replaced by a single (centroid) point. For this model, we have solved a class of special packing problems involving point-symmetric convex objects and have exploited it to design an efficient centroid clustering algorithm. Specifically, we describe a centroid SAHN clustering algorithm that requires 0(n 2) time, in the worst case, for fixedk and for a family of dissimilarity measures including the Manhattan, Euclidean, Chebychev and all other Minkowski metrics.

Stationing guards in rectilinear art galleries

Abstract: A rectilinear polygon can be viewed as an art gallery room whose walls meet at right angles. An algorithm is presented that stations guards in such a room so that every interior point is visible to some guard. The algorithm partitions the polygon into L-shaped pieces, a subclass of star-shaped pieces, and locates one guard within each kernel. The algorithm runs in O ( n log n ) time in the worst case for a polygon of n vertices.

Some Methods of Computational Geometry Applied to Computer Graphics

Abstract: Windowing a two-dimensional picture means to determine those line segments of the picture that are visible through an axis-parallel window. A study of some algorithmic problems involved in windowing a picture is offered. Some methods from computational geometry are exploited to store the picture in a computer such that (1) those line segments inside or partially inside of a window can be determined efficiently, and (2) the set of those line segments can be maintained efficiently while the window is moved parallel to a coordinate axis and/or it is enlarged or reduced.

Computing the connected components of simple rectilinear geometrical objects in D-space

Abstract: Il est montre que l'on peut calculer les composantes connexes d'une famille donnee de n segments de droites horizontaux et verticaux du plan en temps o (n log n) et en place o(n). On discute diverses extensions des resultats a des dimensions plus elevees et a des ensembles dynamiques d'objets

Space Searching For Intersecting Objects

Abstract: Determining or counting geometric objects that intersect another geometric query object is at the core of algorithmic problems in a number of applied areas of computer science. This article presents a family of space-efficient data structures that realize sublinear query time for points, line segments, lines and polygons in the plane, and points, line segments, plaraes, and polyhedra in three dimensions.

Key-Problems and Key-Methods in Computational Geormetry

Abstract: Computational geometry, considered a subfield of computer science, is concerned with the computational aspects of geometric problems. The increasing activity in this rather young field made it split into several reasonably independent subareas. This paper presents several key-problems of the classical part of computational geometry which exhibit strong interrelations. A unified view of the problems is stressed, and the general ideas behind the methods that solve them are worked out.