In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E 4 . Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the first algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f . By a standard lifting map, we obtain outputsensitive algorithms for the Voronoi diagram or Delaunay triangulation in E 3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the "ultimate convex hull algorithm" of Kirkpatrick and Seidel in E 2 and also leads to improved output-sensitive results on constructing convex hulls in E d for any even constant d ? 4. 1 Introduction Geometric structures induced by n points in Euclidean d-dimensional space, such as the convex hull, Voronoi diagram, or Delaunay triangulation, can be of larger size than the point set that defines them. In many practical situat...

An in-place algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. In this paper we describe three in-place algorithms for computing the convex hull of a planar point set. All three algorithms are optimal, some more so than others...

The construction of the convex hull of a finite point set in a low-dimensional Euclidean space is a fundamental problem in computational geometry. This thesis investigates efficient algorithms for the convex hull problem, where complexity is measured as a function of both the size of the input point set and the size of the output polytope. Two new, simple, optimal, output-sensitive algorithms are presented in two dimensions and a simple, optimal, output-sensitive algorithm is presented in three dimensions. In four dimensions, we give the first output-sensitive algorithm that is within a polylogarithmic factor of optimal. In higher fixed dimensions, we obtain an algorithm that is optimal for sufficiently small output sizes and is faster than previous methods for sublinear output sizes; this result is further improved in even dimensions. Although the focus of the thesis is on the convex hull problem, applications of our techniques to many related problems in computational geometry are al...

This paper presents a survey of deterministic algorithms, randomized algorithms and approximation algorithms for the convex hull problem. The algorithms range from almost three decade old ones, such as Graham's and Jarvis's, to modern randomized algorithms, overviewing output-sensitive algorithms and their worst case running times in higher dimensions. 1 Introduction The convex hull of a set of points is the smallest convex set that contains all the points. In the plane, we can visualize the convex hull as a stretched rubber band surrounding the points that, when released, takes a polygonal shape. The extreme points in the set are the vertices of the convex polygon. The convex hull has been extensively studied in computational geometry and its applications spread over an impressive number of fields: analysis of spectrometry data, file searching, cluster analysis, collision detection, crystallography, pattern recognition, image processing, numerical integration, statistics, metal...

Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R B comes from R (respectively, B).