Showing papers by "Herbert Edelsbrunner published in 1987"

Algorithms in Combinatorial Geometry

Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

An Improved Algorithm for Constructing kth-Order Voronoi Diagrams

Abstract: The kth-order Voronoi diagram of a finite set of sites in the Euclidean plane E2 subdivides E2 into maximal regions such that all points within a given region have the same k nearest sites. Two versions of an algorithm are developed for constructing the kth-order Voronoi diagram of a set of n sites in O(n2 log n + k(n - k) log2 n) time, O(k(n - k)) storage, and in O(n2 + k(n - k) log2 n) time, O(n2) storage, respectively.

Linear space data structures for two types of range search

Abstract: This paper investigates the existence of linear space data structures for range searching. We examine thehomothetic range search problem, where a setS ofn points in the plane is to be preprocessed so that for any triangleT with sides parallel to three fixed directions the points ofS that lie inT can be computed efficiently. We also look atdomination searching in three dimensions. In this problem,S is a set ofn points inE3 and the question is to retrieve all points ofS that are dominated by some query point. We describe linear space data structures for both problems. The query time is optimal in the first case and nearly optimal in the second.

Semispaces of Configurations

Abstract: A non-vertical hyperplane h, disjoint from a finite set P of points, partitions P into two sets P+ = P∩h+ and P- = P∩h - called semispaces of P, where h+ is the open half-space above hyperplane h and h- is the open half-space below h. Note that this definition includes the empty set and P itself as semispaces of P. Using geometric transformations and the face counting formulas for arrangements of hyper planes presented in Chapter 1, it is not hard to find tight upper bounds on the number of semispaces of P that depend solely on the cardinality of P. Unfortunately, little is known about the maximum number of semispaces with some fixed cardinality. This chapter addresses the latter counting problem and derives non-trivial upper and lower bounds.

On the lower envelope of bivariate functions and its applications

Abstract: We consider the problem of obtaining sharp (nearly quadratic) bounds for the combinatorial complexity of the lower envelope (i.e. pointwise minimum) of a collection of n bivariate (or generally multi-variate) continuous and "simple" functions, and of designing efficient algorithms for the calculation of this envelope. This problem generalizes the well-studied univariate case (whose analysis is based on the theory of Davenport-Schinzel sequences), but appears to be much more difficult and still largely unsolved. It is a central problem that arises in many areas in computational and combinatorial geometry, and has numerous applications including generalized planar Voronoi diagrams, hidden surface elimination for intersecting surfaces, purely translational motion planning, finding common transversals of polyhedra, and more. In this abstract we provide several partial solutions and generalizations of this problem, and apply them to the problems mentioned above. The most significant of our results is that the lower envelope of n triangles in three dimensions has combinatorial complexity at most O(n2α(n)) (where α(n) is the extremely slowly growing inverse of Ackermann's function), that this bound is tight in the worst case, and that this envelope can be calculated in time O(n2α(n)).

The complexity of cutting convex polytypes

Abstract: Throughout this paper, we use the term subdivision as a shorthand for “a subdivision of E2 into convex regions”. A subdivision is said to be of size n if it is made of n convex (open) regions, and it is of degree d if every region is adjacent to at most d other regions. We define the line span of a subdivision as the maximum number of regions which can be intersected by a single line (section 3).

Constructing Convex Hulls

Abstract: This chapter investigates the problem of constructing the convex hull of a finite set of points in E d , that is, of producing a meaningful representation of the convex hull. If P is a finite set of points in E d , then we write convP for the convex hull of P. By the definitions given in Appendix A, convP is the set of convex combinations of P. Equivalently, convP can be defined as the smallest convex set that contains P, or the intersection of all convex sets that contain P, or the intersection of all half-spaces that contain P.

Testing the necklace condition for shortest tours and optimal factors in the plane

Abstract: A tour τ of a finite set P of points is a necklace-tour if there are disks with the points in P as centers such that two disks intersect if and only if their centers are adjacent in τ. It has been observed by Sanders that a necklace-tour is an optimal traveling salesman tour.

Fundamental Concepts in Combinatorial Geometry

Abstract: Two of the main subjects studied in combinatorial geometry and therefore in this book are finite sets of points and finite sets of hyperplanes. Not all questions about finite sets of points or hyperplanes are combinatorial, though, and one has to keep in mind that a strict classification into combinatorial and non-combinatorial problems is neither reasonable nor desirable. Nevertheless, there are a few characteristics that identify a problem as combinatorial. For example, a typical combinatorial question that can be asked about a set P of n points in d-dimensional Euclidean space E d is the following: “How many partitions of P into two subsets can be defined by hyperplanes?” If H is a set of n hyperplanes in the same space, then it is a combinatorial question if one asks “What is the number of cells the space is cut into by the hyperplanes in H?” We will investigate both problems and many related ones in this chapter and, more generally, in this book.