The complexity of cutting complexes
TLDR
This paper investigates the combinatorial and computational aspects of certain extremal geometric problems in two and three dimensions and is able to prove sharp bounds on the asymptotic behavior of the corresponding extremal functions.Abstract:
This paper investigates the combinatorial and computational aspects of certain extremal geometric problems in two and three dimensions. Specifically, we examine the problem of intersecting a convex subdivision with a line in order to maximize the number of intersections. A similar problem is to maximize the number of intersected facets in a cross-section of a three-dimensional convex polytope. Related problems concern maximum chains in certain families of posets defined over the regions of a convex subdivision. In most cases we are able to prove sharp bounds on the asymptotic behavior of the corresponding extremal functions. We also describe polynomial algorithms for all the problems discussed.read more
Citations
More filters
Journal ArticleDOI
Ray shooting in polygons using geodesic triangulations
Bernard Chazelle,Herbert Edelsbrunner,Michelangelo Grigni,Leonidas J. Guibas,Leonidas J. Guibas,John Hershberger,Micha Sharir,Micha Sharir,Jack Snoeyink +8 more
TL;DR: A simple decomposition scheme that partitions the interior of P intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles can be used to preprocessP in a very simple manner, so any ray-shooting query can be answered in timeO(logn).
Book ChapterDOI
Ray Shooting in Polygons Using Geodesic Triangulations
Bernard Chazelle,Herbert Edelsbrunner,Michelangelo Grigni,Leonidas J. Guibas,Leonidas J. Guibas,John Hershberger,Micha Sharir,Micha Sharir,Jack Snoeyink +8 more
TL;DR: A simple decomposition scheme that partitions the interior of P intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles can be used to preprocessP in a very simple manner, so any ray-shooting query can be answered in timeO(logn).
Journal ArticleDOI
Quasi-phi-functions and optimal packing of ellipses
TL;DR: This work introduces quasi-phi-functions for an analytical description of non-overlapping and containment constraints for 2D- and 3D- objects which can be continuously rotated and translated and develops the phi-function technique for solving cutting and packing problems.
Journal ArticleDOI
Packing of concave polyhedra with continuous rotations using nonlinear optimisation
TL;DR: An efficient solution algorithm is developed, which employs a fast starting point algorithm and a new compaction procedure and derives an exact mathematical model using adjusted radical free quasi phi-functions for concave polyhedra to describe non-overlapping and distance constraints.
Optimal Geometric Data Structures
TL;DR: It is proved that finding an optimal auto-partition is NP-hard and proposed an exact algorithm for finding optimal rectilinear r-partitions whose running time is polynomial when r is a constant, and a faster 2-approximation algorithm.
References
More filters
Book
Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Journal ArticleDOI
Primitives for the manipulation of general subdivisions and the computation of Voronoi
Leonidas J. Guibas,Jorge Stolfi +1 more
TL;DR: The following problem is discussed: given n points in the plane (the sites) and an arbitrary query point q, find the site that is closest to q, which can be solved by constructing the Voronoi diagram of the griven sites and then locating the query point in one of its regions.
Proceedings ArticleDOI
Topologically sweeping an arrangement
TL;DR: The advantages of sweeping with a topological line that is not necessarily straight are demonstrated and an arrangement of n lines in the plane can be swept over in O ( n 2 ) time and O(n) space by a such a line.
Proceedings ArticleDOI
On translating a set of rectangles
TL;DR: In this paper, the authors study the nature of these constraints and exhibit optimal algorithms for finding valid motion ordering for several different classes of objects in the plane for disjoint objects.