Showing papers on "Computational geometry published in 1980"

Multidimensional divide-and-conquer

Abstract: Most results in the field of algorithm design are single algorithms that solve single problems. In this paper we discuss multidimensional divide-and-conquer, an algorithmic paradigm that can be instantiated in many different ways to yield a number of algorithms and data structures for multidimensional problems. We use this paradigm to give best-known solutions to such problems as the ECDF, maxima, range searching, closest pair, and all nearest neighbor problems. The contributions of the paper are on two levels. On the first level are the particular algorithms and data structures given by applying the paradigm. On the second level is the more novel contribution of this paper: a detailed study of an algorithmic paradigm that is specific enough to be described precisely yet general enough to solve a wide variety of problems.

An Optimal Worst Case Algorithm for Reporting Intersections of Rectangles

Abstract: In this paper we investigate the problem of reporting all intersecting pairs in a set of n rectilinearly oriented rectangles in the plane. This problem arises in applications such as design rule checking of very large-scale integrated (VLSI) circuits and architectural databases. We describe an algorithm that solves this problem in worst case time proportional to n lg n + k, where k is the number of interesecting pairs found. This algorithm is optimal to within a constant factor. As an intermediate step of this algorithm, we solve a problem related to the range searching problem that arises in database applications. Although the algorithms that we describe are primarily theoretical devices (being very difficult to code), they suggest other algorithms that are quite practical.

Computational geometry and convexity

Abstract: The purpose of this dissertation is two-fold: To assert the power of convexity as a crucial factor of efficiency in computational geometry and to show how non-convex design can also benefit from this feature. Most of the recent results in computational geometry have relied on the attribute of convexity, and have failed to generalize to arbitrary designs. To remedy this flaw, one general approach consists of decomposing the objects into convex pieces, then applying the procedures to each part. We study the problem of finding minimal convex decompositions in two and three dimensions. Among our major results are an O(n + N('3)) dynamic-programming algorithm for producing minimal decompositions of non-convex polygons and an O(nN('3)) heuristics for decomposing three-dimensional polyhedra. The latter procedure is worst-case optimal in the number of convex parts (within a constant multiplicative factor). In both cases, n denotes the total number of vertices, while N refers to the number of edges which exhibit reflex angles. We further explore the problem of finding minimal decompositions in three dimensions and prove its effective decidability. We also establish an (OMEGA)(N('2)) lower bound on the number of convex parts, and use this result to analyze the performance of the above heuristics. The second purpose of this study is to show how convexity can be used for greater efficiency. We justify this claim by studying one of the most fundamental questions in computational geometry: 'Do two convex objects intersect?' Note that the problem does not call for an actual computation of the intersections, which allows the possibility of sub-linear algorithms. The restriction to a simple detection rather than a complete computation is common in many applications areas where efficiency is the main concern. We present a class of practical algorithms for detecting intersections of lines, planes, polygons, and polyhedra in two and three dimensions. Their run-times range from O(log n) for the planar cases to O(log('3)n) for detecting the intersection of two polyhedra, where n represents the total number of vertices involved.

Computational geometry on an integer grid

Abstract: In t h i s thesis we study a number of geometric problems i n an integer g r i d domain. The worst case time complexity of many of the algorithms that solve geometric problems i n the real plane i s O(nlogn). This lower time bound i s often proved by comparing the problems to Ti (nlogn) time comparison sorting. In the gr i d domain i t i s possible to sort coordinates, distances and angles i n l i n e a r time. By taking advantage of li n e a r integer grid sorting c a p a b i l i t i e s we are able to present l i n e a r time algorithms for the following geometric problems which have 0(nlogn) time algorithms when set i n the real plane: finding the convex h u l l of n points, finding a simple closed polygonal path through n points, finding the diameter of a set of n points, deciding the sep a r a b i l i t y question for two point sets, finding the smallest enclosing c i r c l e for a set of points, finding a triangulation of a set of n points and finding the Voronoi polygon of one of a set of n points. We extend van Emde Boas' O(nloglogn) integer set manipulation tree structure so i t w i l l work on the O(n^) size integer g r i d . Using t h i s extended structure we are able to present O(loglogn) search time algorithms for the problems of searching for a test point i n a set of rectangles, i n a r e c t i l i n e a r planar subdivision and i n a re s t r i c t e d angle subdivision. We are also able to use the extended van Emde Boas tree to present O(nloglogn) time algorithms for the following intersection problems on the g r i d : detecting whether any two of n rectangles intersect, detecting whether any two of n r e c t i l i n e a r polygons intersect and detecting whether any two of n res t r i c t e d angle polygons intersect.

A New Approach to the Fan Beam Geometry Image Reconstruction for a CT Scanner

Abstract: A new derivation of the fan beam geometry image reconstruction formula for CT scanners has been made in this paper. Moreover there are two points discussed here in order to develop fast algorithms based upon this formula. First is to establish a principle of filterered back-projection algorithm for fan beam geometry which enables us to use FFT algorithm. Second is to develop a fast algorithm for fan back-projection process. With these algorithms, the fan beam image reconstruction time to compute can be made comparable with the filtered back-projection algorithm for the parallel beam geometry image reconstruction.

Computational geometry and geography

Abstract: New developments in computational geometry are applicable to geographic data processing. Computationally efficient procedures incorporating logarithmic searches over appropriate data structures, and other shortcuts, can yield important savings in computation time and data storage space. There are “fast”methods for two-dimensional problems such as point inclusion, nearest-neighbor and two-nearest-neighbors determination, and grid overlay. Worst-case analysis of algorithms provides useful performance bounds for the developers of geographic production software.

What is a “good” data structure for 2-d points?

Abstract: This note poses the problem of finding the nearest neighbor of a search point among a given set of points in the plane under the condition that the set must be updated from time to time. Sundry unsuccessful encounters with various aspects of this problem in a geographic data processing environment are traced, and some relevant results from the area of computational geometry and from other sources are summarized.