Results 11  20
of
39
SIMPLIFIED LINEARTIME JORDAN SORTING AND POLYGON CLIPPING
, 1990
"... Given the intersection points of a Jordan curve with the xaxis in the order in which they occur along the curve, the Jordan sorting problem is to sort them into the order in which they occur along the xaxis. This problem arises in clipping a simple polygon against a rectangle (a “window”) and in e ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Given the intersection points of a Jordan curve with the xaxis in the order in which they occur along the curve, the Jordan sorting problem is to sort them into the order in which they occur along the xaxis. This problem arises in clipping a simple polygon against a rectangle (a “window”) and in efficient algorithms for triangulating a simple polygon. Hoffman, Mehlhorn, Rosenstiehl, and Tarjan proposed an algorithm that solves the Jordan sorting problem in time that is linear in the number of intersection points, but their algorithm requires the use of a sophisticated data structure, the levellinked search tree. We propose a variant of the algorithm of Hoffman et al. that retains the lineartime bound but simplifies both the primary data structure and the operations it must perform.
Design of Data Structures for Mergeable Trees
"... Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant allows an operation that merges two tree paths. In contrast to the standard problem, in which only one tree arc at a time changes, a single ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant allows an operation that merges two tree paths. In contrast to the standard problem, in which only one tree arc at a time changes, a single merge operation can change many arcs. In spite of this, we develop a data structure that supports merges and all other standard tree operations in O(log 2 n) amortized time on an nnode forest. For the special case that occurs in the motivating application, in which arbitrary arc deletions are not allowed, we give a data structure with an O(log n) amortized time bound per operation, which is asymptotically optimal. The analysis of both algorithms is not straightforward and requires ideas not previously used in the study of dynamic trees. We explore the design space of algorithms for the problem and also consider lower bounds for it.
The Graham Scan Triangulates Simple Polygons
 Pattern Recogn. Lett
, 1991
"... The Graham scan is a fundamental backtracking technique in computational geometry which was originally designed to compute the convex hull of a set of points in the plane and has since found application in several different contexts. In this note we show how to use the Graham scan to triangulate a s ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
The Graham scan is a fundamental backtracking technique in computational geometry which was originally designed to compute the convex hull of a set of points in the plane and has since found application in several different contexts. In this note we show how to use the Graham scan to triangulate a simple polygon. The resulting algorithm triangulates an n vertex polygon P in O(kn) time where k1 is the number of concave vertices in P. Although the worst case running time of the algorithm is O(n 2 ), it is easy to implement and is therefore of practical interest. 1. Introduction A polygon P is a closed path of straight line segments. A polygon is represented by a sequence of vertices P = (p 0 ,p 1 ,...,p n1 ) where p i has realvalued x,ycoordinates. We assume that no three vertices of P are collinear. The line segments (p i ,p i+1 ), 0 i n1, (subscript arithmetic taken modulo n) are the edges of P. A polygon is simple if no two nonconsecutive edges intersect. A simple polygon part...
Animation of Geometric Algorithms: A Video Review
 DEC SYSTEMS RESEARCH CENTER, RESEARCH REPORT
, 1993
"... Geometric algorithms and data structures are often easiest to understand visually, in terms of the geometric objects they manipulate. Indeed, most papers in computational geometry rely on diagrams to communicate the intuition behind the results. Algorithm animation uses dynamic visual images to expl ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Geometric algorithms and data structures are often easiest to understand visually, in terms of the geometric objects they manipulate. Indeed, most papers in computational geometry rely on diagrams to communicate the intuition behind the results. Algorithm animation uses dynamic visual images to explain algorithms. Thus it is natural to present geometric algorithms, which are inherently dynamic, via algorithm animation. The accompanying videotape presents a video review of geometric animations; the review was premiered at the 1992 ACM Symposium on Computational Geometry. The video review includes singlealgorithm animations and sample graphic displays from "workbench" systems for implementing multiple geometric algorithms. This report contains short descriptions of each video segment.
Surface Triangulation: A Survey
, 1996
"... This paper presents a brief survey of some problems and solutions related to the triangulation of surfaces. A surface (a two dimensional manifold, in the context of this paper) can be represented as a three dimensional function on a planar disk. In that sense, the triangulation of the disk induces a ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
This paper presents a brief survey of some problems and solutions related to the triangulation of surfaces. A surface (a two dimensional manifold, in the context of this paper) can be represented as a three dimensional function on a planar disk. In that sense, the triangulation of the disk induces a triangulation of the surface. Hence the emphasis of this paper is on triangulation on a plane. Apart from the issues in triangulation, this survey talks about the known upper and lower bounds on various triangulation problems. It is intended as a broad compilation of known results rather than an intensive treatise, and the details of most algorithms are skipped. 1 Introduction This survey assumes familiarity with the fundamental concepts of computational geometry. We define the triangulation problem as follows: Input: i. A set S of points, fp i g, such that each p i lies on the surface ii. A set of conditions, fC i g Output: A set S 0 of triples f(p i 1 ; p i 2 ; p i 3 )g such that e...
MultiSplay Trees
, 2006
"... In this thesis, we introduce a new binary search tree data structure called multisplay tree and prove that multisplay trees have most of the useful properties different binary search trees (BSTs) have. First, we demonstrate a close variant of the splay tree access lemma [ST85] for multisplay tree ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In this thesis, we introduce a new binary search tree data structure called multisplay tree and prove that multisplay trees have most of the useful properties different binary search trees (BSTs) have. First, we demonstrate a close variant of the splay tree access lemma [ST85] for multisplay trees, a lemma that implies multisplay trees have the O(log n) runtime property, the static finger property, and the static optimality property. Then, we extend the access lemma by showing the remassing lemma, which is similar to the reweighting lemma for splay trees [Geo04]. The remassing lemma shows that multisplay trees satisfy the working set property and keyindependent optimality, and multisplay trees are competitive to parametrically balanced trees, as defined in [Geo04]. Furthermore, we also prove that multisplay trees achieve the O(log log n)competitiveness and that sequential access in multisplay trees costs O(n). Then we naturally extend the static model to allow insertions and deletions and show how to carry out these operations in multisplay trees to achieve
Polygons are Anthropomorphic
"... al [x i1 ,x i+1 ] that bridges x i lies entirely in P. We say that two ears x i and x j are nonoverlapping if int[x i1 ,x i ,x i+1 ] Ç int[x j1 ,x j ,x j+1 ] = Æ. The following TwoEars Theorem was recently proved by Meisters [Me1]. Theorem 1: (the TwoEars Theorem, Meis ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
al [x i1 ,x i+1 ] that bridges x i lies entirely in P. We say that two ears x i and x j are nonoverlapping if int[x i1 ,x i ,x i+1 ] Ç int[x j1 ,x j ,x j+1 ] = Æ. The following TwoEars Theorem was recently proved by Meisters [Me1]. Theorem 1: (the TwoEars Theorem, Meisters [Me1]) Except for triangles every simple polygon P has at least two nonoverlapping ears. Meisters' proof by induction is both elegant and concise. However, given that a simple polygon can always be triangulated allows a onesentence proof [O'R]. Leaves in the dualtree of the triangulated polygon correspond to ears and every tree of two or more nodes m
Path Planning Algorithms under the LinkDistance Metric
, 2006
"... The Traveling Salesman Problem and the Shortest Path Problem are famous problems in computer science which have been well studied when the objective is measured using the Euclidean distance. Here we examine these geometric problems under a different set of optimization criteria. Rather than consider ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The Traveling Salesman Problem and the Shortest Path Problem are famous problems in computer science which have been well studied when the objective is measured using the Euclidean distance. Here we examine these geometric problems under a different set of optimization criteria. Rather than considering the total distance traversed by a path, this thesis looks at reducing the number of times a turn is made along that path, or equivalently, at reducing the number of straight lines in the path. Minimizing this objective value, known as the linkdistance, is useful in situations where continuing in a given direction is cheap, while turning is a relatively expensive operation. Applications exist in VLSI, robotics, wireless communications, space travel, and other fields where it is desirable to reduce the number of turns. This thesis examines rectilinear and nonrectilinear variants of the Traveling Salesman Problem under this metric. The objective of these problems is to find a path visiting a set of points which has the smallest number of bends. A 2approximation algorithm is given for the rectilinear problem, while for the nonrectilinear problem, an O(log n)approximation algorithm is given. The latter problem is also shown to be NPComplete.
Computational Geometry for the Gourmet  Old Fare and New Dishes
"... This is a personal account of some of the most exciting challenges computational geometry faces today. Needless to say, my choice of open problems reflects my own taste and bias. I did not wish to cover the whole spectrum of the field and give a bland, comprehensive assessment of what’s ahead for co ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This is a personal account of some of the most exciting challenges computational geometry faces today. Needless to say, my choice of open problems reflects my own taste and bias. I did not wish to cover the whole spectrum of the field and give a bland, comprehensive assessment of what’s ahead for computational geometers. Rather, I chose to discuss a few problems that sprang to my mind as being both fundamental in the issues they raise, and plain fun to work on. Some of these problems are as old as the field and can be stated in one or two sentences. In this category, one of my favorites is the problem of sorting X + X, the set obtained by adding elements pairwise from a set X of n numbers: Can it be done in o � n 2 log n � ? After Fredman’s 1976 paper [31] that showed that, in essence, O(n 2) comparisons suffice, precious little progress has been made. What makes Fredman’s result impractical is that to find the right sequence of comparisons can only be done at this point with an exponentialsize lookup table. The importance of this problem in geometry is that without a solution to it, hopes to sort the slopes formed by n points in the plane optimally or equivalently, sort the vertices of a line arrangement along a given direction, all vanish. This is old, hard and still open, but hardly an isolated case by a long shot. I will discuss more of these old unsolved cases and then gradually move on to newer trends, e.g., probabilistic algorithms
Slicing an Ear in Linear Time
 Pattern Recognition Letters
, 1989
"... It remains as one of the major open problems in computational geometry, whether there exists a lineartime algorithm for triangulating a simple polygon P. Yet it is well known that a diagonal of P can easily be found in linear time. In this note we show that an ear of P can be found in linear time. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
It remains as one of the major open problems in computational geometry, whether there exists a lineartime algorithm for triangulating a simple polygon P. Yet it is well known that a diagonal of P can easily be found in linear time. In this note we show that an ear of P can be found in linear time. An ear is a triangle such that one of its edges is a diagonal of P and the remaining two edges are edges of P. Applications of this result are indicated. 1. Introduction The triangulation of simple polygons has received much attention in the computational geometry literature because of its many applications in such areas as pattern recognition, computer graphics, CAD and solid modeling. Nevertheless, it remains as one of the major open problems in computational geometry, whether there exists a lineartime algorithm for triangulating a simple polygon P. The fastest algorithm to date is due to Tarjan & Van Wyk [TV] and runs in O(n log log n) time, where n is the number of vertices of P. On th...