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38
Design of data structures for mergeable trees
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant requires merging two paths in a single operation. In contrast to the standard problem, in which only one tree arc changes at a time, a singl ..."
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Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant requires merging two paths in a single operation. In contrast to the standard problem, in which only one tree arc changes at a time, a single merge operation can change many arcs. In spite of this, we develop a data structure that supports merges on an nnode forest in O(log 2 n) amortized time and all other standard tree operations in O(log n) time (amortized, worstcase, or randomized depending on the underlying data structure). For the special case that occurs in the motivating application, in which arbitrary arc deletions (cuts) are not allowed, we give a data structure with an O(log n) time bound per operation. This is asymptotically optimal under certain assumptions. For the evenmore special case in which both cuts and parent queries are disallowed, we give an alternative O(log n)time solution that uses standard dynamic trees as a black box. This solution also applies to the motivating application. Our methods use previous work on dynamic trees in various ways, but the analysis of each algorithm requires novel ideas. We also investigate lower bounds for the problem under various assumptions. 1
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 ..."
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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 ..."
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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.
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 ..."
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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 ..."
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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
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 ..."
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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...
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 ..."
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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.
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. ..."
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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...
Linear Time Triangulation of Simple Polygons
, 2009
"... From the early days of computational geometry, practitioners have looked for faster ways to triangulate a simple polygon. Several nearlinear time algorithms have been devised and implemented. However, the true linear time algorithms of Chazelle and Amato et al. are considered impractical for actual ..."
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From the early days of computational geometry, practitioners have looked for faster ways to triangulate a simple polygon. Several nearlinear time algorithms have been devised and implemented. However, the true linear time algorithms of Chazelle and Amato et al. are considered impractical for actual use despite their faster asymptotic running time. In this paper, I examine the latter of these in detail and attempt to implement it. 1
In Applied Mathematics
"... Typeset by LATEX andredblue c ○ 2002, a document class designed by José Luis Ruiz. AlamemoriadeRamón y Luz. Dedicado a Carles, Robert, Irena y Andreu ..."
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Typeset by LATEX andredblue c ○ 2002, a document class designed by José Luis Ruiz. AlamemoriadeRamón y Luz. Dedicado a Carles, Robert, Irena y Andreu