We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle's algorithm and of simple non-optimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991). As in Chazelle's algorithm, it is indispensable to include a bottom-up preprocessing phase, in addition to the actual top-down construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done.

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...

al [x i-1 ,x i+1 ] that bridges x i lies entirely in P. We say that two ears x i and x j are non-overlapping if int[x i-1 ,x i ,x i+1 ] Ç int[x j-1 ,x j ,x j+1 ] = Æ. The following Two-Ears Theorem was recently proved by Meisters [Me1]. Theorem 1: (the Two-Ears Theorem, Meisters [Me1]) Except for triangles every simple polygon P has at least two non-overlapping ears. Meisters' proof by induction is both elegant and concise. However, given that a simple polygon can always be triangulated allows a one-sentence proof [O'R]. Leaves in the dual-tree of the triangulated polygon correspond to ears and every tree of two or more nodes m

It remains as one of the major open problems in computational geometry, whether there exists a linear-time 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 linear-time 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...

Abstract. To computer circular visibility inside a simple polygon, circular arcs that emanate from a given interior point are classified with respect to the edges of the polygon they first intersect. Representing these sets of circular arcs by their centers results in a planar partition called the circular visibility diagram. An O(n) algorithm is given for constructing the circular visibility diagram for a simple polygon with n vertices.

This thesis introduces the concept of a heterogeneous decomposition of a balanced search tree and apply it to the following problems: • How can finger search be implemented without changing the representation of a Red-Black Tree, such as introducing extra storage to the nodes? (Answer: Any degree-balanced search tree can support finger search without modification in its representation by maintaining an auxiliary data structure of logarithmic size and suitably modifying the search algorithm to make use of this auxiliary data structure.) • Do Multi-Splay Trees, which is known to be O(log log n)-competitive to the optimal binary search trees, have the Dynamic Finger property? (Answer: This is work in progress. We believe the answer is yes.)

Introduction. The one-dimensional channel compaction problem with automatic jog insertion may be described informally as follows. We are given n horizontal wire segments organized into t tracks. Segments on the same track have the same y-coordinate and the ranges of their x-coordinates do not overlap. Each segment has a via at each end, connecting it to vertical wires on another layer. The goal is to minimize channel height subject to design rule constraints on the distances between wires and between wires and vias. The relative vertical position of wires is not allowed to change during compaction. Figure 1(a) shows an initial layout with 7 tracks. The horizontal wires to be compacted are shown as thin solid lines while vertical wires on another layer are thick dashed lines. Dots represent vias. Relative vertical position must be maintained due to what are called vertical constraints (see e.g. [PL88]). The left part of net

This paper describes two approaches to triangulate a simple polygon. Emphasis is on practical and easy to implement algorithms, especially the first algorithm is straightforward and intuitive but, however, quite efficient. Further, it does not require the sorting or the use of balanced tree structures. Its worst running time complexity is O(n 2 ), but for special classes of polygons it runs in linear time. The second approach requires some more sophisticated concepts of computational geometry but yields a better worst running time complexity of O(n log n). Both algorithms do not introduce new vertices and triangulate in a greedy fashion, that is they never remove edges once inserted. Further, they are designed to find an arbitrary triangulation and they do not optimize the result in any way. Keywords: computational geometry, polygon, triangulation, computational complexity, monotone polygon, trapezoidation 1 Introduction The problem of triangulating a polygon can be stated as: "giv...

2 1 1 Figure 1: A red-black tree. The darkened nodes are black nodes. The external nodes are denoted by squares. Shown with each node is its rank. Wyk give another, simpler, implementation of finger trees. They describe a finger data structure which is a modification of red-black trees, but other forms of balanced trees could be used as a basis for the structure. The two problems presented in Chapters 3 and 4 rely on the use of redblack and finger trees respectively. In this chapter we give a fairly complete overview of red-black trees, of the finger trees introduced by Tarjan and Van Wyk, and of a variant of these which we use in Chapter 4. The material here is intended to be comprehensive and useful as an introduction to these two types of data structures. Re - ack rees A red-black tree is a full binary tree in which each node is assigned a color, either red or black. The leaves are called

Typeset by LATEX andredblue c ○ 2002, a document class designed by José Luis Ruiz. AlamemoriadeRamón y Luz. Dedicado a Carles, Robert, Irena y Andreu