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26
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 180 (8 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons
 Comput. Geom. Theory Appl
, 1991
"... This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log n). This leads to a simple algorithm of t ..."
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Cited by 99 (2 self)
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This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log n). This leads to a simple algorithm of the same complexity for triangulating polygons. More generally, if S is presented as a plane graph with k connected components, then the expected running time of the algorithm is O(n log n k log n). As a byproduct our algorithm creates a search structure of expected linear size that allows point location queries in the resulting trapezoidation in logarithmic expected time. The analysis of the expected performance is elementary and straightforward. All expectations are with respect to "coinflips" generated by the algorithm and are not based on assumptions about the geometric distribution of the input. Large Portions of the research reported here were conducted while the author visit...
Planar Separators and Parallel Polygon Triangulation
, 1992
"... We show how to construct an O( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O( p n)separator of that subgraph. We also show how to construct an O(n ffl )way decomposition tree ..."
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Cited by 51 (7 self)
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We show how to construct an O( p n)separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree where each node corresponds to a subgraph of G and stores an O( p n)separator of that subgraph. We also show how to construct an O(n ffl )way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n 1=2+ffl )separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n= log n) processors on a CRCW PRAM. Keywords: Computational geometry, algorithmic graph theory, planar graphs, planar separators, polygon triangulation, parallel algorithms, PRAM model. 1 Introduction Let G = (V; E) be an nnode graph. An f(n)separator is an f(n)sized subset of V whose removal disconnects G into two subgraphs G 1 and G 2 each...
Compaction Algorithms for NonConvex Polygons and Their Applications
, 1994
"... Given a twodimensional, nonoverlapping layout of convex and nonconvex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial twodimensional packing applications, compaction can improve the material utilization of already ti ..."
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Cited by 27 (2 self)
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Given a twodimensional, nonoverlapping layout of convex and nonconvex polygons, compaction refers to a simultaneous motion of the polygons that generates a more densely packed layout. In industrial twodimensional packing applications, compaction can improve the material utilization of already tightly packed layouts. Efficient algorithms for compacting a layout of nonconvex polygons are not previously known. This dissertation offers the first systematic study of compaction of nonconvex polygons. We start by formalizing the compaction problem as that of planning a motion that minimizes some linear objective function of the positions. Based on this formalization, we study the complexity of compaction and show it to be PSPACEhard. The major contribution of this dissertation is a positionbased optimization model that allows us to calculate directly new polygon positions that constitute a locally optimum solution of the objective via linear programming. This model yields the first ...
The searchlight scheduling problem
, 1988
"... Abstract. The problem of searching for a mobile robber in a simple polygon by a number of searchlights is considered. A searchlight is a stationary point which emits a single ray that cannot penetrate the boundary of the polygon. The direction of the ray can be changed continuously, and a point is d ..."
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Cited by 19 (2 self)
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Abstract. The problem of searching for a mobile robber in a simple polygon by a number of searchlights is considered. A searchlight is a stationary point which emits a single ray that cannot penetrate the boundary of the polygon. The direction of the ray can be changed continuously, and a point is detected by a searchlight at a given time if and only if it is on the ray. A robber is a point that can move continuously with unbounded speed. First, it is shown that the problem of obtaining a search schedule for an instance having at least one searchlight on the polygon boundary can be reduced to that for instances having no searchlight on the polygon boundary. The reduction is achieved by a recursive search strategy called the oneway sweep strategy. Then various sufficient conditions for the existence of a search schedule are presented by using the concept of a searchlight visibility graph. Finally, a simple necessary and sufficient condition for the existence of a search schedule for instances having exactly two searchlights in the interior is presented. Key words, geometry, searchlight, visibility AMS(MOS) subject classification. 68E99 1. Introduction. We
Purely Functional Representations of Catenable Sorted Lists.
 In Proceedings of the 28th Annual ACM Symposium on Theory of Computing
, 1996
"... The power of purely functional programming in the construction of data structures has received much attention, not only because functional languages have many desirable properties, but because structures built purely functionally are automatically fully persistent: any and all versions of a structur ..."
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Cited by 16 (5 self)
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The power of purely functional programming in the construction of data structures has received much attention, not only because functional languages have many desirable properties, but because structures built purely functionally are automatically fully persistent: any and all versions of a structure can coexist indefinitely. Recent results illustrate the surprising power of pure functionality. One such result was the development of a representation of doubleended queues with catenation that supports all operations, including catenation, in worstcase constant time [19].
FIST: Fast industrialstrength triangulation of polygons
 Algorithmica
, 1998
"... A preliminary version of this paper has appeared as an extended abstract at CGI'98; see [26]. y ..."
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Cited by 11 (2 self)
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A preliminary version of this paper has appeared as an extended abstract at CGI'98; see [26]. y
LinearTime Triangulation of a Simple Polygon Made Easier Via Randomization
 In Proc. 16th Annu. ACM Sympos. Comput. Geom
, 2000
"... 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 wellknown and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm ..."
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Cited by 10 (0 self)
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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 wellknown 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 and, hence, positively answers his question of whether a simpler randomized algorithm for the problem exists. The new algorithm can be viewed as a combination of Chazelle's algorithm and of nonoptimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991), with the essential innovation that sampling is performed on subchains of the initial polygonal chain, rather than on its edges. It is also essential, as in Chazelle's algorithm, to include a bottomup preprocessing phase previous to the topdown construction phase. 1 Introduction Polygon triangulation is a classic problem in comp...
A Randomized Algorithm for Triangulating a Simple Polygon in Linear Time
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2001
"... 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 wellknown and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm ..."
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Cited by 7 (1 self)
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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 wellknown 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 nonoptimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991). As in Chazelle's algorithm, it is indispensable to include a bottomup preprocessing phase, in addition to the actual topdown 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.