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47
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...
Art gallery and illumination problems
 In Handbook on Computational Geometry, Elsevier Science Publishers, J.R. Sack and
, 2000
"... How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke pub ..."
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Cited by 86 (3 self)
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How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke published his book Art Gallery Theorems and Algorithms which has further fueled this area of research. The present book is being written almost 10 years since the publication of O’Rourke’s book, and the need for an uptodate manuscript on Art Gallery or Illumination Problems is evident. Some important open problems stated in O’Rourke’s book, such as... have been solved. New directions of research have since been investigated, including: watchman routes, floodlight illumination problems, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this book, we study these results and try to give a complete
Geometric Pattern Matching under Euclidean Motion
, 1993
"... Given two planar sets A and B, we examine the problem of determining the smallest " such that there is a Euclidean motion (rotation and translation) of A that brings each member of A within distance " of some member of B. We establish upper bounds on the combinatorial complexity of this subproblem i ..."
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Cited by 73 (2 self)
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Given two planar sets A and B, we examine the problem of determining the smallest " such that there is a Euclidean motion (rotation and translation) of A that brings each member of A within distance " of some member of B. We establish upper bounds on the combinatorial complexity of this subproblem in modelbased computer vision, when the sets A and B contain points, line segments, or (filledin) polygons. We also show how to use our methods to substantially improve on existing algorithms for finding the minimum Hausdorff distance under Euclidean motion. 1 Author's address: Department of Computer Science, Cornell University, Ithaca, NY 14853. This work was supported by the Advanced Research Projects Agency of the Department of Defense under ONR Contract N0001492J1989, and by ONR Contract N0001492J1839, NSF Contract IRI9006137, and AFOSR Contract AFOSR910328. 2 Author's address: Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218. This work was suppo...
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...
On Compatible Triangulations of Simple Polygons
 Computational Geometry: Theory and Applications
, 1993
"... It is well known that, given two simple nsided polygons, it may not be possible to triangulate the two polygons in a compatible fashion, if one's choice of triangulation vertices is restricted to polygon corners. Is it always possible to produce compatible triangulations if additional vertices insi ..."
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Cited by 46 (4 self)
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It is well known that, given two simple nsided polygons, it may not be possible to triangulate the two polygons in a compatible fashion, if one's choice of triangulation vertices is restricted to polygon corners. Is it always possible to produce compatible triangulations if additional vertices inside the polygon are allowed? We give a positive answer and construct a pair of such triangulations with O(n 2 ) new triangulation vertices. Moreover, we show that there exists a "universal" way of triangulating an nsided polygon with O(n 2 ) extra triangulation vertices. Finally, we also show that creating compatible triangulations requires a quadratic number of extra vertices in the worst case. 1 The Problem Given two simple polygons P 1 and P 2 , each with n vertices, it is not always possible to find compatible triangulations of the two polygons. In other words, there may not exist a circular labeling of the corners of each polygon by consecutive numbers 1 through n and a set of n \G...
Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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Cited by 46 (11 self)
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
Triangulation and shapecomplexity
 ACM Trans. Graph
, 1984
"... This paper describes a new method for triangulating a simple nsided polygon. The algorithm runs in time O(n log s), with s _< n. The quantity s measures the sinuosity of the polygon, that is, the number of times the boundary alternates between complete spirals of opposite orientation. The value of ..."
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Cited by 41 (1 self)
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This paper describes a new method for triangulating a simple nsided polygon. The algorithm runs in time O(n log s), with s _< n. The quantity s measures the sinuosity of the polygon, that is, the number of times the boundary alternates between complete spirals of opposite orientation. The value of s is in practice a very small constant, even for extremely winding polygons. Our algorithm is the first method whose performance is linear in the number of vertices, up to within a factor that depends only on the shapecomplexity of the polygon. Informally, this notion of shapecomplexity measures how entangled a polygon is, and is thus highly independent of the number of vertices. A practical advantage of the algorithm is that it does not require sorting or the use of any balanced tree structure. Aside from the notion of sinuosity, we are also able to characterize a large class of polygons for which the algorithm can be proven to run in O(n log log n) time. The algorithm has been implemented, tested, and empirical evidence has confirmed its theoretical claim to efficiency.
AN O(n log log n)TIME ALGORITHM FOR TRIANGULATING A SIMPLE POLYGON
, 1988
"... Given a simple nvertex polygon, the triangulation problem is to partition the interior of the polygon into n2 triangles by adding n3 nonintersecting diagonals. We propose an O(n log logn)time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that tria ..."
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Cited by 37 (4 self)
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Given a simple nvertex polygon, the triangulation problem is to partition the interior of the polygon into n2 triangles by adding n3 nonintersecting diagonals. We propose an O(n log logn)time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that triangulation is not as hard as sorting. Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from our result.
Optimal Doubly Logarithmic Parallel Algorithms Based On Finding All Nearest Smaller Values
, 1993
"... The all nearest smaller values problem is defined as follows. Let A = (a 1 ; a 2 ; : : : ; an ) be n elements drawn from a totally ordered domain. For each a i , 1 i n, find the two nearest elements in A that are smaller than a i (if such exist): the left nearest smaller element a j (with j ! i) a ..."
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Cited by 37 (7 self)
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The all nearest smaller values problem is defined as follows. Let A = (a 1 ; a 2 ; : : : ; an ) be n elements drawn from a totally ordered domain. For each a i , 1 i n, find the two nearest elements in A that are smaller than a i (if such exist): the left nearest smaller element a j (with j ! i) and the right nearest smaller element a k (with k ? i). We give an O(log log n) time optimal parallel algorithm for the problem on a CRCW PRAM. We apply this algorithm to achieve optimal O(log log n) time parallel algorithms for four problems: (i) Triangulating a monotone polygon, (ii) Preprocessing for answering range minimum queries in constant time, (iii) Reconstructing a binary tree from its inorder and either preorder or postorder numberings, (vi) Matching a legal sequence of parentheses. We also show that any optimal CRCW PRAM algorithm for the triangulation problem requires \Omega\Gammauir log n) time. Dept. of Computing, King's College London, The Strand, London WC2R 2LS, England. ...