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Christopher van Wyk

Bio: Christopher van Wyk is an academic researcher from Princeton University. The author has contributed to research in topics: Polygon triangulation & Diagonal. The author has an hindex of 1, co-authored 1 publications receiving 165 citations.

Papers
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Journal ArticleDOI
TL;DR: Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from the proposed O(n\log \log n)-time algorithm, improving on the previously best bound and showing that triangulation is not as hard as sorting.
Abstract: Given a simple n-vertex polygon, the triangulation problem is to partition the interior of the polygon into $n - 2$ triangles by adding $n - 3$ nonintersecting diagonals. We propose an $O(n\log \log n)$-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.

166 citations


Cited by
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Proceedings ArticleDOI
Kenneth L. Clarkson1
06 Jan 1988
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.

1,163 citations

Journal Article
TL;DR: A deterministic algorithm for triangulating a simple polygon in linear time is given, using the polygon-cutting theorem and the planar separator theorem, whose role is essential in the discovery of new diagonals.
Abstract: We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

632 citations

Journal ArticleDOI
TL;DR: In this paper, a deterministic algorithm for triangulating a simple polygon in linear time is presented. But the main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals.
Abstract: We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

592 citations

Book ChapterDOI
01 Jan 2000
TL;DR: Since Victor Klee's question, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others.
Abstract: In 1973, Victor Klee posed the following question: 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 naive question of combinatorial geometry has, since its formulation, stimulated a plethora of papers, surveys and a book, most of them written in the last fifteen years. The first result in this area, due to V. Chvatal, asserts that n 3 guards are occasionally necessary and always sufficient to guard an art gallery represented by a simple polygon with n vertices. Since ChvataFs result, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this paper, we survey most of these results.

474 citations

Journal ArticleDOI
TL;DR: This work develops a new approach to solving minimum-cost circulation problems that combines methods for solving the maximum flow problem with successive approximation techniques based on cost scaling and shows that a minimum- cost circulation can be computed by solving a sequence of On lognC blocking flow problems.
Abstract: We develop a new approach to solving minimum-cost circulation problems. Our approach combines methods for solving the maximum flow problem with successive approximation techniques based on cost scaling. We measure the accuracy of a solution by the amount that the complementary slackness conditions are violated. We propose a simple minimum-cost circulation algorithm, one version of which runs in On3lognC time on an n-vertex network with integer arc costs of absolute value at most C. By incorporating sophisticated data structures into the algorithm, we obtain a time bound of Onm logn2/mlognC on a network with m arcs. A slightly different use of our approach shows that a minimum-cost circulation can be computed by solving a sequence of On lognC blocking flow problems. A corollary of this result is an On2log nlognC-time, m-processor parallel minimum-cost circulation algorithm. Our approach also yields strongly polynomial minimum-cost circulation algorithms. Our results provide evidence that the minimum-cost circulation problem is not much harder than the maximum flow problem. We believe that a suitable implementation of our method will perform extremely well in practice.

331 citations