Triangulating a simple polygon in linear time
TLDR
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.read more
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Geometric Tools for Computer Graphics
TL;DR: Geometric Tools for Computer Graphics is an extensive, conveniently organized collection of proven solutions to fundamental problems that you'd rather not solve over and over again, including building primitives, distance calculation, approximation, containment, decomposition, intersection determination, separation, and more.
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Art Gallery and Illumination Problems
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.
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On a class of O(n2) problems in computational geometry
Anka Gajentaan,Mark H. Overmars +1 more
TL;DR: A large class of problems is described for which it is proved that they are all at least as difficult as the following base problem 3sum: Given a set S of n integers, are there three elements of S that sum up to 0.
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Wiener Index of Hexagonal Systems
TL;DR: In this paper, the authors present the results known for W of the HS: method for computing W, expressions relating W with the structure of the respective HS, results on HS's extremal w.r.t. W, and on integers that cannot be the W-values of HS's.
References
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Art gallery theorems and algorithms
TL;DR: In this paper, the authors proposed a visibility algorithm based on three-dimensions and miscellany of the polygons, and showed that minimal guard covers threedimensions of the polygon.
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A Separator Theorem for Planar Graphs
TL;DR: In this paper, it was shown that the vertices of a planar graph can be partitioned into three sets A, B, C such that no edge joins a vertex in A with another vertex in B, neither A nor B contains more than ${2n/3}$ vertices, and C contains no more than $2.
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Primitives for the manipulation of general subdivisions and the computation of Voronoi
Leonidas J. Guibas,Jorge Stolfi +1 more
TL;DR: The following problem is discussed: given n points in the plane (the sites) and an arbitrary query point q, find the site that is closest to q, which can be solved by constructing the Voronoi diagram of the griven sites and then locating the query point in one of its regions.
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Optimal Search in Planar Subdivisions
TL;DR: This work presents a practical algorithm for subdivision search that achieves the same (optimal) worst case complexity bounds as the significantly more complex algorithm of Lipton and Tarjan, namely $O(\log n)$ search time with $O(n)$ storage.
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Applications of a Planar Separator Theorem
TL;DR: Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only O(√n) vertices, and this separator theorem in combination with a divide-and-conquer strategy leads to many new complexity results for planar graphs problems.