Visibility and Ray Shooting Queries in Polygonal Domains

Chen, Danny Z.; Wang, Haitao

doi:10.1007/978-3-642-40104-6_22

Cited by 11 publications

(24 citation statements)

References 20 publications

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“…However, without any preprocessing, the query can be answered by computing the visibility polygon of query point in O (n log n) time which is also high. There are also some other results with a trade-off between the query time and the preprocessing cost [8][9][10][11][12] (a complete survey is presented in the visibility book of [13]). …”

Section: Introductionmentioning

confidence: 96%

Visibility testing and counting

Alipour

Ghodsi

Zarei

et al. 2015

Information Processing Letters

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Section: Introductionmentioning

confidence: 96%

Visibility testing and counting

Alipour

Ghodsi

Zarei

et al. 2015

Information Processing Letters

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“…The corridor structure and its extended version for polygonal domains have been used for solving shortest path and visibility problems [Chen et al 2014;Chen and Wang 2011b, 2012, 2013a, 2013b, 2013cInkulu and Kapoor 2009;Kapoor and Maheshwari 1988;Kapoor et al 1997]. For our splinegonal domain, we generalize the approach in Kapoor et al [1997] for the polygonal domain case.…”

Section: The Corridor Structurementioning

confidence: 99%

Computing Shortest Paths among Curved Obstacles in the Plane

Chen

Wang

2015

ACM Trans. Algorithms

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A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this article, we consider the problem version on curved obstacles, which are commonly modeled as splinegons. A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge (polygons are special splinegons), and the combinatorial complexity of each curved edge is assumed to be O(1). Given in the plane two points s and t and a set S of h pairwise disjoint splinegons with a total of n vertices, after a bounded degree decomposition of S is obtained, we compute a shortest s-to-t path avoiding the splinegonswhere k is a parameter sensitive to the geometric structures of the input and is upper bounded by O(h 2 ). The bounded degree decomposition of S, which is similar to the triangulation of the polygonal domains, can be computed in O(n log n) time or O(n + h log 1+ h) time for any > 0. In particular, when all splinegons are convex, the decomposition can be computed in O(n + h log h) time and k is linear to the number of common tangents in the free space (called "free common tangents") among the splinegons. Our techniques also improve several previous results: (1) For the polygon case (i.e., when all splinegons are polygons), the shortest path problem was previously solved in O(n log n) time, or in O(n+ h 2 log n) time. Thus, our algorithm improves the O(n+ h 2 log n) time result, and is faster than the O(n log n) time solution for sufficiently small h, for example, h = o( n log n). (2) Our techniques produce an optimal output-sensitive algorithm for a basic visibility problem of computing all free common tangents among h pairwise disjoint convex splinegons with a total of n vertices. Our algorithm runs in O(n+h log h+k) time and O(n) working space, where k is the number of all free common tangents. Note that k = O(h 2 ). Even for the special case where all splinegons are convex polygons, the previously best algorithm for this visibility problem takes O(n + h 2 log n) time. (3) We improve the previous work for computing the shortest path between two points among convex pseudodisks of O(1) complexity each.In addition, a by-product of our techniques is an optimal O(n + h log h) time and O(n) space algorithm for computing the Voronoi diagram of a set of h pairwise disjoint convex splinegons with a total of n vertices.

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“…We use the same classification of largest rectangles and find the LMRs of the six types. We construct a ray-shooting data structure, such as the one by Chen and Wang [7] in O(n + h 2 polylog h) time using O(n + h 2 ) space, which supports a ray-shooting query in O(log n) time. We also construct the visibility region from each vertex of P , which can be done in O(n 2 log n) time using O(n 2 ) space by using the algorithm in [7].…”

Section: Computing a Largest Rectangle Of Type Ementioning

confidence: 99%

Maximum-Area Rectangles in a Simple Polygon

Choi¹,

Lee²,

Ahn³

2019

Preprint

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We study the problem of finding maximum-area rectangles contained in a polygon in the plane. There has been a fair amount of work for this problem when the rectangles have to be axis-aligned or when the polygon is convex. We consider this problem in a simple polygon with n vertices, possibly with holes, and with no restriction on the orientation of the rectangles. We present an algorithm that computes a maximum-area rectangle in O(n 3 log n) time using O(kn 2 ) space, where k is the number of reflex vertices of P . Our algorithm can report all maximum-area rectangles in the same time using O(n 3 ) space. We also present a simple algorithm that finds a maximum-area rectangle contained in a convex polygon with n vertices in O(n 3 ) time using O(n) space.

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Visibility and Ray Shooting Queries in Polygonal Domains

Cited by 11 publications

References 20 publications

Visibility testing and counting

Visibility testing and counting

Computing Shortest Paths among Curved Obstacles in the Plane

Maximum-Area Rectangles in a Simple Polygon

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