Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

Bennett, Huck; Papadopoulou, Evanthia; Yap, Christina

doi:10.1111/cgf.12979

Cited by 9 publications

(11 citation statements)

References 25 publications

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“…Besides being of theoretical interest, our smooth quadtree data structure is useful in applications, such as the authors' recent work with Papadopoulou on computing a general class of Voronoi diagrams using subdivision [BPY16]. We have implemented it as part of the Core Library [Cor] 3 .…”

Section: Our Resultsmentioning

confidence: 99%

Amortized analysis of smooth quadtrees in all dimensions

Bennett

Yap

2017

Computational Geometry

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Quadtrees are a well-known data structure for representing geometric data in the plane, and naturally generalize to higher dimensions. A basic operation is to expand the tree by splitting any given leaf. A quadtree is smooth if any two adjacent leaf boxes differ by at most one in depth. In this paper, we analyze quadtrees that restore smoothness after each split operation, and also maintain neighbor pointers. Our main result shows that the smooth split operation has an amortized cost of at most 2 D • (D + 1)! auxiliary split operations, which corresponds to amortized constant time in quadtrees of any fixed dimension D. We also show that the exponential dependence on the dimension is unavoidable via a lower bound construction. We additionally give a lower bound construction showing an amortized cost of Ω(log n) for splits in a related quadtree model that does not maintain smoothness.

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Section: Our Resultsmentioning

confidence: 99%

Amortized analysis of smooth quadtrees in all dimensions

Bennett

Yap

2017

Computational Geometry

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“…E.g., users can choose > 0 to be correlated with the uncertainty in the environment and the precision of the robot sensors. By using weighted Voronoi diagrams [4], we can achieve practical planners that have obstacle-dependent clearances (larger clearance for "dangerous" obstacles).…”

Section: Remarkmentioning

confidence: 99%

Soft subdivision motion planning for complex planar robots

Zhou

Chiang

Yap

2021

Computational Geometry

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The design and implementation of theoretically-sound robot motion planning algorithms is challenging. Within the framework of resolution-exact algorithms, it is possible to exploit soft predicates for collision detection. The design of soft predicates is a balancing act between easily implementable predicates and their accuracy/effectivity. In this paper, we focus on the class of planar polygonal rigid robots with arbitrarily complex geometry. We exploit the remarkable decomposability property of soft collision-detection predicates of such robots. We introduce a general technique to produce such a decomposition. If the robot is an m-gon, the complexity of this approach scales linearly in m. This contrasts with the O(m 3) complexity known for exact planners. It follows that we can now routinely produce soft predicates for any rigid polygonal robot. This results in resolution-exact planners for such robots within the general Soft Subdivision Search (SSS) framework. This is a significant advancement in the theory of sound and complete planners for planar robots. We implemented such decomposed predicates in our open-source Core Library. The experiments show that our algorithms are effective, perform in real time on non-trivial environments, and can outperform many sampling-based methods.

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“…The computation of φ(C) is hereditary, since φ(C) ⊆ φ(C ), if C is the parent of C. But it is rather costly; given φ(C ) with |φ(C )| = κ, it takes O(κ 2 ) to compute φ(C), since the relative position of C to the bisector of every pair of sites in φ(C ) must be specified. Alternatively, following the work of [22,5], instead of implementing the exact predicate φ(·), we compute an approximate one. We denote by p C , r C the center and the L ∞ -radius of C and define the active set of C as:…”

Section: Subdivision Phasementioning

confidence: 99%

“…where δ C = min s D s (p C ), if p C ∈ P, and 0 otherwise. We now explain how φ approximates φ by adapting [5,Lem.2], where φ appears as a soft version of φ.…”

Section: Subdivision Phasementioning

confidence: 99%

Voronoi Diagram of Orthogonal Polyhedra in Two and Three Dimensions

Emiris¹,

Katsamaki²

2019

Lecture Notes in Computer Science

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Voronoi diagrams are a fundamental geometric data structure for obtaining proximity relations. We consider collections of axisaligned orthogonal polyhedra in two and three-dimensional space under the max-norm, which is a particularly useful scenario in certain application domains. We construct the exact Voronoi diagram inside an orthogonal polyhedron with holes defined by such polyhedra. Our approach avoids creating full-dimensional elements on the Voronoi diagram and yields a skeletal representation of the input object. We introduce a complete algorithm in 2D and 3D that follows the subdivision paradigm relying on a bounding-volume hierarchy; this is an original approach to the problem. The complexity is adaptive and comparable to that of previous methods. Under a mild assumption it is O(n/∆ + 1/∆ 2 ) in 2D or O(nα 2 /∆ 2 + 1/∆ 3 ) in 3D, where n is the number of sites, namely edges or facets resp., ∆ is the maximum cell size for the subdivision to stop, and α bounds vertex cardinality per facet. We also provide a numerically stable, open-source implementation in Julia, illustrating the practical nature of our algorithm.

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Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

Cited by 9 publications

References 25 publications

Amortized analysis of smooth quadtrees in all dimensions

Amortized analysis of smooth quadtrees in all dimensions

Soft subdivision motion planning for complex planar robots

Voronoi Diagram of Orthogonal Polyhedra in Two and Three Dimensions

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