Direct Diffusion Method for the Construction of Generalized Voronoi Diagrams

Nishida, Tetsushi; Ono, Shunsuke; Sugihara, Kōkichi

doi:10.1109/isvd.2007.17

Cited by 7 publications

(8 citation statements)

References 18 publications

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“…Note that Eq. (15) implies that z(x) belongs to the hyperplane in R 4 that is perpendicular to the constant vector ξ (χ ) and its distance from the origin is equal to |m|/|ξ (χ )| = |m|/|χ|. …”

Section: Analysis Of the Partitioning Problemmentioning

confidence: 99%

“…Subsequently, we assign to each node, n G , of the mesh G the index i of the vehicle for which J • (x(n G ); τ ,z i ) ≤ J • (x(n G ); τ ,z j ), for all j ∈ I n \ {i}, where x(n G ) denotes the location of the node n G in the plane. This naive partitioning scheme has time complexity O(n|G|), where |G| denotes the number of nodes of the spatial mesh [15]. Typically, the size of the mesh depends on the number of vehicles; the higher the number of vehicles, the finer the mesh should be.…”

Section: Numerical Simulationsmentioning

confidence: 99%

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Decentralized spatial partitioning algorithms for multi-vehicle systems based on the minimum control effort metric

Bakolas

2014

Systems & Control Letters

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Section: Analysis Of the Partitioning Problemmentioning

confidence: 99%

Section: Numerical Simulationsmentioning

confidence: 99%

Decentralized spatial partitioning algorithms for multi-vehicle systems based on the minimum control effort metric

Bakolas

2014

Systems & Control Letters

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“…A straightforward approach to address such partitioning problems is to use algorithms that compute approximations of the desired partitions by utilizing a discretization grid, say G, over the space to be partitioned. For example, one such approach involves the use of computational techniques that are commonly employed for the numerical solution of partial differential equations, and in particular, direct diffusion methods [27]. The time complexity of the approach presented in [27] is O(card(G)), where card(G) denotes the cardinality of the grid G, which is, in turn, equal to the number of its nodes.…”

Section: B Computation Of the Solution To The Evading Roadmap Problemmentioning

confidence: 99%

“…For example, one such approach involves the use of computational techniques that are commonly employed for the numerical solution of partial differential equations, and in particular, direct diffusion methods [27]. The time complexity of the approach presented in [27] is O(card(G)), where card(G) denotes the cardinality of the grid G, which is, in turn, equal to the number of its nodes. An alternative approach is to compute the so-called lower envelope function that bounds from below the graphs of the proximity metric associated with each generator in S × [0, ∞) (see, for example, [28]).…”

Section: B Computation Of the Solution To The Evading Roadmap Problemmentioning

confidence: 99%

Evasion from a group of pursuers with double integrator kinematics

Bakolas

2013

52nd IEEE Conference on Decision and Control

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We consider the problem of characterizing an evading strategy for an agent traversing a convex polygon populated by a group of pursuers. We address the problem by associating it with a generalized Voronoi partitioning problem, which encodes information about the proximity relations between the evader and the pursuers based on the value function of a pursuit-evasion game involving the evader and each pursuer from the group individually. The generalized Voronoi partition furnishes a collection of continuous paths which have the following property: When the evader travels along any of these paths, none of the pursuers will have a unilateral incentive to initiate the pursuit against it. With the proposed approach, the problem of evasion from the group of pursuers admits an elegant geometric solution, which can be computed by means of known computational techniques. Numerical simulations that illustrate the theoretical developments are presented.

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“…While this situation may occur in reality and cause a serious computational problem, such a degeneracy has been recently well-solved by the exact computation technique proposed by Sugihara's group. [47] There is a rich set of literature on the Voronoi diagram of the threedimensional sphere set regarding on its definition, algorithms, and applications. [22,[48][49][50][51][52][53][54][55][56][57] Given the Voronoi diagram, the quasi-triangulation QT of A is the dual structure of VD.…”

Section: Voronoi Diagram Quasi-triangulation and The Beta-complexmentioning

confidence: 99%

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

et al. 2012

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Given a set of spherical balls, called atoms, in three-dimensional space, its mass properties such as the volume and the boundary area of the union of the atoms are important for many disciplines, particularly for computational chemistry/biology and structural molecular biology. Despite many previous studies, this seemingly easy problem of computing mass properties has not been well-solved. If the mass properties of the union of the offset of the atoms are to be computed as well, the problem gets even harder. In this article, we propose algorithms that compute the mass properties of both the union of atoms and their offsets both correctly and efficiently. The proposed algorithms employ an approach, called the Beta-decomposition, based on the recent theory of the beta-complex. Given the beta-complex of an atom set, these algorithms decompose the target mass property into a set of primitives using the simplexes of the beta-complex. Then, the molecular mass property is computed by appropriately summing up the mass property corresponding to each simplex. The time complexity of the proposed algorithm is O(m) in the worst case where m is the number of simplexes in the beta-complex that can be efficiently computed from the Voronoi diagram of the atoms. It is known in ℝ(3) that m = O(n) on average for biomolecules and m = O(n(2)) in the worst case for general spheres where n is the number of atoms. The theory is first introduced in ℝ(2) and extended to ℝ(3). The proposed algorithms were implemented into the software BetaMass and thoroughly tested using molecular structures available in the Protein Data Bank. BetaMass is freely available at the Voronoi Diagram Research Center web site.

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Direct Diffusion Method for the Construction of Generalized Voronoi Diagrams

Cited by 7 publications

References 18 publications

Decentralized spatial partitioning algorithms for multi-vehicle systems based on the minimum control effort metric

Decentralized spatial partitioning algorithms for multi-vehicle systems based on the minimum control effort metric

Evasion from a group of pursuers with double integrator kinematics

Beta‐decomposition for the volume and area of the union of three‐dimensional balls and their offsets

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