158 research outputs found

    An Efficient, Practical Algorithm and Implementation for Computing Multiplicatively Weighted Voronoi Diagrams

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    We present a simple wavefront-like approach for computing multiplicatively weighted Voronoi diagrams of points and straight-line segments in the Euclidean plane. If the input sites may be assumed to be randomly weighted points then the use of a so-called overlay arrangement [Har-Peled&Raichel, Discrete Comput. Geom. 53:547-568, 2015] allows to achieve an expected runtime complexity of O(nlog4n)O(n\log^4 n), while still maintaining the simplicity of our approach. We implemented the full algorithm for weighted points as input sites, based on CGAL. The results of an experimental evaluation of our implementation suggest O(nlog2n)O(n\log^2 n) as a practical bound on the runtime. Our algorithm can be extended to handle also additive weights in addition to multiplicative weights, and it yields a truly simple O(nlogn)O(n\log n) solution for solving the one-dimensional version of this problem

    Vorosweep: a fast generalized crystal growing Voronoi diagram generation algorithm

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    We propose a new algorithm for generating quickly approximate generalized Voronoi diagrams of point sites associated to arbitrary convex distance metric in the Euclidian plane. This algorithm produces connected cells by emulating the growth of crystals starting at the point sites, in order to reduce the complexity of the diagram. The main practical contribution is the Vorosweep package which is the reference implementation of the algorithm. Experimental results and benchmarks are given to demonstrate the versatility of this approach.WIST 3 grant 1017074 DOMHEX (Dominant Hexahedral Mesh Generation

    Generalized offsetting of planar structures using skeletons

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    We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams

    Zone design of specific sizes using adaptive additively weighted voronoi diagrams

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    Territory or zone design processes entail partitioning a geographic space, organized as a set of areal units, into different regions or zones according to a specific set of criteria that are dependent on the application context. In most cases, the aim is to create zones of approximately equal sizes (zones with equal numbers of inhabitants, same average sales, etc.). However, some of the new applications that have emerged, particularly in the context of sustainable development policies, are aimed at defining zones of a predetermined, though not necessarily similar, size. In addition, the zones should be built around a given set of seeds. This type of partitioning has not been sufficiently researched; therefore, there are no known approaches for automated zone delimitation. This study proposes a new method based on a discrete version of the adaptive additively weighted Voronoi diagram that makes it possible to partition a two-dimensional space into zones of specific sizes, taking both the position and the weight of each seed into account. The method consists of repeatedly solving a traditional additively weighted Voronoi diagram, so that each seed?s weight is updated at every iteration. The zones are geographically connected using a metric based on the shortest path. Tests conducted on the extensive farming system of three municipalities in Castile-La Mancha (Spain) have established that the proposed heuristic procedure is valid for solving this type of partitioning problem. Nevertheless, these tests confirmed that the given seed position determines the spatial configuration the method must solve and this may have a great impact on the resulting partition

    A Method Using GIS Integrated Voronoi Diagrams for Commuter Rail Station Identification: A Case Study from Brasilia (Brazil)

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    AbstractThis article describes a station location method for a commuter rail system applying a GIS integrated Voronoi diagrams. The method comes from a previously defined track line and considers the stations area coverage. As parameters to define area coverage we first used the point density representing the maximum concentrated activity area. We also use the trip generating rate weights for point density. The method was applied to Brasilia Metropolitan Area, and the final product was a “T” Trunk-Feeder framework that allows an integrated transportation system planning considering others existing transit systems. The stations were classified according to their degree of activities density and its importance for integration. The “T” Trunk-Feeder is a representation of the commuter rail network and the Trunk-Distribution is the transport network responsible for distributing the passengers in central business district - CDB. The proposed model features 69km extension, with the estimated total travel time of 62minutes, and 17 stations being integrated into 2 metro, 4 integrated into LRT stations and 3 integrated into the bus stops
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