VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

Improving Distance-Join Query Processing with Voronoi-Diagram based Partitioning in SpatialHadoop

SpatialHadoop is an extended MapReduce framework supporting global indexing techniques that partition spatial datasets across several machines and improve spatial query processing performance compared to traditional Hadoop systems. SpatialHadoop supports several spatial operations (e.g., Nearest Neighbor search, range query, spatial intersection join, etc.) and seven spatial partitioning techniques (Grid, Quadtree, STR, STR+, -d tree, Z-curve and Hilbert-curve). Distance-Join Queries (DJQs), like the Nearest Neighbors Join Query (NNJQ) and Closest Pairs Query (CPQ), are common operations used in numerous spatial applications. DJQs are costly operations, since they combine spatial joins with distance-based search. Data partitioning improves the management of large datasets and speeds up query performance. Therefore, performing DJQs efficiently with new partitioning methods in SpatialHadoop is a challenging task. In this paper, a new data partitioning technique based on Voronoi-Diagrams is designed and implemented in SpatialHadoop. Moreover, improved NNJQ and CPQ MapReduce algorithms, using the new partitioning mechanism, are also designed and developed for SpatialHadoop. Finally, the results of an extensive set of experiments with real-world datasets are presented, demonstrating that the new partitioning technique and the improved DJQ MapReduce algorithms are efficient, scalable and robust in SpatialHadoop

VGQ-Vor: extending virtual grid quadtree with Voronoi diagram for mobile k nearest neighbor queries over mobile objects

Abstract Performing mobile k nearest neighbor (MkNN) queries whilst also being mobile is a challenging problem. All the mobile objects issuing queries and/or being queried are mobile. The performance of this kind of query relies heavily on the maintenance of the current locations of the objects. The index used for mobile objects must support efficient update operations and efficient query handling. This study aims to improve the performance of the MkNN queries while reducing update costs. Our approach is based on an observation that the frequency of one region changing between being occupied or not by mobile objects is much lower than the frequency of the position changes reported by the mobile objects. We first propose an virtual grid quadtree with Voronoi diagram (VGQ-Vor), which is a two-layer index structure that indexes regions occupied by mobile objects in a quadtree and builds a Voronoi diagram of the regions. Then we propose a moving k nearest neighbor (kNN) query algorithm on the VGQ-Vor and prove the correctness of the algorithm. The experimental results show that the VGQ-Vor outperforms the existing techniques (Bx-tree, Bdual-tree) by one to three orders of magnitude in most cases

Smooth Distance Approximation

Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities. In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation. We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in ?^d from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute ?-approximation to this query in time O(log (1/?)) using O(1/?^{d/2}) storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian

Smooth Distance Approximation

Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities. In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation. We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in $\mathbb{R}^d$ from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute $\varepsilon$-approximation to this query in time $O(\log (1/\varepsilon))$ using $O(1/\varepsilon^{d/2})$ storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian.Comment: To appear in the European Symposium on Algorithms (ESA) 202

Voronoi classfied and clustered constellation data structure for three-dimensional urban buildings

In the past few years, the growth of urban area has been increasing and has resulted immense number of urban datasets. This situation contributes to the difficulties in handling and managing issues related to urban area. Huge and massive datasets can degrade the performance of data retrieval and information analysis. In addition, urban environments are very difficult to manage because they involved with various types of data, such as multiple types of zoning themes in urban mixeduse development. Thus, a special technique for efficient data handling and management is necessary. In this study, a new three-dimensional (3D) spatial access method, the Voronoi Classified and Clustered Data Constellation (VOR-CCDC) is introduced. The VOR-CCDC data structure operates on the basis of two filters, classification and clustering. To boost up the performance of data retrieval, VORCCDC offers a minimal percentage of overlap among nodes and a minimal coverage area in order to avoid repetitive data entry and multi-path queries. Besides that, VOR-CCDC data structure is supplemented with an extra element of nearest neighbour information. Encoded neighbouring information in the Voronoi diagram allows VOR-CCDC to optimally explore the data. There are three types of nearest neighbour queries that are presented in this study to verify the VOR-CCDC’s ability in finding the nearest neighbour information. The queries are Single Search Nearest Neighbour query, k Nearest Neighbour (kNN) query and Reverse k Nearest Neighbour (RkNN) query. Each query is tested with two types of 3D datasets; single layer and multi-layer. The test demonstrated that VOR-CCDC performs the least amount of input/output than their best competitor, the 3D R-Tree. Besides that, VOR-CCDC is also tested for performance evaluation. The results indicate that VOR-CCDC outperforms its competitor by responding 60 to 80 percent faster to the query operation. In the future, VOR-CCDC structure is expected to be expanded for temporal and dynamic objects. Besides that, VOR-CCDC structure can also be used in other applications such as brain cell database for analysing the spatial arrangement of neurons or analysing the protein chain reaction in bioinformatics applications