On the Analysis of Spatially Constrained Power of Two Choice Policies

We consider a class of power of two choice based assignment policies for allocating users to servers, where both users and servers are located on a two-dimensional Euclidean plane. In this framework, we investigate the inherent tradeoff between the communication cost, and load balancing performance of different allocation policies. To this end, we first design and evaluate a Spatial Power of two (sPOT) policy in which each user is allocated to the least loaded server among its two geographically nearest servers sequentially. When servers are placed on a two-dimensional square grid, sPOT maps to the classical Power of two (POT) policy on the Delaunay graph associated with the Voronoi tessellation of the set of servers. We show that the associated Delaunay graph is 4-regular and provide expressions for asymptotic maximum load using results from the literature. For uniform placement of servers, we map sPOT to a classical balls and bins allocation policy with bins corresponding to the Voronoi regions associated with the second order Voronoi diagram of the set of servers. We provide expressions for the lower bound on the asymptotic expected maximum load on the servers and prove that sPOT does not achieve POT load balancing benefits. However, experimental results suggest the efficacy of sPOT with respect to expected communication cost. Finally, we propose two non-uniform server sampling based POT policies that achieve the best of both the performance metrics. Experimental results validate the effectiveness of our proposed policies.

A Parallelizing Method for Generation of Voronoi Diagram Using Contact Zone

Due to the recent popularization of the Geographic Information System (GIS), spatial network environments that can display the changes of spatial axes on mobile devices are receiving great attention. In spatial network environments, since a query object that seeks location information selects several candidate target objects based on the search conditions, we often use a k-nearest neighbor (kNN) search, which seeks several target objects near the query object. However, since a kNN search needs to find the kNN by calculating the distance from the query to all the objects, the computational complexity might become too large based on the number of objects. To reduce this computation time in a kNN search, many researchers have proposed a search method that divides regions using a Voronoi diagram. However, since conventional methods generate Voronoi diagrams for objects in order, the processing time for generating Voronoi diagrams might become too large when the number of objects is increased. In this paper, we propose a generation method of the Voronoi diagram by parallelizing the generation of Voronoi regions using a contact zone. Our proposed method can reduce the processing time of generating the Voronoi diagram by generating Voronoi regions in parallel based on the number of targets. Our evaluation confirmed that the processing time under the proposed method was reduced about 15.9\% more than conventional methods that are not parallelized.

Computation of Compact Distributions of Discrete Elements

In our daily lives, many plane patterns can actually be regarded as a compact distribution of a number of elements with certain shapes, like the classic pattern mosaic. In order to synthesize this kind of pattern, the basic problem is, with given graphics elements with certain shapes, to distribute a large number of these elements within a plane region in a possibly random and compact way. It is not easy to achieve this because it not only involves complicated adjacency calculations, but also is closely related to the shape of the elements. This paper attempts to propose an approach that can effectively and quickly synthesize compact distributions of elements of a variety of shapes. The primary idea is that with the seed points and distribution region given as premise, the generation of the Centroidal Voronoi Tesselation (CVT) of this region by iterative relaxation and the CVT will partition the distribution area into small regions of Voronoi, with each region representing the space of an element, to achieve a compact distribution of all the elements. In the generation process of Voronoi diagram, we adopt various distance metrics to control the shape of the generated Voronoi regions, and finally achieve the compact element distributions of different shapes. Additionally, approaches are introduced to control the sizes and directions of the Voronoi regions to generate element distributions with size and direction variations during the Voronoi diagram generation process to enrich the effect of compact element distributions. Moreover, to increase the synthesis efficiency, the time-consuming Voronoi diagram generation process was converted into a graphical rendering process, thus increasing the speed of the synthesis process. This paper is an exploration of elements compact distribution and also carries application value in the fields like mosaic pattern synthesis.

Design of a New Parametric Path Plan for Additive Manufacturing of Hollow Porous Structures With Functionally Graded Materials

In this paper, a novel path planning approach is proposed to generate porous structures with internal features. The interconnected and continuous deposition path is designed to control the internal material composition in a functionally graded manner. The proposed layer-based algorithmic solutions generate a bilayer pattern of zigzag and spiral toolpath consecutively to construct heterogeneous three-dimensional (3D) objects. The proposed strategy relies on constructing Voronoi diagrams for all bounding curves in each layer to decompose the geometric domain and discretizing the associated Voronoi regions with ruling lines between the boundaries of the associated Voronoi regions. To avoid interference among ruling lines, reorientation and relaxation techniques are introduced to establish matching for continuous zigzag path planning. In addition, arc fitting is used to reduce over-deposition, allowing nonstop deposition at sharp turns. Layer-by-layer deposition progresses through consecutive layers of a ruling-line-based zigzag pattern followed by a spiral path deposition. A biarc fitting technique is employed through isovalues of ruling lines to generate G1 continuity along the spiral deposition path plan. Functionally graded material properties are then mapped based on a parametric distance-based weighting technique. The proposed approach enables elimination or minimization of over-deposition of materials, nonuniformity on printed strands and discontinuities on the toolpath, which are shortcomings of traditional zigzag-based toolpath plan in additive manufacturing (AM). In addition, it provides a practical path for printing functionally graded materials.

A Level Set Method for the Construction of Boundary Conforming Voronoi Regions and Delaunay Triangulations Governed by a Spatial Distribution of Metrics

A Voronoi region can be interpreted as the shape achieved by a crystal that grows from a seed and stops growing when it reaches either the domain boundary or another crystal. This analogy is exploited here to devise a method for the generation of anisotropic boundary-conforming Voronoi regions for a set of points. This is achieved by simulating the propagation of crystals as evolving fronts modeled by a level set method. The techniques to detect the collision of fronts (crystals), formation of interfaces between seeds, and treatment of boundaries as additional (inner or outer) restricting seeds are described in detail. The generation of anisotropic Voronoi regions consistent with a user-prescribed Riemannian metric is achieved by re-interpreting the metric tensor in terms of the speed of propagation normal to the boundary of the crystal. This re-interpretation offers a better means of restricting metric fields for mesh generation.