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Register a new userAn analysis of a random algorithm for estimating all the matchings
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Jinshan Zhang
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Jinshan Zhang
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Counting the number of all the matchings on a bipartite graph has been transformed into calculating the permanent of a matrix obtained from the extended bipartite graph by Yan Huo, and Rasmussen presents a simple approach (RM) to approximate the permanent, which just yields a critical ratio O($nomega(n)$) for almost all the 0-1 matrices, provided its a simple promising practical way to compute this #P-complete problem. In this paper, the performance of this method will be shown when its applied to compute all the matchings based on that transformation. The critical ratio will be proved to be very large with a certain probability, owning an increasing factor larger than any polynomial of $n$ even in the sense for almost all the 0-1 matrices. Hence, RM fails to work well when counting all the matchings via computing the permanent of the matrix. In other words, we must carefully utilize the known methods of estimating the permanent to count all the matchings through that transformation.
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Conformal Geometric Algebra (CGA) is a framework that allows the representation of objects, such as points, planes and spheres, and deformations, such as translations, rotations and dilations as uniform vectors, called multivectors. In this work, we demonstrate the merits of multivector usage with a novel, integrated rigged character simulation framework based on CGA. In such a framework, and for the first time, one may perform real-time cuts and tears as well as drill holes on a rigged 3D model. These operations can be performed before and/or after model animation, while maintaining deformation topology. Moreover, our framework permits generation of intermediate keyframes on-the-fly based on user input, apart from the frames provided in the model data. We are motivated to use CGA as it is the lowest-dimension extension of dual-quaternion algebra that amends the shortcomings of the majority of existing animation and deformation techniques. Specifically, we no longer need to maintain objects of multiple algebras and constantly transmute between them, such as matrices, quaternions and dual-quaternions, and we can effortlessly apply dilations. Using such an all-in-one geometric framework allows for better maintenance and optimization and enables easier interpolation and application of all native deformations. Furthermore, we present these three novel algorithms in a single CGA representation which enables cutting, tearing and drilling of the input rigged model, where the output model can be further re-deformed in interactive frame rates. These close to real-time cut,tear and drill algorithms can enable a new suite of applications, especially under the scope of a medical VR simulation.
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Toolpath planning is an important task in laser aided additive manufacturing (LAAM) and other direct energy deposition (DED) processes. The deposition toolpaths for complex geometries with slender structures can be further optimized by partitioning the sliced 2D layers into sub-regions, and enable the design of appropriate infill toolpaths for different sub-regions. However, reported approaches for 2D layer segmentation generally require manual operations that are tedious and time-consuming. To increase segmentation efficiency, this paper proposes an autonomous approach based on evolutional computation for 2D layer segmentation. The algorithm works in an identify-and-segment manner. Specifically, the largest quasi-quadrilateral is identified and segmented from the target layer iteratively. Results from case studies have validated the effectiveness and efficacy of the developed algorithm. To further improve its performance, a roughing-finishing strategy is proposed. Via multi-processing, the strategy can remarkably increase the solution variety without affecting solution quality and search time, thus providing great application potential in LAAM toolpath planning. To the best of the authors knowledge, this work is the first to address automatic 2D layer segmentation problem in LAAM process. Therefore, it may be a valuable supplement to the state of the art in this area.
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