16505104
Approximate minimum weight Steiner triangulation in three dimensions
ApproximateMinimumWeightSteinerTriangulationinThreeDimensionsSiu-WingChengyTamal K. DeyAbstractDicultyofminimumweighttriangulationap oinsetinR2is well known.In this pap er we study the minimumweight triangulation problem for p olyhedra and general ob-stacle set in three dimensions.The weight of a triangulationin three dimensionsis assumed to b e the total surface areaof alltrianglesinvolved.It isshownthat ap olyhedronPof sizencan b e triangulated withO(2log) tetrahedra intimeO(n2log3) approximating the minimum weight trian-gulation ofPwithin a constant factor. No such prior resultisknown.Thesameb oundsalsoholdfor a3Dp ointsettriangulationallowingSteiner p oints.We consider anothersetting called general obstacle set, where the convex hull ofa set ofntrianglesis triangulatedconforming to the inputtriangles.In thiscase we show thatour metho dpro ducesa triangulati onof sizeO(n3log) in time) ap-proximating the weight of the minimum wt triangulationwithin a constant factor. This is a considerable improvementover theO(n6) b ound known for this case [3]. It is shown in[3] that minimumweighttriangulationisgo o dfor averagecase ray sho oting in three dimensions.This further substan-tiates the signi cance of our result.1Intro ductionThe minimum weight triangulation (MWT) problem ismostwell-knownintheplane,whereatriangulationwiththeminimumtotaledgelengthissoughtfora2Dp ointset.Theproblemisnotknowntob eNP-hard but no p olynomial-time algorithm is known.Re-centlyanapproximation algorithm withconstantfac-tor p erformance guarantee has b een designed for MWTby Levcop oulos and Krznaric[10].In another versionof theproblem Steinerp ointsareallowed andamini-mum weight Steinertriangulation (MWST)among allp ossibleSteinertriangulationsisdesired.Inparticu-lar, MWT is also a Steiner triangulation but with zeroSteiner p oints.Eppstein [8] showed how to approximatethe MWST of a 2Dp oint set within a constant factor.Instead of a 2Dp oint set, general inputs such as planarstraight line graphs (PSLG) have also b eenconsideredDepartment of Computer Science, HKUST, Hong Kong.e-mail: scheng@cs.ust.hkyDepartmentofCSE,I ITKharagpur,Kharagpur721302,India. e-mail: dey@cse.iitkgp.ernet.infor their applications in ray sho oting [3, 7]. A triangula-tion of a PSLG is the decomp osition of the convex hullof its verticeswhere the input line segments app ear asedges.When Steiner p oints are allowed these segmentsmay app ear as union of edges.Aronov and Fortune [3]observe that the technique of [8] applies to this case topro duce a p olynomial size triangulation approximatingMWSTwithinaconstantfactor.Inthesp ecialcasewhen the input PSLG is a p olygon p ossibly with holes,Cheng and Lee [7] presented an algorithm with b oth sizeand running timeO(nlog), whereis the numb er ofedges.Thetriangulationofap ointsetin3Disthedecomp osition of its convex hull into disjoint tetrahedra.The weight of the triangulation is the sum of the area ofthe triangles in it.Instead of p oint sets, the input mayconsist of a set of triangles with disjoint relative interior,acounterpartofPSLGin3D.Inthissettingcalledgeneral obstacle set, a minimum weight triangulation ofthe convex hull is sought where the input triangles haeto b e conformed.A sp ecial case arises when the inputtrianglesformtheb oundaryofap olyhedronwhoseinterioristob etriangulatedwithminimumweight.Noticethattheformulationofminimumweighttriangulation problem in these cases must allow Steinerp oints since there are arrangements of triangles in threespace such as the b oundary of Schonardt p olyhedron forwhich no triangulation is p ossible without Steiner p oints[12].WeconsiderMWSTproblemforthreedi erentcategories of inputs:p olyhedra, p oint sets, and generalobstaclesets.Of course,thisleaves op entheproblemof MWT of a 3Dp oint set since a p oint set can alwaysb e triangulated without Steiner p oints.ResultonMWSTprobleminthreedimensionsisscarce.Eppstein, in his pap er [8], acknowledges the dif- culty of the problem and observes the obstacles in gen-eralizing a metho d working in two dimension. When thelengths of the edges are taken as weights, he shows thatthemetho dcanb emo di edtopro duceanO(logn)-factorapproximation toMWSTofap ointset.How-ever,withareasoftrianglestakenastheweights,metho d could not b e generalized.Aronov and Fortune[3], building up on the technique of Eppstein showed hotopro duceanO(n6)sizetriangulation approximatingthe MWST of a general obstacle set.They argued that1
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Manifold reconstruction from point samples
We present an algorithm to "reconstruct" a smooth <i>k</i>-dimensional manifold <i>M</i> embedded in an Euclidean space R<sup>d</sup> from a "sufficiently dense" point sample from the manifold. The algorithm outputs a simplicial manifold that is homeomorphic and geometrically close to <i>M.</i> The running time is <i>O</i>(<i>n</i> log <i>n</i>) where <i>n</i> is the number of points in the sample (the multiplicative constant depends exponentially on the dimension though).
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Curve reconstruction from noisy samples
We present an algorithm to reconstruct a collection of disjoint smooth closed curves from n noisy samples. Our noise model assumes that the samples are obtained by first drawing points on the curves according to a locally uniform distribution followed by a uniform perturbation of each point in the normal direction with a magnitude smaller than the minimum local feature size. The reconstruction is faithful with a probability that approaches 1 as n increases.We expect that our approach can lead to provable algorithms under less restrictive noise models and for handling non-smooth features.
5215240
Sliver exudation
A sliver is a tetrahedron whose four vertices lie close to a plane and whose projection to that plane is a convex quadrilateral with no short edge. Slivers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that if the Delaunay triangulation has the ratio property introduced in [15] then there is an assignment of weights so the weighted Delaunay triangulation contains no slivers. We also give an algorithm to compute such a weight assignment.
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Quality meshing with weighted Delaunay refinement
Delaunay meshes with bounded circumradius to shortest edge length ratio have been proposed in the past for quality meshing. The only poor quality tetrahedra called slivers that can occur in such a mesh can be eliminated by the sliver exudation method. This method has been shown to work for periodic point sets, but not with boundaries. Recently a randomized point-placement strategy has been proposed to remove silvers while conforming to a given boundary. In this paper we present a deterministic algorithm for generating a weighted Delaunay mesh which respects the input boundary and has no poor quality tetrahedron including silvers. This success is achieved by combining the weight pumping method for sliver exudation and the Delaunay refinement method for boundary conformation. We show that an incremental weight pumping can be mixed seamlessly with vertex insertions in our weighted Delaunay refinement paradigm.
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Curve reconstruction from noisy samples
574471
Allocation and scheduling of real-time periodic tasks with relative timing constraints
Allocation problem has always been one of the fundamental issues of building the applications in real-time computing systems. For real-time applications, the allocation problem should directly address the issues of task and communication scheduling. In this context, the allocation of tasks has to fully utilize the available processors and the scheduling of tasks has to meet the specified timing constraints. Clearly, the execution of tasks under the allocation and schedule has to satisfy the precedence, resources, and other synchronization constraints among them. Recently, the timing requirements of the real-time systems emerge that the relative timing constraints are imposed on the consecutive executions of each task and the inter-task temporal relationships are specified across task periods. In this paper we consider the allocation and scheduling problem of the periodic tasks with such timing requirements. Given a set of periodic tasks, we consider the least common multiple (LCM) of the task periods. Each task is extended to several instances within the LCM. The scheduling window for each task instance is derived to satisfy the timing constraints. We develop a simulated annealing algorithm as the overall control algorithm. An example problem of the sanitized version of the Boeing 777 Aircraft Information Management System is solved by the algorithm. Experimental results show that the algorithm solves the problem in a reasonable time complexity.
1362947
Optimal Replication of Series-Parallel Graphs for Computation-Intensive Applications
Abstract We consider the replication problem of series-parallel (SP) task graphs where each task may run on more than one processor. The objective of the problem is to minimize the total cost of task execution and interprocessor communication. We call it the minimum cost replication problem for SP graphs (MCRP-SP). In this paper, we adopt a new communication model where the purpose of replication is to reduce the total cost. The class of applications we consider is computation-intensive applications in which the execution cost of a task is greater than its communication cost. The complexity of MCRP-SP for such applications is proved to be NP-complete. We present a branch-and-bound method to find an optimal solution as well as an approximation approach for a suboptimal solution. The numerical results show that such replication may lead to a lower cost than the optimal assignment problem (in which each task is assigned to only one processor) does. The proposed optimal solution has complexity O ( n 2 2 n M ), while the approximation solution has O ( n 4 M 2 ), where n is the number of processors in the system and M is the number of tasks in the graph.
1009077
An architecture for coordinating multiple self-management systems
A common approach to adding self-management capabilities to a system is to provide one or more external control modules, whose responsibility is to monitor system behavior, and adapt the system at run time to achieve various goals (configure the system, improve performance, recover from faults, etc.). An important problem arises when there is more than one such self-management module: how can one make sure that they are composed to provide consistent and complementary benefits? In this paper we describe a solution that introduces a self-management coordination architecture and infrastructure to support such composition. We focus on the problem of coordinating self-configuring and self-healing capabilities, particularly with respect to global configuration and incremental repair. We illustrate the approach in the context of a self-managing video teleconference system that composes two preexisting adaptation modules to achieve synergistic benefits of both.
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Trellis coded color quantization of images
We examine color quantization of images using trellis coded quantization (TCQ). Together with a simple dithering scheme, an 8-bit trellis coded color quantizer reproduces images that are visually indistinguishable from the 24-bit originals. The proposed algorithm can be viewed as a predictive trellis coded color quantization scheme. It is universal in the sense that no training or lookup table is needed. The complexity of TCQ is linear with respect to image size, making trellis coded color quantization suitable for interactive graphics and a window-based display environment.
16467768
Automatic Leveling System for E-Learning Examination Pool Using Entropy-Based Decision Tree
In this paper, we propose an automatic leveling system for e-learning examination pool using the algorithm of the decision tree. The automatic leveling system is built to automatically level each question in the examination pool according its difficulty. Thus, an e-learning system can choose questions that are suitable for each learner according to individual background. Not all attributes are relevant to the classification, in other words, the decision tree tells the importance of each attribute.