18074193
Dynamic Diffusion Load Balancing
We consider the problem of dynamic load balancing in arbitrary (connected) networks on n nodes. Our load generation model is such that during each round, n tasks are generated on arbitrary nodes, and then (possibly after some balancing) one task is deleted from every non-empty node. Notice that this model fully saturates the resources of the network in the sense that we generate just as many new tasks per round as the network is able to delete. We show that even in this situation the system is stable, in that the total load remains bounded (as a function of n alone) over time. Our proof only requires that the underlying “communication” graph be connected. (It of course also works if we generate less than n new tasks per round, but the major contribution of this paper is the fully saturated case.) We further show that the upper bound we obtain is asymptotically tight (up to a moderate multiplicative constant) by demonstrating a corresponding lower bound on the system load for the particular example of a linear array (or path). We also show some simple negative results (i.e., instability) for work-stealing based diffusion-type algorithms in this setting.
94046
Rapidly mixing Markov chains for sampling contingency tables with a constant number of rows
We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. (2002) have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which only employs Markov chain methods indirectly. But they leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we answer this question in the affirmative, and hence provide a very different proof of the main result of Cryan and Dyer. We show that the "2 /spl times/ 2 heat-bath" Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris and Sinclair (2002) (see also Morris (2002)) for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste (1993) to show that the 2 /spl times/ 2 chain is rapidly mixing. As part of our analysis, we give the first proof that the 2 /spl times/ 2 chain mixes in time polynomial in the input size when both the number of rows and the number of columns is constant.
13827959
Turning Shape Decision Problems into Measures
This paper considers the problem of constructing shape measures; we start by giving a short overview of areas of practical application of such measures. 
Shapes can be characterised in terms of a set of geometric properties, some of which are Boolean in nature. We give two general principles for turning Boolean properties into continuous measures of that property. (For example, we may turn the Boolean property conves into a convexity value in the range [0–1].) We demonstrate how these principles can be applied to construct various shape measures, including ones for convexity, circularity, ellipticity, triangularity, rectilinearity, rectangularity and symmetry in two dimensions, and 2.5D-ness, stability, and imperforateness in three dimensions. Some of these measures are new; others are well known and we show how they fit into this general framework. 
We also show how such measures for a single shape can be generalised to multiple shapes, and consider as examples measures for containment, resemblance, congruence, and similarity.
14838227
Sweeping of three-dimensional objects
Evaluating the volume swept out by a three-dimensional (3D) object as it moves along an arbitrary path is of interest to many areas of CAD and CAM, such as mechanism design and robot path planning. This paper shows how envelope theory from differential geometry can be used to find the volumes swept out by the individual surfaces of a solid body, and how computer algebra methods may be of use to perform the computations involved. Finally, a new algorithm is presented which shows how the results of sweeping the individual surfaces of a solid body can be combined to form a new 3D model of the swept volume. This algorithm has strong resemblance to hidden line algorithms, but works in one dimension higher.
18253899
Density-Controlled Sampling of Parametric Surfaces Using Adaptive Space-Filling Curves
Low-discrepancy point distributions exhibit excellent uniformity properties for sampling in applications such as rendering and measurement. We present an algorithm for generating low-discrepancy point distributions on arbitrary parametric surfaces using the idea of converting the 2D sampling problem into a 1D problem by adaptively mapping a space-filling curve onto the surface. The 1D distribution takes into account the parametric mapping by employing a corrective approach similar to histogram equalisation to ensure that it gives a 2D low-discrepancy point distribution on the surface. This also allows for control over the local density of the distribution, e.g. to place points more densely in regions of higher curvature. To allow for parametric distortion, the space-filling curve is generated adaptively to cover the surface evenly. Experiments show that this approach efficiently generates low-discrepancy distributions on arbitrary parametric surfaces and creates nearly as good results as well-known low-discrepancy sampling methods designed for particular surfaces like planes and spheres. However, we also show that machine-precision limitations may require surface reparameterisation in addition to adaptive sampling.
2505791
Morphing based on strain field interpolation
Strain fields provide a method of deformation measurement based on physics. Using these as a tool, we can analyze deformation of objects in a measurable way. We have developed a new morphing technique based on strain field interpolation. Shape shaking and squeezing, which often happen when using linear interpolation for morphing, do not arise in our approach. We have also developed a new method to create isomorphic meshes from corresponding objects in two images. Meshes generated by this method have much fewer triangles than other methods, which greatly decreases calculation loads in the morphing process. Copyright # 2004 John Wiley & Sons, Ltd.
37238502
Computing Volume Properties Using Low-Discrepancy Sequences
This paper considers the use of low-discrepancy sequences for computing volume integrals in solid modelling. An introduction to low-discrepancy point sequences is presented which explains how they can be used to replace random points in Monte Carlo methods. The relative advantages of using low-discrepancy methods compared to random point sequences are discussed theoretically, and then practical results are given for a series of test objects which clearly demonstrate the superiority of the low-discrepancy method when used in a simple approach. Finally, the performance of such methods is assessed when used in conjunction with spatial subdivision in the svLis geometric modeller.