There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Computational geometry

TetGen is a Cpp program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available.This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.

TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator

combinatorial auctions What to say and what to do when mostly your friends love reading? Are you the one that don't have such hobby? So, it's important for you to start having that hobby. You know, reading is not the force. We're sure that reading will lead you to join in better concept of life. Reading will be a positive activity to do every time. And do you know our friends become fans of combinatorial auctions as the best book to read? Yeah, it's neither an obligation nor order. It is the referred book that will not make you feel disappointed.

Combinatorial Auctions

Irregular applications, which manipulate large, pointer-based data structures like graphs, are difficult to parallelize manually. Automatic tools and techniques such as restructuring compilers and run-time speculative execution have failed to uncover much parallelism in these applications, in spite of a lot of effort by the research community. These difficulties have even led some researchers to wonder if there is any coarse-grain parallelism worth exploiting in irregular applications.In this paper, we describe two real-world irregular applications: a Delaunay mesh refinement application and a graphics application thatperforms agglomerative clustering. By studying the algorithms and data structures used in theseapplications, we show that there is substantial coarse-grain, data parallelism in these applications, but that this parallelism is very dependent on the input data and therefore cannot be uncoveredby compiler analysis. In principle, optimistic techniques such asthread-level speculation can be used to uncover this parallelism, but we argue that current implementations cannot accomplish thisbecause they do not use the proper abstractions for the data structuresin these programs.These insights have informed our design of the Galois system, an object-based optimistic parallelization system for irregular applications. There are three main aspects to Galois: (1) a small number of syntactic constructs for packaging optimistic parallelism as iteration over ordered and unordered sets, (2)assertions about methods in class libraries, and (3) a runtime scheme for detecting and recovering from potentially unsafe accesses to shared memory made by an optimistic computation.We show that Delaunay mesh generation and agglomerative clustering can be parallelized in a straight-forward way using the Galois approach, and we present experimental measurements to show that this approach is practical. These results suggest that Galois is a practical approach to exploiting data parallelismin irregular programs.

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Optimistic parallelism requires abstractions

Combinatorial auctions where agents can bid on bundles of items are desirable because they allow the agents to express complementarity and substitutability between the items. However, expressing one's preferences can require bidding on all bundles. Selective incremental preference elicitation by the auctioneer was recently proposed to address this problem [4], but the idea was not evaluated. In this paper we show, experimentally and theoretically, that automated elicitation provides a large benefit. In all of the elicitation schemes under study, as the number of items for sale increases, the amount of information elicited is a vanishing fraction of the information collected in traditional "direct revelation mechanisms" where bidders reveal all their valuation information. Most of the elicitation schemes also maintain the benefit as the number of agents increases. We develop more effective elicitation policies for existing query types. We also present a new query type that takes the incremental nature of elicitation to a new level by allowing agents to give approximate answers that are refined only on an as-needed basis. In the process, we present methods for evaluating different types of elicitation policies.

Effectiveness of Preference Elicitation in Combinatorial Auctions

We present a new algorithm, Sparse Voronoi Refinement, that produces a conformal Delaunay mesh in arbitrary dimension with guaranteed mesh size and quality. Our algorithm runs in output-sensitive time O(n log(L/s)+m), with constants depending only on dimension and on prescribed element shape quality bounds. For a large class of inputs, including integer coordinates, this matches the optimal time bound of Θ(n log n + m). Our new technique uses interleaving: we maintain a sparse mesh as we mix the recovery of input features with the addition of Steiner vertices for quality improvement. This technical report is the long version of an article [HMP06] presented at IMR 2006, and contains full proofs.

/pdf/sparse-voronoi-refinement-nd58luoa3x.pdf

Sparse Voronoi Refinement

Combinatorial auctions, where agents can bid on bundles of items (resources, tasks, etc.), are desirable because the agents can express complementarity and substitutability among the items. However, expressing oneýs preferences can require bidding on all bundles. We evaluate an approach known as incremental preference elicitation [3] and show that as the number of items increases, the amount of information required to clear the auction is a vanishing fraction of the information collected in direct revelation mechanisms. Most of the elicitors also maintain the benefit as the number of agents increases. We prove that randomization helps, in that no deterministic elicitor is a universal revelation reducer. Finally, we present a new query type that allows agents to use anytime algorithms to give approximate answers that are refined only as needed.

/pdf/effectiveness-of-query-types-and-policies-for-preference-41v280bwe4.pdf

Effectiveness of Query Types and Policies for Preference Elicitation in Combinatorial Auctions

We apply ideas from mesh generation to improve the time and space complexities of computing the full persistent homological information associated with a point cloud P in Euclidean space ℜd. Classical approaches rely on the Cech, Rips, ±-complex, or witness complex filtrations of P, whose complexities scale up very badly with d. For instance, the ±-complex filtration incurs the n Ω(d) size of the Delaunay triangulation, where n is the size of P. The common alternative is to truncate the filtrations when the sizes of the complexes become prohibitive, possibly before discovering the most relevant topological features. In this paper we propose a new collection of filtrations, based on the Delaunay triangulation of a carefully-chosen superset of P, whose sizes are reduced to 2O(d2)n. Our filtrations interleave multiplicatively with the family of offsets of P, so that the persistence diagram of P can be approximated in 2O(d2)n3 time in theory, with a near-linear observed running time in practice. Thus, our approach remains tractable in medium dimensions, say 4 to 10.

/pdf/topological-inference-via-meshing-121cwx15rc.pdf

Topological inference via meshing

In a well-spaced point set, when there is a bounding hypercube, the Voronoi cells all have bounded aspect ratio, i.e., the distance from the Voronoi site to the farthest point in the Voronoi cell divided by the distance to the nearest neighbor in the set is bounded by a small constant. Well-spaced point sets satisfy some important geometric properties and yield quality Voronoi or simplicial meshes that can be important in scientific computations. In this paper, we consider the dynamic well-spaced point sets problem, which requires computing the well-spaced superset of a dynamically changing input set, e.g., as input points are inserted or deleted. We present a dynamic algorithm that allows inserting/deleting points into/from the input in worst-case O(log Δ) time, where Δ is the geometric spread, a natural measure that is bounded by O(log n) when input points are represented by log-size words. We show that the runtime of the dynamic update algorithm is optimal in the worst case. Our algorithm generates size-optimal outputs: the resulting output sets are never more than a constant factor larger than the minimum size necessary. A preliminary implementation indicates that the algorithm is indeed fast in practice. To the best of our knowledge, this is the first time- and size-optimal dynamic algorithm for well-spaced point sets.

/pdf/dynamic-well-spaced-point-sets-3n14wg6o2t.pdf

Benoît Hudson

Papers

Effectiveness of Preference Elicitation in Combinatorial Auctions

Sparse Voronoi Refinement

Effectiveness of Query Types and Policies for Preference Elicitation in Combinatorial Auctions

Topological inference via meshing

Dynamic well-spaced point sets