Summary. 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.

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 drastic 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.

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 R d . Classical approaches rely on theČech, Rips, α-complex, or witness complex filtrations of P , whose complexities scale up very badly with d. For instance, the α-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 2 O(d 2 ) n. A nice property of these filtrations is to be interleaved multiplicatively with the family of offsets of P , so that the persistence diagram of P can be approximated in 23 time in theory, with a near-linear observed running time in practice (ignoring the constant factors depending exponentially on d). Thus, our approach remains tractable in medium dimensions, say 4 to 10.Key-words: Topological persistence, Delaunay triangulation, offsets, sparse Voronoi refinement. * bhudson@tti-c.edu † glmiller@cs.cmu.edu ‡ steve.oudot@inria.fr § dsheehy@cs.cmu.edu Inférence topologique par maillageRésumé : Nous appliquons des idées issues de la litérature sur la génération de maillages afin d'eméliorer la complexité du calcul de l'information topologique persistante associéesà un nuage de poins P dans l'espace euclidien R d . Les méthodes classiques reposent sur l'utilisation de filtrations telles que celle deČech, celle de Rips-Vietoris, celle de l'α-complex ou celle du witness complex, dont les complexités se comportent très mal lorsque la dimension d augmente. Par exemple, la filtration de l'α-complex a la même taille que la triangulation de Delaunay de P , qui est de l'ordre de n Ω(d( dans le pire cas, où n est la taille de P . La solution communément adoptée consisteà tronquer les filtrations avant que leur coût ne devienne prohibitif, mais bien sûr peut-êtreégalement avant que les données topologiques les plus pertinentes n'aientété saisies. Dans cet article nous proposons une nouvelle famille de filtrations, basée sur la triangulation de Delaunay d'un sur-ensemble fini M de P , dont la taille est réduiteà 2 O(d 2 ) n. Une propriété intéressante de ces filtrations est d'être entrelacées multiplicativement avec la famille des offsets de P , si bien que le diagramme de persistance de P peutêtre approché en temps 2en théorie, avec un comportement quasi-linéaire en pratique (en laissant de côté les facteurs dépendant exponentiellement en la dimension d). Ainsi, notre approche demeure praticable en dimensions moyennes, disons entre 4 et 10.

The authors recently introduced the technique of sparse mesh refinement to produce the first near-optimal sequential time bounds of O(n lg L/s+m) for inputs in any fixed dimension with piecewiselinear constraining (PLC) features. This paper extends that work to the parallel case, refining the same inputs in time O(lg(L/s) lg m) on an EREW PRAM while maintaining the work bound; in practice, this means we expect linear speedup for any practical number of processors. This is faster than the best previously known parallel Delaunay mesh refinement algorithms in two dimensions. It is the first technique with work bounds equal to the sequential case. In higher dimension, it is the first provably fast parallel technique for any kind of quality mesh refinement with PLC inputs. Furthermore, the algorithm's implementation is straightforward enough that it is likely to be extremely fast in practice.