pollack@GEOMETRY.CIMS.NYU.EDU (Ricky Pollack) (10/25/90)
FIFTEENTH COMPUTATIONAL GEOMETRY DAY Friday, November 9, 1988 New York University Courant Institute of Mathematical Sciences Room 109, Warren Weaver Hall 251 Mercer St., New York, NY 10012 10:00-10:30 - Coffee (13th floor lounge) 10:30-11:15 Pankaj K. Agarwal, Duke University Applications of a new Space Partitioning Technique 11:30-12:15 Godfried Toussaint, McGill University Computing Extremal Transversals of Sets 12:30-2:00 - Lunch 2:00-3:00 Open Problem Session 3:00-3:45 Ketan Mulmuley, University of Chicago Dynamic Problems in Randomized Coimputational Geometry 4:00-5:00 - Wine and Cheese Reception in the 13th floor lounge For more information contact: Richard Pollack (212) 998-3167 pollack@geometry.nyu.edu **************************************************************************** Pankaj K. Agarwal, Duke University Applications of a New Partitioning Technique Recently Chazelle, Sharir and Welzl proposed a new algorithm for simplex range searching based on a clever partitioning scheme. In this talk we show that their partitioning scheme cane be applied to a variety of problems including ray shooting in 2D and 3D, hidden surface removal, and computing spanning trees of low stabbing number. * This is a joint work with Micha Sharir ***************************************************************************** Godfried Toussaint, McGill University Computing Extremal Transversals of Sets We will present a divide-and-conquer strategy for triangulating a simple polygon efficiently. The algorithm is deterministic and runs in time $O(nlog^*n)$, where n is the number of vertices. The same method can also be used to check whether a polygonal curve is simple. ***************************************************************************** Ketan Mulmuley, University of Chicago Dynamic Problems in Randomized Computational Geometry In recent years randomized incrementation has turned out to be a highly successful paradigm in computational geometry. There are two general theories underlying this paradigm: 1) the theory of random sampling (Clarkson-Shor) and 2) the theory of probabilistic geometric games. These techniques can be used to analyze the algorithms which are randomized, and incremental, i.e. to say the algorithms which add objects one at a time in a random order. Unfortunately, both theories become inapplicable when deletions of objects are allowed. This happens to be the case when one is interested in designing fully dynamic algorithms, allowing additions as well as deletions, for the geometric problems under consideration. Here we shall extend the theory of probabilistic games to problems involving deletions. This enables us to design fully dynamic algorithms for several important goemtric problems and show that they have good expected behaviour. These problems include: * Dynamic construction of planar partitions, * Semi-dynamic hidden surface removal, * Dynamic construction of Voronoi diagrams in $R^d$, * Dynamic point location in arrangements of hyperplanes (dimension three and four), * Dynamic construction of three dimensional partitions, i.e. partitions induced by polygons in $R^3$, and dynamic point location in such paritions, * Dynamic point location in Voronoi diagrams in $R^2$ and $R^3$. A crucial ingradient in this extension is a certain Theta-series that can be associated with abstract combinatorial systems. Previously we had associated such a series with arrangements of hyperplanes and polytopes in $R^d$ and had studied semi-dynamic (only additions allowed), random evolutions of geometric configurations using this series. Now we study fully dynamic, random evolutions.