The power crust is a construction which takes a sample of points from the surface of a three-dimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce the surface representation from the MAT.

Abstract. This paper gives an algcmthm for solving linear programming problems. For a problem with tz constraints and d variables, the algorithm requires an expected O(d’n) + (log n)o(d)d’’+(’(’) + o(dJA log n) arithmetic operations, as rz ~ ~. The constant factors do not depend on d. Also, an algorlthm N gwen for integer hnear programmmg. Let p bound the number of bits required to specify the ratmnal numbers defmmg an input constraint or the ob~ective function vector. Let n and d be as before. Then, the algorithm requires expected 0(2d dn + S~dm In n) + dc)’d) ~ in H operations on numbers with O(1~p bits d ~ ~ ~z + ~, where the constant factors do not depend on d or p. The expectations are with respect to the random choices made by the algorithms, and the bounds hold for any gwen input. The techmque can be extended to other convex programming problems. For example, m algorlthm for finding the smallest sphere enclosing a set of /z points m Ed has the same t]me bound

Let P be a set of n points in R d (where d is a small fixed positive integer), and let \Gamma be a collection of subsets of R d , each of which is defined by a constant number of bounded degree polynomials. We consider the following \Gamma-range searching problem: Given P , build a data structure for efficient answering of queries of the form `Given a fl 2 \Gamma, count (or report) the points of P lying in fl'. Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and with O(n 1\Gamma1=b+ffi ) query time, where d b 2d \Gamma 3 and ffi ? 0 is an arbitrarily small constant. The actual value of b is related to the problem of partitioning arrangements of algebraic surfaces into constant-complexity cells. We present some of the applications of \Gamma-range searching problem, including improved ray shooting among triangles in R³.

We present an efficient O(n + 1/ε^4.5)-time algorithm for computing a (1 + 1/ε)-approximation of the minimum-volume bounding box of n points in R³. We also present a simpler algorithm (for the same purpose) whose running time is O(n log n+n/ε³). We give some experimental results with implementations of various variants of the second algorithm. The implementation of the algorithm described in this paper is available online [Har00].

With the transformation of the Internet into a commercial infrastructure, the ability to provide differentiated services to users with widely varying requirements is rapidly becoming as important as meeting the massive increases in bandwidth demand. Hence, while deploying routers, switches, and transmission systems of ever increasing capacity, Internet service providers would also like to provide customer-specific differentiated services using the same shared network infrastructure. In this article, we describe router architectures that can support the two trends of rising bandwidth demand and rising demand for differentiated services. We focus on router mechanisms that can support differentiated services at a level not contemplated in proposals currently under consideration due to concern regarding their implementability at high speeds. We consider the types of differentiated services that service providers may want to offer and then discuss the mechanisms needed in routers to support them. We describe plausible implementations of these mechanisms (the scalability and performance of which have been demonstrated by implementation in a prototype system) and argue that it is

We define the antiumbra and the antipenumbra of aconvex area light source shining through a sequence of convex areal holes in three dimensions. The antiumbra is the volume from which all points on the light source can be seen. The antipenumbra is the volume from which some, but not all, of the light source can be seen. We show that the antipenumbra is, in general, a disconnected set bounded by portions of quadric surfaces, and describe an implemented O(n 2 ) time algorithm that computes this boundary, where n is the total number of edges comprising the light source and holes. The antipenumbra computation is motivated by a visibility scheme in which we wish to determine the volume visible to an observer looking through a sequenceof transparent convex holes, or portals, connecting adjacent cells in a spatial subdivision. Knowledge of the antipenumbra should also prove useful for rendering shadowed objects. Finally, we have extended the algorithm to compute the planar and quadratic su...

The theme of this paper is a rather simple method that has proved very potent in the analysis of the expected performance of various randomized algorithms and data structures in computational geometry. The method can be described as "analyze a randomized algorithm as if it were running backwards in time, from output to input." We apply this type of analysis to a variety of algorithms, old and new, and obtain solutions with optimal or near optimal expected performance for a plethora of problems in computational geometry, such as computing Delaunay triangulations of convex polygons, computing convex hulls of point sets in the plane or in higher dimensions, sorting, intersecting line segments, linear programming with a fixed number of variables, and others. 1 Introduction The curious phenomenon that randomness can be used profitably in the solution of computational tasks has attracted a lot of attention from researchers in recent years. The approach has proved useful in such diverse area...

We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to 1 � O(�log n)/k).

This paper presents an experimental comparison of a number of different algorithms for computing the Deluanay triangulation. The algorithms examined are: Dwyer’s divide and conquer algorithm, Fortune’s sweepline algorithm, several versions of the incremental algorithm (including one by Ohya, Iri, and Murota, a new bucketing-based algorithm described in this paper, and Devillers’s version of a Delaunay-tree based algorithm that appears in LEDA), an algorithm that incrementally adds a correct Delaunay triangle adjacent to a current triangle in a manner similar to gift wrapping algorithms for convex hulls, and Barber’s convex hull based algorithm. Most of the algorithms examined are designed for good performance on uniformly distributed sites. However, we also test implementations of these algorithms on a number of non-uniform distibutions. The experiments go beyond measuring total running time, which tends to be machine-dependent. We also analyze the major high-level primitives that algorithms use and do an experimental analysis of how often implementations of these algorithms perform each operation.

Let be a set of points in in general position, i.e., no points on a common-flat,. A-set of is a set of points in that can be separated from by a hyperplane. A-facet of is an oriented-simplex spanned by points in which has exactly points from on the positive side of its affine hull. If is a planar point set and is even, a halving edge is an undirected edge between two points, such that the connecting line has the same number of points on either side. The number! "$ # of-sets is twice the number of halving edges. Inspired by Dey’s recent proof of a new bound on the number of-sets we show that