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151
The Power Crust
, 2001
"... The power crust is a construction which takes a sample of points from the surface of a threedimensional 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 ..."
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Cited by 216 (7 self)
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The power crust is a construction which takes a sample of points from the surface of a threedimensional 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.
Las Vegas algorithms for linear and integer programming when the dimension is small
 J. ACM
, 1995
"... 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 algor ..."
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Cited by 104 (3 self)
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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
On Range Searching with Semialgebraic Sets
 DISCRETE COMPUT. GEOM
, 1994
"... 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 \Gammarange searching problem: Given P , build a data structur ..."
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Cited by 80 (23 self)
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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 \Gammarange 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 constantcomplexity cells. We present some of the applications of \Gammarange searching problem, including improved ray shooting among triangles in R³.
Efficiently Approximating the MinimumVolume Bounding Box of a Point Set in Three Dimensions
 In Proc. 10th ACMSIAM Sympos. Discrete Algorithms
, 2001
"... We present an efficient O(n + 1/ε^4.5)time algorithm for computing a (1 + 1/ε)approximation of the minimumvolume 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). ..."
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Cited by 79 (12 self)
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We present an efficient O(n + 1/&epsilon;^4.5)time algorithm for computing a (1 + 1/&epsilon;)approximation of the minimumvolume bounding box of n points in R&sup3;. We also present a simpler algorithm (for the same purpose) whose running time is O(n log n+n/&epsilon;&sup3;). 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].
RANDOM SAMPLING IN CUT, FLOW, AND NETWORK DESIGN PROBLEMS
, 1999
"... 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 pro ..."
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Cited by 75 (11 self)
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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 cutapproximation algorithms extend unchanged to weighted graphs while our weightedgraph 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 nearlinear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cutbased problems, including approximating the best balanced cut of a graph, finding a kconnected orientation of a 2kconnected 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 kconnected subgraph problem from 1.85 to 1 � O(�log n)/k).
Beyond Best Effort: Router Architectures for the Differentiated Services of Tomorrow’s Internet
 IEEE Communications Magazine
, 1998
"... 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 tran ..."
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Cited by 73 (1 self)
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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 customerspecific 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
Computing the Antipenumbra of an Area Light Source
 Computer Graphics
, 1992
"... 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 ..."
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Cited by 71 (0 self)
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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...
Backwards Analysis of Randomized Geometric Algorithms
 Trends in Discrete and Computational Geometry, volume 10 of Algorithms and Combinatorics
, 1992
"... 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 backward ..."
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Cited by 60 (0 self)
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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...
A Comparison of Sequential Delaunay Triangulation Algorithms
, 1996
"... 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, an ..."
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Cited by 60 (0 self)
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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 bucketingbased algorithm described in this paper, and Devillers’s version of a Delaunaytree 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 nonuniform distibutions. The experiments go beyond measuring total running time, which tends to be machinedependent. We also analyze the major highlevel primitives that algorithms use and do an experimental analysis of how often implementations of these algorithms perform each operation.
Improved Approximation Algorithms for Geometric Set Cover
, 2005
"... Given a collection S of subsets of some set U, and M ⊂ U, the set cover problem is to find the smallest subcollection C ⊂ S such that M is a subset of the union of the sets in C. While the general problem is NPhard to solve, even approximately, here we consider some geometric special cases, where u ..."
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Cited by 57 (5 self)
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Given a collection S of subsets of some set U, and M ⊂ U, the set cover problem is to find the smallest subcollection C ⊂ S such that M is a subset of the union of the sets in C. While the general problem is NPhard to solve, even approximately, here we consider some geometric special cases, where usually U = ℜ d. Extending prior results[BG95], we show that approximation algorithms with provable performance exist, under a certain general condition: that for a random subset R ⊂ S and function f(), there is a decomposition of the complement U \ ∪Y∈RY into an expected f(R) regions, each region of a particular simple form. We show that under this condition, a cover of size O(f(C)) can be found. Our proof involves the generalization of shallow cuttings [Mat92] to more general geometric situations. We obtain constantfactor approximation algorithms for covering by unit cubes in ℜ³, for guarding a onedimensional terrain, and for covering by similarsized fat triangles in ℜ². We also obtain improved approximation guarantees for fat triangles, of arbitrary size, and for a class of fat objects.