New applications of random sampling in computational geometry
TL;DR: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry by creating a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer.
Abstract: This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requiresO(sd+?) expected preprocessing time to build a search structure for an arrangement ofs hyperplanes ind dimensions. The expectation, as with all expected times reported here, is with respect to the random behavior of the algorithm, and holds for any input. Given the data structure, and a query pointp, the cell of the arrangement containingp can be found inO(logs) worst-case time. (The bound holds for any fixed ?>0, with the constant factors dependent ond and ?.) Using point-plane duality, the algorithm may be used for answering halfspace range queries. Another algorithm finds random samples of simplices to determine the separation distance of two polytopes. The algorithm uses expectedO(n[d/2]) time, wheren is the total number of vertices of the two polytopes. This matches previous results [10] for the cased = 3 and extends them. Another algorithm samples points in the plane to determine their orderk Voronoi diagram, and requires expectedO(s1+?k) time fors points. (It is assumed that no four of the points are cocircular.) This sharpens the boundO(sk2 logs) for Lee's algorithm [21], andO(s2 logs+k(s?k) log2s) for Chazelle and Edelsbrunner's algorithm [4]. Finally, random sampling is used to show that any set ofs points inE3 hasO(sk2 log8s/(log logs)6) distinctj-sets withj≤k. (ForS ?Ed, a setS? ?S with |S?| =j is aj-set ofS if there is a half-spaceh+ withS? =S ?h+.) This sharpens with respect tok the previous boundO(sk5) [5]. The proof of the bound given here is an instance of a "probabilistic method" [15].
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TL;DR: The Voronoi diagram as discussed by the authors divides the plane according to the nearest-neighbor points in the plane, and then divides the vertices of the plane into vertices, where vertices correspond to vertices in a plane.
Abstract: Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer graphics, computer-aided design, robotics, pattern recognition, and operations research—give rise to problems that inherently are geometrical. This is one reason computational geometry has attracted enormous research interest in the past decade and is a well-established area today. (For standard sources, we refer to the survey article by Lee and Preparata [19841 and to the textbooks by Preparata and Shames [1985] and Edelsbrunner [1987bl.) Readers familiar with the literature of computational geometry will have noticed, especially in the last few years, an increasing interest in a geometrical construct called the Voronoi diagram. This trend can also be observed in combinatorial geometry and in a considerable number of articles in natural science journals that address the Voronoi diagram under different names specific to the respective area. Given some number of points in the plane, their Voronoi diagram divides the plane according to the nearest-neighbor
4,236 citations
06 Jan 1988
TL;DR: Asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets, are given.
Abstract: Random sampling is used for several new geometric algorithms. The algorithms are “Las Vegas,” and their expected bounds are with respect to the random behavior of the algorithms. One algorithm reports all the intersecting pairs of a set of line segments in the plane, and requires O(A + n log n) expected time, where A is the size of the answer, the number of intersecting pairs reported. The algorithm requires O(n) space in the worst case. Another algorithm computes the convex hull of a point set in E3 in O(n log A) expected time, where n is the number of points and A is the number of points on the surface of the hull. A simple Las Vegas algorithm triangulates simple polygons in O(n log log n) expected time. Algorithms for half-space range reporting are also given. In addition, this paper gives asymptotically tight bounds for a combinatorial quantity of interest in discrete and computational geometry, related to halfspace partitions of point sets.
1,163 citations
01 Jan 1993
TL;DR: The up-tree (vantage point tree) is introduced in several forms, together‘ with &&ciated algorithms, as an improved method for these difficult search problems in general metric spaces.
Abstract: We consider the computational problem of finding nearest neighbors in general metric spaces. Of particular interest are spaces that may not be conveniently embedded or approximated in Euclidian space, or where the dimensionality of a Euclidian representation 1s very high. Also relevant are high-dimensional Euclidian settings in which the distribution of data is in some sense of lower dimension and embedded in the space. The up-tree (vantage point tree) is introduced in several forms, together‘ with &&ciated algorithms, as an improved method for these difficult search nroblems. Tree construcI tion executes in O(nlog(n i ) time, and search is under certain circumstances and in the imit, O(log(n)) expected time. The theoretical basis for this approach is developed and the results of several experiments are reported. In Euclidian cases, kd-tree performance is compared.
1,145 citations
Cites methods from "New applications of random sampling..."
...More recently, the Voronoi digram [21] has provided a useful tool in low- dimensional Euclidian settings { and Figure 1: vp-tree decomposition Figure 2: kd-tree decomposition the overall eld and outlook of Computational Geometry has yielded many interesting results such as those of [22, 23, 24, 25] and earlier [26]....
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01 Oct 1998
TL;DR: New packet classification schemes are presented that, with a worst-case and traffic-independent performance metric, can classify packets, by checking amongst a few thousand filtering rules, at rates of a million packets per second using range matches on more than 4 packet header fields.
Abstract: The ability to provide differentiated services to users with widely varying requirements is becoming increasingly important, and Internet Service Providers would like to provide these differentiated services using the same shared network infrastructure. The key mechanism, that enables differentiation in a connectionless network, is the packet classification function that parses the headers of the packets, and after determining their context, classifies them based on administrative policies or real-time reservation decisions. Packet classification, however, is a complex operation that can become the bottleneck in routers that try to support gigabit link capacities. Hence, many proposals for differentiated services only require classification at lower speed edge routers and also avoid classification based on multiple fields in the packet header even if it might be advantageous to service providers. In this paper, we present new packet classification schemes that, with a worst-case and traffic-independent performance metric, can classify packets, by checking amongst a few thousand filtering rules, at rates of a million packets per second using range matches on more than 4 packet header fields. For a special case of classification in two dimensions, we present an algorithm that can handle more than 128K rules at these speeds in a traffic independent manner. We emphasize worst-case performance over average case performance because providing differentiated services requires intelligent queueing and scheduling of packets that precludes any significant queueing before the differentiating step (i.e., before packet classification). The presented filtering or classification schemes can be used to classify packets for security policy enforcement, applying resource management decisions, flow identification for RSVP reservations, multicast look-ups, and for source-destination and policy based routing. The scalability and performance of the algorithms have been demonstrated by implementation and testing in a prototype system.
741 citations
IBM1
TL;DR: These results are tied together, stronger convergence results are reviewed, and several coreset bounds are generalized or strengthened.
Abstract: The problem of maximizing a concave function f(x) in the unit simplex Δ can be solved approximately by a simple greedy algorithm. For given k, the algorithm can find a point x(k) on a k-dimensional face of Δ, such that f(x(k) ≥ f(xa) − O(1/k). Here f(xa) is the maximum value of f in Δ, and the constant factor depends on f. This algorithm and analysis were known before, and related to problems of statistics and machine learning, such as boosting, regression, and density mixture estimation. In other work, coming from computational geometry, the existence of ϵ-coresets was shown for the minimum enclosing ball problem by means of a simple greedy algorithm. Similar greedy algorithms, which are special cases of the Frank-Wolfe algorithm, were described for other enclosure problems. Here these results are tied together, stronger convergence results are reviewed, and several coreset bounds are generalized or strengthened.
456 citations
References
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Book•
01 Jan 1987TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
Abstract: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.
2,284 citations
"New applications of random sampling..." refers background or methods or result in this paper
...time for d > 3. This improves known results for odd dimensions [36, 40, 41, 20 ]....
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...To nd the set SnI (S), we use an algorithm for point location in a planar subdivision (see [37, 20 ])....
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...Proof. The algorithm is similar to that above; we maintainP(R) as halfspaces are added to R. The incidence graph ofP(R) is maintained, as in the beneath-beyond method[ 20 ], and the conict lists of edges (one-dimensional faces) are maintained....
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...This is no loss of generality, as gk(S) attains its maximum when S is nondegenerate [ 20 ]....
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...Proof. Omitted. Cyclic polytopes[ 20 ] realize the bound, as can be shown using the techniques of the theorem, or constructively [19]....
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TL;DR: A generalization of the convex hull of a finite set of points in the plane leads to a family of straight-line graphs, "alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets.
Abstract: A generalization of the convex hull of a finite set of points in the plane is introduced and analyzed. This generalization leads to a family of straight-line graphs, " \alpha -shapes," which seem to capture the intuitive notions of "fine shape" and "crude shape" of point sets. It is shown that a-shapes are subgraphs of the closest point or furthest point Delaunay triangulation. Relying on this result an optimal O(n \log n) algorithm that constructs \alpha -shapes is developed.
1,648 citations
"New applications of random sampling..." refers background in this paper
...Like polytopes, spherical intersections have duals, which were introduced as -hulls in [ 21 ]....
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PARC1
TL;DR: The following problem is discussed: given n points in the plane (the sites) and an arbitrary query point q, find the site that is closest to q, which can be solved by constructing the Voronoi diagram of the griven sites and then locating the query point in one of its regions.
Abstract: The following problem is discussed: given n points in the plane (the sites) and an arbitrary query point q, find the site that is closest to q. This problem can be solved by constructing the Voronoi diagram of the griven sites and then locating the query point inone of its regions. Two algorithms are given, one that constructs the Voronoi diagram in O(n log n) time, and another that inserts a new sit on O(n) time. Both are based on the use of the Voronoi dual, or Delaunay triangulation, and are simple enough to be of practical value. the simplicity of both algorithms can be attributed to the separation of the geometrical and topological aspects of the problem and to the use of two simple but powerful primitives, a geometric predicate and an operator for manipulating the topology of the diagram. The topology is represented by a new data structure for generalized diagrams, that is, embeddings of graphs in two-dimensional manifolds. This structure represents simultaneously an embedding, its dual, and its mirror image. Furthermore, just two operators are sufficients for building and modifying arbitrary diagrams.
1,201 citations
TL;DR: The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.
Abstract: We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ .
These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.
799 citations
"New applications of random sampling..." refers background or methods in this paper
...Intuitively, the fact that an edge e has no points of the random sample R beyond it is good evidence that e has few points of S beyond it. This kind of tail estimate has been the basis of several previous applications of random sampling to computational geometry [12, 28 ]....
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...In recent years, random sampling has seen increasing use in discrete and computational geometry, with applications in proximity problems, point location, and range queries [11, 12, 28 ]....
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...The algorithm is a variant of Haussler and Welzl’s [ 28 ]....
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...With these two facts, by [ 28 , 4] the resulting query time is O(A+n ), where = 1 1=(1 + B), and...
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...From previous analysis [ 28 , 4], there are two key properties of this algorithm that imply a bound on the query time....
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TL;DR: The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls to ensure optimal time complexity within a multiplicative constant.
Abstract: The convex hulls of sets of n points in two and three dimensions can be determined with O(n log n) operations. The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls. Since any convex hull algorithm requires at least O(n log n) operations, the time complexity of the proposed algorithms is optimal within a multiplicative constant.
731 citations
"New applications of random sampling..." refers methods or result in this paper
...time for d > 3. This improves known results for odd dimensions [ 36 , 40, 41, 20]....
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...In particular, the divide-and-conquer technique of Preparata and Hong[ 36 ] does not seem to lead to a fast algorithm for computing spherical intersections....
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