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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01 Oct 1987
TL;DR: Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage.
Abstract: Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.
53 citations
"New applications of random sampling..." refers background or methods in this paper
...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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...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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01 Jun 1985
TL;DR: The kth-order Voronoi diagram of a finite set of sites in the Euclidean plane E2 subdivides into maximal regions such that all points within a given region have the same k nearest sites.
Abstract: The kth-order Voronoi diagram of a set of points in E2 (called sites) subdivides E2 into maximal regions such that each point within a given region has the same k nearest sites. Two versions of an algorithm are developed for constructing the kth-order Voronoi diagram of a set of n sites in O(n2logn+k(n-k)log2n) time, O(k(n-k)) storage, and O(n2+k(n-k)log2n) time, O(n2) storage, respectively.
53 citations
"New applications of random sampling..." refers background or methods in this paper
...Proof. The work performed at each call is as follows: If s ≤ k(r − 7), then a (k + 1)-VoD of no more than k(r − 7) sites is constructed using the [ 4 ] algorithm....
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...(It is assumed that no four of the points are cocircular.) This sharpens the bound O(sk2 log s) for Lee’s algorithm [21], and O(s2 log s + k(s k)log2 s) for Chazelle and Edelsbrunner’s algorithm [ 4 ]....
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...first order Voronoi diagram, and then uses that to build the second order diagram, and so on. Chazelle and Edelsbrunner [ 4 ] have given an algorithm requiring O(s2 log s + k(s − k) log2 s)...
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...co r is a (sufficiently large) constant, and S′ = γ(S) for some S ⊂ E2 oc; s ← |S′|; if s ≤ k(r − 7) then Determine the Vk(S′)-triples by finding the (k + 1)-VoD of S using the [ 4 ] procedure; else repeat...
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01 Dec 1985
TL;DR: The algorithm employs random sampling, so the expected time holds for any set of points, and approaches the preprocessing time required for any algorithm constructing the Voronoi diagram of the input points.
Abstract: The post office problem is the following: points in d-dimensional space, so that given an arbitrary point p, the closest points in S to p can be found quickly.We consider the case of this problem where the Euclidean norm is the measure of distance. The previous best algorithm for this problem for d>2 requires O(n2d+1) preprocessing time to build a data structure allowing an O(log n query time. We will show that a data structure can be built in expected O(n(d-1)(1+k)) time, for any fixed k;>O, so that closest-point queries can be answered in O(log n) worstcase time. (The constant factors depend on d and k.) The algorithm employs random sampling, so the expected time holds for any set of points. A variant of this algorithm (for the variant problem where only one closest point of S to the query point is desired) requires O(n⌈d/2⌉) o(n⌈d/2⌉) preprocessing time for o(nt) worst-case query time, for any fixed e>0. These results approach the O(n⌈d/2⌉) preprocessing time required for any algorithm constructing the Voronoi diagram of the input points. Implementation of these algorithms requires not too much more than a random sampling procedure and a procedure for constructing the Voronoi diagram of that random sample.
47 citations
"New applications of random sampling..." refers background or methods in this paper
...A more detailed discussion of this triangulation procedure is given in [6], with a more rigorous discussion of the triangulation of polyhedral sets that are unbounded....
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...A combinatorial question relevant to several algorithms [5, 6, 14] concerns the quantity...
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...In general the geometric notation used here follows [6], which in general follows [17]....
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TL;DR: By generalizing to a combinatorial problem, it is shown that for 2 k n the number of sets of size at most k is at most 2 nk − 2 k 2 − k.
45 citations
"New applications of random sampling..." refers result in this paper
...Our bound is within a small constant factor of the tight bounds known for the plane [ 25 , 3, 42], and it improves previous results for d = 3 [18, 8, 12]; apparently no interesting bounds were known before for higher dimensions....
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21 Aug 1983
TL;DR: This paper presents a new scheme for recording a history of h updates over an ordered set S of n objects, which allows fast neighbor computation at any time in the history, and shows that with only O(n2) preprocessing, which of n given points in E3 is closest to an arbitrary query point.
Abstract: This paper considers the problem of granting a dynamic data structure the capability of remembering the situation it held at previous times. We present a new scheme for recording a history of h updates over an ordered set S of n objects, which allows fast neighbor computation at any time in the history. This scheme requires O(n + h) space and O(log n log h) query response-time, which saves a factor of log n space over previous structures. Aside from its improved performance, the novelty of our method is to allow the set S to be only partially ordered with respect to queries and the time-measure to be multi-dimensional. The generality of our method makes it useful to a number of problems in three-dimensional geometry. For example, we are able to give fast algorithms for locating a point in a 3d-complex, using linear space, or for finding which of n given points is closest to a query plane. Using a simpler, yet conceptually similar technique, we show that with only O(n2) preprocessing, we can determine in O(log2 n) time which of n given points in E3 is closest to an arbitrary query point.
17 citations