T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th ACM Sympos. Comput. Geom., pages 284-290, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of H that minimizes c. Using multi dimensional parametric searching and data structures for answering halfspaceemptiness queries, Matousek [201] presented an efficient algorithm for answering linearoptimization queries. A slightly faster randomized algorithm has recently been proposed by Chan [55]. Linear optimization queries can be used to answer many other queries. For example, using Matousek s technique and a dynamic data structure for halfspace range searching, the 1 center of a set S of points in R can be maintained dynamically, as points are inserted into or deleted from S. See ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284--290.

....the value of the linear function while satisfying all the constraints can be determined eciently. This was rst solved almost completely by Matou sek [14] using a multidimensional version of parametric search together with data structures for halfspace range emptiness queries. Later, Chan [3] presented an alternative approach which through randomization reduced the problem to halfspace range reporting queries. Chan s approach is conceptually simpler and achieves better query times in the case of small storage space; however, comparatively, Max Planck Institut f ur Informatik, ....

.... is a (1=r) net from H, with r = f(n) to be determined, ii) the partition children of H consist of all the singleton sets and they are all leaves (thus, it maintains a single set in L) and (iii) the procedure violated(v; H) is implemented as follows: 2 2 This follows a construction in [3]. 10 1. Determine those i that intersect v . 2. For each surviving i , determine whether P i H = using a halfspace emptiness data structure. 3. For each surviving i , obtain P i H by checking each element of P i . The reason for doing Steps 2 and 3 rather than using ....

[Article contains additional citation context not shown here]

T. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th Annu. ACM Sympos. Comput. Geom., 284-290, 1996.

....if Step ( is executed. It can be shown that an iteration is successful with probability at least 1=2 and that there are at most d 1 successful iterations. Using these observations it can be shown that the expected running time of the algorithm is O(d 2 n d d=2 log log n O(1) Chan [34] showed that RECURSIVE lp algorithm can be modi ed to answer linear programming queries. That is, preprocess a set H of n constraints, so that for a query objective function c, the point x 2 T H that minimizes cx can be computed Geometric Optimization September 7, 2000 Linear Programming 8 ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284-290.

....and data structures for answering halfspaceemptiness queries, Matousek [172] presented an efficient algorithm for answering linear Geometric Optimization January 24, 1997 Query Type Problems 39 optimization queries. A slightly faster randomized algorithm has recently been proposed by Chan [47]. Linear optimization queries can be used to answer many other queries. For example, using Matousek s technique and a dynamic data structure for halfspace range searching, the 1 center of a set S of points in R d can be maintained dynamically, as points are inserted into or deleted from S. See ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284--290.

....it for point location queries. For higher dimensions, Matousek [181] showed that, using multidimensional parametric searching and a data structure for answering halfspace emptiness queries, a linear programming query can be answered in O( n=m 1=bd=2c ) polylog n) with O(m) storage. Recently Chan [51] has described a randomized procedure whose expected query time is n 1 Gamma1=bd=2c 2 O(log n) using linear space. 7.4 Segment dragging queries Preprocess a set S of objects in the plane so that for a query segment e and a ray ae, the first position at which e intersects any object of S ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284--290.

....if Step ( is executed. It can be shown that an iteration is successful with probability at least 1=2 and that there are at most d 1 successful iterations. Using these observations it can be shown that the expected running time of the algorithm is O(d 2 n d d=2 log log n O(1) Chan [36] showed that RECURSIVE lp algorithm can be modified to answer linear programming queries. That is, preprocess a set H of n constraints, so that for a query objective function c, the point x 2 T H that minimizes cx can be computed efficiently. He sets r = n=b where b 9d 2 is a sufficiently ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284--290.

....and double description implementations in lrs and cdd respectively [Avi00, Fuk99] to solve linear programs. Important work has been done for linear programming queries in xed dimension, where the constraints do not change between successive optimizations of dioeerent objective functions [Mat93, Cha96, Ram00] This is very relevant here, except that dimension is an input parameter. Possibly relevant work concerns incremental approaches, such as those that could be derived from [AS93, Cla95, GW00] In the latter work, the idea is to randomly choose a subset (ffl net) of constraints so that, ....

T. Chan. Fixed-dimensional linear programming queries made easy. In Proc. Annual ACM Symp. on Comput. Geometry, pages 284290. ACM Press, 1996. 17

....asking whether any of the rays intersects a query quadrilateral the portion of the original query triangle that lies below some horizontal plane. To answer a lowest intersection query using this data structure, we apply parametric search [52] or one of Chan s recent randomized reduction techniques [19, 20] to nd the largest value z so that no ray crosses the intersection of the query triangle and the halfspace z z . The query algorithm we describe below can easily be executed in parallel in O(log n) time using O(n 1 =s 1=4 ) processors, so the additional cost of the parametric ....

T. M. Chan. Fixed-dimensional linear programming queries made easy. Proc. 12th Annu. ACM Sympos. Comput. Geom., pp. 284-290. 1996.

....query [6, 45] which asks whether the query range contains any points in the set. Emptiness query data structures have been used to solve several geometric problems, including point location [20] ray shooting [2, 20, 41, 44] nearest and farthest neighbor queries [2] linear programming queries [40, 11], depth ordering [25] collision detection [19] and output sensitive convex hull construction [40, 12] This paper presents the rst nontrivial lower bounds on the complexity of data structures that support online emptiness queries, where the query ranges are either arbitrary hyperplanes or ....

....for these problems. Is there a reduction from hyperplane queries to halfspace queries that only increases the dimension by a constant factor (preferably two) Finally, can our techniques be applied to other closely related problems, such as nearest neighbor queries [2] linear programming queries [40, 11] and ray shooting queries [2, 20, 41, 44] ################ I thank Pankaj Agarwal for suggesting studying the complexity of online emptiness problems. ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, in Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284-290.

....query [6, 45] which asks whether the query range contains any points in the set. Emptiness query data structures have been used to solve several geometric problems, including point location [20] ray shooting [2, 20, 41, 44] nearest and farthest neighbor queries [2] linear programming queries [40, 11], depth ordering [25] collision detection [19] and output sensitive convex hull construction [40, 12] This paper presents the rst nontrivial lower bounds on the complexity of data structures that support online emptiness queries, where the query ranges are either arbitrary hyperplanes or ....

....for these problems. Is there a reduction from hyperplane queries to halfspace queries that only increases the dimension by a constant factor (preferably two) Finally, can our techniques be applied to other closely related problems, such as nearest neighbor queries [2] linear programming queries [40, 11] and ray shooting queries [2, 20, 41, 44] Acknowledgments. I thank Pankaj Agarwal for suggesting studying the complexity of online emptiness problems. ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, in Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284-290.

....O(n 1 ) preprocessing time. Combining Clarkson s and Matousek s data structures, for a fixed parameter n s n bd=2c , one can answer queries in time O( n log n) s 1=bd=2c ) after O(s polylog n) preprocessing time [19,1,7] For extensions and applications of halfspace range reporting, see [1,2,6,7,22,20]. Given n points and m halfspaces, we can solvethe offline halfspace emptiness problem in time O i n log m (nm) bd=2c= bd=2c 1) polylog(n m) m log n j # using either Clarkson s data structure or Matousek s combined data structure, depending on the relative growth rates of n and m.Intwo ....

....and the size of the subgraph induced by a query range (time) A problem closely related to halfspace range searching is linear programming. The best known data structures of linear programming queries are based on data structures for halfspace emptiness [22] and halfspace reporting queries [6]. However, no nontrivial lower bounds are known for linear programming queries in any model of computation. One application of particular interest is deciding, given a set of points, whether every pointisavertex of the set s convex hull. Bounds for this problem closely match the best known bounds ....

T. M. Chan. Fixed-dimensional linear programming queries made easy.InProc. 12th Annu. ACM Sympos. Comput. Geom., pages 284--290, 1996.

.... time, O( n= log n) bd=2c (log n) c ) it is possible to achieve storage O( n= log n) bd=2c 2 c log n ) We use the same approach as in [23] with some variations, together with the derandomization work of Chazelle [13] and Br onnimann et al. [9] and some recent work of Chan [10]. Ray Shooting among Hyperplanes. Let H be a set of n hyperplanes in R d . The problem is to construct a data structure that for a given query ray , the rst hyperplane in H hit by the ray can be quickly determined. The particular case in which the ray is vertical was solved optimally with ....

....time O(nr bd=2c 1 ) that consists of a (1=r) cutting of size O(r bd=2c ) covering the 0 level of H, and that given a query ray returns in time O(log r) a simplex in the cutting whose con ict list contains the solution to the ray shooting query. We also need the following result from [21] and [10] (faster query time) Lemma 5 Given a set H of n hyperplanes in R d , a data structure can be constructed deterministically using preprocessing time O(n log n) resp. O(n c 0 2 ) and storage O(n) so that given a query ray with origin inside the 0 level of H, it rerports the rst ....

T. Chan. Fixed-dimensional linear programming queries made easy. SoCG'96, 284-290.

.... time, O( n= log n) bd=2c (log n) c ) it is possible to achieve storage O( n= log n) bd=2c 2 c log n ) We use the same approach as in [21] with some variations, together with the derandomization work of Chazelle [11] and Br onnimann et al. 7] and some recent work of Chan [8]. Ray Shooting among Hyperplanes. Let H be a set of n hyperplanes in R d . The problem is to construct a data structure so that for a given query ray , the rst hyperplane in H hit by can be determined quickly. The particular case in which the ray is vertical was solved optimally with ....

....time O(nr bd=2c 1 ) that consists of a (1=r) cutting of size O(r bd=2c ) covering the 0 level of H, and that given a query ray returns in time O(log r) a simplex in the cutting whose con ict list contains the solution to the ray shooting query. We also need the following result from [19] and [8] (faster query time) Lemma 5 Given a set H of n hyperplanes in R d , a data structure can be constructed deterministically using preprocessing time O(n log n) resp. O(n c 0 2 ) and storage O(n) so that given a query ray with origin inside the 0 level of H, it reports the rst ....

T. Chan. Fixed-dimensional linear programming queries made easy. SoCG'96, 284-290.

....for point location queries. For higher dimensions, Matousek [201] showed that, using multidimensional parametric searching and a data structure for answering halfspace emptiness queries, a linear programming query can be answered in O( n=m 1=bd=2c ) polylog n) with O(m) storage. Recently Chan [54] has described a randomized procedure whose expected query time is n 1 Gamma1=bd=2c 2 O(log n) using linear space. 7.4 Segment dragging queries Preprocess a set S of objects in the plane so that for a query segment e and a ray ae, the first position at which e intersects any object of S ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284--290.

....for point location queries. For higher dimensions, Matousek [84] showed that, using multidimensional parametric searching and the data structure for answering halfspace emptiness queries, a linear programming query can be answered in O( n=m 1=bd=2c ) log c n) with O(m) storage. Recently Chan [24] has described a randomized procedure whose expected query time is slightly faster. SEGMENT DRAGGING QUERIES Preprocess a set S of objects in the plane so that for a query segment e and a ray ae, the first position at which e intersects any object of S as it is translated (dragged) along ae can ....

T. Chan, Fixed-dimensional linear programming queries made easy, Proc. 12th ACM Symp. Comput. Geom., 1996, p. to appear.

....1 ) preprocessing time. Combining Clarkson s and Matousek s data structures, for a fixed parameter n s n bd=2c , one can answer queries in time O( n log n) s 1=bd=2c ) after O(s polylog n) preprocessing time [107, 3, 29] For extensions and applications of halfspace range reporting, see [3, 4, 27, 29, 110, 108]. 93 Given n points and m halfspaces, we can solve the offline halfspace emptiness problem in time O i n log m (nm) bd=2c= bd=2c 1) polylog(n m) m log n j ; using either Clarkson s data structure or one of Matousek s data structures, depending on the relative growth rates of n and m. In ....

....and the size of the subgraph induced by a query range (time) A problem closely related to halfspace range searching is linear programming. The best known data structures of linear programming queries are based on data structures for halfspace emptiness [110] and halfspace reporting queries [27]. However, no nontrivial lower bounds are known for linear programming queries in any model of computation. One application of particular interest is deciding, given a set of points, whether every point is a vertex of the set s convex hull. Bounds for this problem closely match the best known ....

Timothy M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 284--290, 1996.

....S. Army Research Office under grant DAAH04 961 0013. and halfspace emptiness queries. Emptiness query data structures are used to solve several geometric problems, including point location [12] ray shooting [2, 12, 24, 27] nearest and farthest neighbor queries [2] linear programming queries [23, 7], depth ordering [5] collision detection [11] and output sensitive convex hull construction [23, 8] Most previous range searching lower bounds are presented in the so called semigroup arithmetic model, originally introduced by Fredman [20] and later refined by Yao [31] In this model, the ....

....of optimal data structures for these problems. Is there a reduction from hyperplane queries to halfspace queries that only increases the dimension by a constant factor (preferably two) Finally, can our techniques be applied to other closely related problems, such as linear programming queries [23, 7] and ray shooting queries [2, 12, 24, 27] Acknowledgments I thank Pankaj Agarwal for suggesting this problem for study. ....

T. M. Chan. Fixed-dimensional linear programming queries made easy. Proc. 12th Annu. ACM Sympos. Comput. Geom., pp. 284--290, 1996.

....query [6, 43] which asks whether the query range contains any points in the set. Emptiness query data structures have been used to solve several geometric problems, including point location [19] ray shooting [2, 19, 39, 42] nearest and farthest neighbor queries [2] linear programming queries [38, 11], depth ordering [9] collision detection [18] and output sensitive convex hull construction [38, 12] This paper presents the first nontrivial lower bounds on the complexity of data structures that support emptiness queries, where the query ranges are either arbitrary hyperplanes or arbitrary ....

....for these problems. Is there a reduction from hyperplane queries to halfspace queries that only increases the dimension by a constant factor (preferably two) Finally, can our techniques be applied to other closely related problems, such as nearest neighbor queries [2] linear programming queries [38, 11] and ray shooting queries [2, 19, 39, 42] Acknowledgments I thank Pankaj Agarwal for suggesting studying the complexity of online emptiness problems. ....

T. M. Chan. Fixed-dimensional linear programming queries made easy. Proc. 12th Annu. ACM Sympos. Comput. Geom., pp. 284--290. 1996.

....1 ) preprocessing time. Combining Clarkson s and Matousek s data structures, for a fixed parameter n s n bd=2c , one can answer queries in time O( n log n) s 1=bd=2c ) after O(s polylog n) preprocessing time [19, 1, 7] For extensions and applications of halfspace range reporting, see [1, 2, 6, 7, 22, 20]. Given n points and m halfspaces, we can solve the offline halfspace emptiness problem in time O i n log m (nm) bd=2c= bd=2c 1) polylog(n m) m log n j ; using either Clarkson s data structure or Matousek s combined data structure, depending on the relative growth rates of n and m. In ....

....(space) and the size of the subgraph induced by a query range (time) A problem closely related to halfspace range searching is linear programming. The best known data structures of linear programming queries are based on data structures for halfspace emptiness [22] and halfspace reporting queries [6]. However, no nontrivial lower bounds are known for linear programming queries in any model of computation. One application of particular interest is deciding, given a set of points, whether every point is a vertex of the set s convex hull. Bounds for this problem closely match the best known ....

T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 284--290, 1996.

....that, for any n m n bd=2c , can answer linear optimization queries in O(n=m 1=bd=2c log 2d Gamma1 n) time using O(m log O(1) n) space and preprocessing. A simpler randomized algorithm with O(n) space and O(n 1 Gamma1=bd=2c 2 O(log n) expected query time was proposed by Chan [55]. Linear optimization queries can be used to answer many other queries. For example, using Matousek s technique and a dynamic data structure for halfspace range searching, the 1 center of a set S of points in R d can be maintained dynamically, as points are inserted into or deleted from S. See ....

T. M. Chan, Fixed-dimensional linear programming queries made easy, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 284--290.

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T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th ACM Sympos. Comput. Geom., pages 284-290, 1996.

No context found.

T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th ACM Sympos. Comput. Geom., pages 284--290, 1996.

....by the cost of O(k ) dynamic LP operations. The general 2 d problem can be lifted to a feasible 3 d problem, and with the appropriate data structures [45] these O(k ) operations can be carried out in O(n log n k n) time. The first term has been lowered to O(n log k) by this author [9, 8]; the second term can probably also be lowered using recent dynamic data structures [13] though the particular queries needed were not explicitly considered in [13] Modulo these small improvements, the rough bounds of O(nk) by Everett et al. for large k) and O(n k ) by Matou sek (for ....

....lower bound n k ) it is not far off (see Figure 1) and in particular, improves both Efrat et al. s near O(nk ) bound and Matousek s near O(n k problem best previous result(s) refs. new result 2 d feasible O(n k 1 log k) 48, 11] 2 d general O(n log n nk) 32] k) [39, 8] O( n k ) log n) 3 3 d general O(nk (log n log (n=k) 29] 4 8=3 2=3 1=3 Table 1. Time bounds for LP with at most k violations. In this paper, 0 denotes an arbitrary small constant, and c denotes a specific constant. bound (the ....

[Article contains additional citation context not shown here]

T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th ACM Sympos. Comput. Geom., pages 284--290, 1996.

....constructions of expander graphs (e.g. 45] following a technique of Katz and Sharir [42, 43] Without additional ideas, all of these techniques increase the running time by at least a logarithmic factor. Randomized techniques have also been suggested as an alternative in several isolated cases [2, 10, 13, 15, 29, 34, 47, 50]. Unfortunately, since these techniques are not unified and as straightforward to apply as parametric search, potential applications are sometimes missed. One important exception though is the class of LP type optimization problems [65] where general randomized linear time solutions have been ....

....ray shooting problem to the decision problem (which they called segment emptiness) The reduction uses parametric search and is not likely to be practical, besides increasing the query time by a polylogarithmic factor and requiring a parallel version of the decision algorithm. In an earlier paper [13], the author gave a randomized reduction of linear programming queries to halfspace range reporting. This reduction, when specialized to ray shooting, yields a different approach in which the preprocessing algorithm employs random sampling. Arya and Mount [8] noted a similar reduction in the ....

T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th ACM Sympos. Comput. Geom., pages 284--290, 1996.

....method, we can also permit deletions of halfspaces, if during each insertion of a halfspace, we are told the time at which the halfspace will be deleted. For the fully dynamic case though, the best results currently known are obtained from Matousek s linear optimization techniques [17] see also [4]) combined with data structures by Agarwal and Matousek [1] The remainder of this paper is organized as follows. In the next section, we present our convex programming algorithm in two dimensions. In Section 3, we consider the case where the constraints are convex n gons. Section 4 discusses the ....

T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th ACM Sympos. Comput. Geom., pages 284--290, 1996.

....In other instances, one can employ constructions of expander graphs by a technique of Katz and Sharir [35] Without additional ideas, all of these techniques increase the running time by logarithmic factors. Randomized techniques have also been suggested as an alternative in several isolated cases [2, 10, 23, 27, 37, 40]. Unfortunately, since these techniques are not unified and as straightforward to apply as parametric search, potential applications are sometimes missed. One important exception though is the class of LP type optimization problems [52] where general randomized solutions have been developed. ....

.... described a general deterministic reduction of the above ray shooting problem to the decision problem (which they called segment emptiness) The reduction uses parametric search and is not likely to be practical (besides increasing the query time by a polylogarithmic factor) In an earlier paper [10], the author gave a randomized reduction of linear programming queries to halfspace range reporting. This reduction, when specialized to ray shooting, yields a different approach in which the preprocessing algorithm employs random sampling. Arya and Mount [6] noted a similar reduction in the ....

T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th ACM Sympos. Comput. Geom., pages 284--290, 1996.

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T. M. Chan. Fixed-dimensional linear programming queries made easy. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 284--290, 1996.

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