P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. Algorithmica, 9:495--514, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....expected running time will be lower than the running time of the deterministic version. Cole [Co1] uses probabilistic parallel median and minimum finding algorithms that use O(n) processors and constant expected time to improve some parametric search algorithms. Agarwal, Aronov, Sharir and Suri [AASS] use this idea extensively. They devise randomized algorithms for both the parallel generic algorithm and the sequential algorithm to reduce considerably the complexity of the deterministic version of the algorithm. 1.7 A Matrix Search Approach In a number of papers, Frederickson and Johnson ....

P. K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting distances in the plane, Proc. 6th ACM Symp. on Computational Geometry, 1990, 321--331.

....[9, 10, 25, 26] have studied methods for nding rigid motions that minimize either the directed or undirected Hausdor distance between 2 the two point sets. All of these methods are based on intersecting higher degree curves and or surfaces, which are then searched (sometimes parametrically [1, 11, 12, 13, 30]) to nd a global minimum. This reliance upon intersection computations leads to algorithms that are potentially numerically unstable, are conceptually complex, and have running times that are high for all but the most trivial motions. Indeed, Rucklidge [35] gives evidence that such methods must ....

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. Algorithmica, 9:495-514, 1993.

....branches accordingly to these values. The algorithm completes the simulation and determines a final interval whose left endpoint is exactly t . The total time of the simulation is given by O(pT Ppar T P 0 T Ppar log p) Applications of Megiddo s parametric search to geometric problems are in [2] [4] 9] 3 The HSRA Algorithm In this section we give a skeleton algorithm for solving HSR queries. This algorithm uses the off line data structures as black boxes (i.e asking queries and receiving answers) and is independent of any particular implementation of these data structures. The ....

P. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 321--331, 1990.

....branches accordingly to these values. The algorithm completes the simulation and determines a final interval whose left endpoint is exactly t . The total time of the simulation is given by O(pT P T P 0 T P log p) Applications of Megiddo s parametric search to geometric problems are in [AASS90] AM92b] CEGS92] 2.7 Plucker Coordinates of lines To solve problems on lines and polyhedra we use the Plucker coordinates of lines. Algorithmic uses of Plucker coordinates can be found in [CEGS89a] and in [Pel91, PS92] a classical treatment of Plucker coordinates can be found in [Som51] A ....

P.K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. In Proceedings of the 6th ACM Symposium on Computational Geometry, pages 321--331, 1990.

....k is not known in advance (the enumeration is terminated when the triangulation is complete) and the pairs are required in nondecreasing order of distance. Closely related to Problem 2 is the following problem recently investigated by Chazelle [8] and by Agarwal, Aronov, Sharir, and Suri [1]: Problem 4 (Selecting Distances) Given a finite set S of n points, let d 1 # # d ( n 2 ) be the distances determined by the pairs of points in S. For a given positive integer k # # n 2 # , determine the value of d k and find a pair of points that realizes d k . A solution to ....

.... solution to Problem 4 based on the batching technique of Yao [27] The algorithm works in any dimension in time O(n 2 #(d) log 1 #(d) n) where#(d) 1 (2 d 1) Thus for dimension 2 2 the running time is O(n 9 5 log 4 5 n) More recently, Agarwal, Aronov, Sharir, and Suri [1] have improved that result for the planar case providing a deterministic algorithm that runs in time O(n 3 2 log 5 2 n) and a randomized algorithm with expected running time of O(n 4 3 log 8 3 n) Agarwal, in a personal communication, has claimed that the randomized algorithm can be made ....

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P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri, "Selecting distances in the plane", Proc. 6th Annual ACM Symposium on Computational Geometry, June 1990, 321-- 331.

....k farthest pairs (see Section 4) Our algorithms work under the algebraic decision tree model and are thus optimal in this model. Distances are not enumerated in sorted order. A related problem is how to select the k th smallest distance. In the plane (under the Euclidean metric) Agarwal et al. [1] solved a decision version of the problem in O(n 4=3 log 2=3 n) expected time. Applying parametric search, they then showed that the k th smallest distance can be selected in O(n 4=3 log 8=3 n) expected time in dimension 2. Deterministic variants were considered by Goodrich [19] and Katz ....

.... Gamma j Deltaj; 0g copies of r to the output. As j Deltaj k 0 , the running time is O(TR (n; k 0 ) 2 2.2 From Selection to Counting An ordinary binary search reduces the counting problem to the selection problem. In this subsection, we explore the reverse direction. As Agarwal et al. [1] noted, a fairly general reduction of selection to counting is possible by parametric search. We point out a simpler approach by randomization. Our approach is similar to a scheme that Matousek [25] called randomized halving (see also [16] The idea is a binary search that repeatedly cuts the ....

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P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. Algorithmica, 9:495--514, 1993.

....k farthest pairs (see Section 4) Our algorithms work under the algebraic decision tree model and are thus optimal in this model. Distances are not enumerated in sorted order. A related problem is how to select the k th smallest distance. In the plane (under the Euclidean metric) Agarwal, et al. [1] solved a decision version of the problem in O(n 4=3 log 2=3 n) expected time. Applying parametric search, they then showed that the k th smallest distance can be selected in O(n 4=3 log 8=3 n) expected time in dimension 2. Deterministic variants were con1 sidered by Goodrich [18] and Katz ....

.... Gamma j Deltaj; 0g copies of r to the output. As j Deltaj k 0 , the running time is O(TR (n; k 0 ) 2 2.2 From Selection to Counting An ordinary binary search reduces the counting problem to the selection problem. In this subsection, we explore the reverse direction. As Agarwal, et al. [1] noted, a fairly general reduction of selection to counting is possible by parametric search. We point out a simpler approach by randomization. Our approach is similar to a scheme that Matou sek [24] called randomized halving (see also [16] The idea is a binary search that repeatedly cuts the ....

[Article contains additional citation context not shown here]

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. Algorithmica, 9:495--514, 1993.

....in a point set, and gave optimal algorithms for these problems. The problem of selecting only the k th smallest distance turns out to be much more difficult. Salowe [103] shows how to select the k th smallest L1 distance in a set of n points in IR D , in O(n log D n) time. Agarwal et al. [1] give an efficient randomized algorithm for selecting the k th Euclidean distance in a planar set of points. Both algorithms use the parametric search technique. The problem of selecting the largest distance, i.e. the diameter, in a set of n points in IR D , can be solved in O(n log n) time if ....

P.K. Agarwal, B. Aronov, M. Sharir and S. Suri. Selecting distances in the plane. Algorithmica 9 (1993), pp. 495--514.

....objects of bounded complexity, or a set of n points, and L p be the underlying norm, for 1 p 1. Let 1 k n 2 . Then d (k) can be found in time O(n 1:5 log 3 n) Proof: We extend in a trivial manner the O(n 1:5 log 3 n) solution to k th distance selection problem that appeared in [1]. For completeness, let us repeat the basic ideas. Bottleneck Matching 17 08 1997 10:24 Computing the Matching 15 We use the parametric searching technique of Megiddo [45] with which we assume the reader is familiar, and refer to [1] for a full explanation. We first introduce an oracle whose ....

....solution to k th distance selection problem that appeared in [1] For completeness, let us repeat the basic ideas. Bottleneck Matching 17 08 1997 10:24 Computing the Matching 15 We use the parametric searching technique of Megiddo [45] with which we assume the reader is familiar, and refer to [1] for a full explanation. We first introduce an oracle whose input is sets A and B as above, an integer 1 k n 2 and a fixed parameter r. The oracle decides in time O(n 1:5 log n) if d (k) is smaller, larger, or equal to r. Let B 1 : B d p ne , be a decomposition of B into disjoint ....

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P.K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting distances in the plane, Algorithmica 9 (1993), 495--514.

....standard incremental algorithm. Then we decide by traversing every feature (face, edge, vertex) of the arrangement, whether there is a feature not covered by any forbidden region. This can be accomplished within the same time bounds. In order to parallelize the algorithm we adapt a technique of [2]. The arrangement can be constructed by n 2 processors in O(log n) parallel time on a CRCW PRAM. This is done by computing all vertices of the arrangement, sorting the edges incident to a vertex around this vertex and sorting the vertices incident to an edge along this edge. For every feature of ....

P. Agarwal, B. Aranov, M. Sharir, S. Suri. Selecting distances in the plane, Algorithmica, 9 (1993) pp. 495-514.

....Proof: The binary search algorithm described in the proof of Theorem 4.1 can also be used to solve the distance ranking problem. To select the kth smallest distance, we can perform a parametric search over the space of interpoint distances, using a distance ranking algorithm as an oracle [2]. Theorem 4.3. Distance ranking is harder than unit distance detection in the algebraic decision tree model. Proof: To detect the presence or absence of unit distances in a set of points, we call a distance ranking algorithm twice, once on the original set of points, and once on the set of ....

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. Algorithmica, 9:495--514, 1993.

....have shown that the algorithm of [77] can be extended to compute, in O(n log n) time, the k th leftmost vertex in an arrangement of n line segments. The slope selection problem is only one of many problems in geometric optimization that have been efficiently solved using parametric searching. See [5, 9, 14, 18, 60, 233] for a sample of other problems, many of which are described in Part II, that benefit from parametric searching. The parametric searching technique can be extended to higher dimensions in a natural manner. Suppose we have a d variate (strictly) concave function F ( where varies over . We ....

....of S. This can be done using parametric searching. The decision problem is to compute, for a given real r, the sum p2S jD r (p) S Gamma fpg)j, where D r (p) is the closed disk of radius r centered at p. This sum is twice the number of pairs of points of S at distance r. Agarwal et al. [5] gave an O(n expected time randomized algorithm for the decision problem, using the random sampling technique of [73] which yields an O(n 8=3 n) expected time algorithm for the distanceselection problem. Goodrich [135] derandomized this algorithm, at a cost of an additional ....

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri, Selecting distances in the plane, Algorithmica, 9 (1993), 495--514.

....vertical decomposition A ) For each face 2 ) we compute the subset L L of pseudolines whose dual points lie inside . This step takes O( m ) time. Next, we compute an Eulerian tour of the planar graph dual to A ) so that each face of A ) is visited O(1) times; see [3]. Each node of corresponds to a face of A ) If an edge e of crosses two adjacent faces of A(P ) we set (e) to be the point of P whose dual pseudo line separates these two faces. Otherwise, i.e. e connects two faces of ) separated by a vertical line, we set (e) Next, we ....

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri, Selecting distances in the plane, Algorithmica, 9 (1993), 495-514.

....first problem we develop an O(n log n) solution, which, although suboptimal, is very simple. The other two problems are more typical examples of our approach. Our solutions have running time O(n n) and O(n n) respectively, slightly better than the previous respective solutions of [2, 6]. We also mention briefly two other problems that can be solved efficiently by our technique. In solving these problems, we also obtain some auxiliary results concerning batched range searching, where the ranges are congruent discs or annuli. For example, we show that it is possible to compute ....

....More than 10 years ago, Megiddo [34] has proposed the ingenious technique of parametric searching for solving efficiently a variety of optimization problems. The technique has recently been applied to many problems in geometric optimization, thus demonstrating its power and versatility (see [2, 3, 4, 6, 7, 12, 18, 20, 37] for some of these applications) The general idea behind parametric searching is that we are given a decision problem P ( that depends on a real parameter , and we seek an optimal value of , where a certain extremal condition is satisfied. For example, in the slope selection problem, we ....

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P.K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting distances in the plane, Algorithmica 9 (1993), 495--514.

....k th nearest neighbor problem for a set of points in convex position. We conclude by mentioning some open problems. Selection in monotone matrices April 26, 1994 References 18 1. Can the time complexity of the array selection problem be improved to O(n 4=3 log O(1) n) Agarwal et al. [1] have given such an algorithm for selecting the k th smallest distance in a set of points in the plane. 2. Can the k th smallest distance in a set of planar points in convex position be computed in close to linear time The best known algorithm to date is the same as for an arbitrary set of ....

....such an algorithm for selecting the k th smallest distance in a set of points in the plane. 2. Can the k th smallest distance in a set of planar points in convex position be computed in close to linear time The best known algorithm to date is the same as for an arbitrary set of points (cf. [1]) 3. Can the all k th nearest neighbor problem for an arbitrary set of points in the plane be solved in time O(n 4=3 log c n) ....

P.K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting distances in the plane, Algorithmica 9 (1993), 495--514.

....points of S. This can be done using parametric searching. The decision problem is to compute, for a given real r, the sum P p2S jD r (p) S fpg)j, where D r (p) is the closed disk of radius r centered at p. This sum is twice the number of pairs of points of S at distance r. Agarwal et al. [4] gave a randomized algorithm, with O(n 4=3 log 4=3 n) expected time, for the decision problem, using the random sampling technique of [50] which yields an O(n 4=3 log 8=3 n) expected time algorithm for the distance selection problem. Matou sek [132] showed that the randomized halving ....

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri, Selecting distances in the plane, Algorithmica, 9 (1993), 495-514.

....have shown that the algorithm of [67] can be extended to compute, in O(n log n) time, the k th leftmost vertex in an arrangement of n line segments. The slope selection problem is only one of many problems in geometric optimization that have been efficiently solved using parametric searching. See [5, 9, 13, 17, 51, 200] for a sample of other problems, many of which are described in Part II, that benefit from parametric searching. The parametric searching technique can be extended to higher dimensions in a natural manner. Suppose we have a d variate (strictly) concave function F ( where varies over R d . We ....

....of S. This can be done using parametric searching. The decision problem is to compute, for a given real r, the sum P p2S jD r (p) S Gamma fpg)j, where D r (p) is the closed disk of radius r centered at p. This sum is twice the number of pairs of points of S at distance r. Agarwal et al. [5] gave an O(n 4=3 log 4=3 n) expected time randomized algorithm for the decision problem, Geometric Optimization January 24, 1997 Proximity Problems 29 using the random sampling technique of [64] which yields an O(n 4=3 log 8=3 n) expected time algorithm for the distance selection problem. ....

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri, Selecting distances in the plane, Algorithmica, 9 (1993), 495--514.

....points of S. This can be done using parametric searching. The decision problem is to compute, for a given real r, the sum P p2S jD r (p) S n fpg)j, where D r (p) is the closed disk of radius r centered at p. This sum is twice the number of pairs of points of S at distance r. Agarwal et al. [4] gave a randomized algorithm, with O(n 4=3 log 4=3 n) expected time, for the decision problem, using the random sampling technique of [54] which yields an O(n 4=3 log 8=3 n) expected time algorithm for the distance selection problem. Matousek [138] showed that the randomized halving ....

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri, Selecting distances in the plane, Algorithmica, 9 (1993), 495--514.

....the k th smallest distance between a pair of points of S. The decision problem is to compute, for a given real r, the sum P p2S jD r (p) S Gamma fpg)j, where D r (p) is the disk of radius r centered at p. This sum is twice the number of pairs of points of S at distance r. Agarwal et al. [2] gave an O(n 4=3 log 4=3 n) expected time randomized algorithm for the decision problem, which yielded an O(n 4=3 log 8=3 n) time algorithm for the distance selection problem. Goodrich [56] derandomized this algorithm. Katz and Sharir [71] gave an expander based O(n 4=3 log 3 n) time ....

P. Agarwal, B. Aronov, M. Sharir, and S. Suri, Selecting distances in the plane, Algorithmica 9 (1993), 495--514.

....(D; P ) 2m 2 n; as asserted. 2 Proof of Theorem 3.1. Let r 1 be a fixed parameter, to be specified later. We choose a random subset R D of size r, where each subset of size r is chosen with equal probability, and consider the vertical decomposition A (R) of the arrangement A(R) [2, 4]. For each Discrete 2 Center March 10, 1998 Complexity of K 13 cell Delta 2 A (R) let D Delta D be the set of disks whose boundaries intersect Delta (including the edges of Delta) let E Delta D be the set of disks that are disjoint from Delta, and let P Delta P be the set of points ....

....(R) and two nodes corresponding to cells Delta; Delta 0 are connected by an edge if the boundaries of Delta and Delta 0 overlap along (a portion of) an edge. We compute a path Pi in G that visits each node of G at least once and at most 4 times. The existence of such a path was proved in [2]. We traverse Pi, and at each node corresponding to a cell Delta, we maintain I Delta , as follows. When we move from a node corresponding to Delta to the next node in Pi, corresponding to a cell Delta 0 2 A (R) we delete all the disks of D Delta n D Delta 0 from the intersection, ....

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P.K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting distances in the plane, Algorithmica 9 (1993), 495--514.

....geometric optimization problems, including many of those that involve arrangements. Some specific results of this kind include: Selecting distances in the plane: Given a set S of n points in IR 2 and a parameter k Gamma n 2 Delta , find the k th largest distance among the points of S [2]. Here the problem reduces to the construction and searching in 2 dimensional arrangements of congruent disks. The segment center problem: Given a set S of n points in IR 2 , and a line segment e, find a placement of e that minimizes the largest distance from the points of S to e [35] Using ....

P.K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting distances in the plane, Algorithmica 9 (1993), 495--514.

....asserted. 2 Proof of Theorem 3.1. Let r 1 be a fixed parameter, to be specified later. We choose a random subset R D of size r, where each subset of size r is chosen with equal probability, Discrete 2 Center July 18, 13 and consider the vertical decomposition A (R) of the arrangement A(R) [2, 4]. For each cell Delta 2 A (R) let D Delta D be the set of disks whose boundaries intersect Delta (including the edges of Delta) let E Delta D be the set of disks that are disjoint from Delta, and let P Delta P be the set of points that lie in Delta (a point lying on an edge (or ....

....(R) and two nodes corresponding to cells Delta; Delta 0 are connected by an edge if the boundaries of Delta and Delta 0 overlap along (a portion of) an edge. We compute a path Pi in G that visits each node of G at least once and at most 4 times. The existence of such a path was proved in [2]. We traverse Pi, and at each node corresponding to a cell Delta, we maintain I Delta , as follows. When we move from a node corresponding to Delta to the next node in Pi, corresponding to a cell Delta 0 2 A (R) we delete all the disks of D Delta n D Delta 0 from the intersection, ....

[Article contains additional citation context not shown here]

P.K. Agarwal, B. Aronov, M. Sharir and S. Suri, Selecting distances in the plane, Algorithmica 9 (1993), 495--514.

....based on (1=r) cuttings, was later proposed by Arrangements May 26, Applications 55 Bronnimann and Chazelle [82] See also [226, 251] Distance selection. Given a set S of n points in R 2 and a parameter k Gamma n 2 Delta , find the k th largest distance among the points of S [12, 227]. The corresponding decision problem reduces to point location in a set of congruent disks in R 2 . Specifically, given a set Gamma of m congruent disks in the plane, we wish to count efficiently the number of containments between disks of Gamma and points of S. This problem can be solved ....

....problem reduces to point location in a set of congruent disks in R 2 . Specifically, given a set Gamma of m congruent disks in the plane, we wish to count efficiently the number of containments between disks of Gamma and points of S. This problem can be solved using parametric searching [12], expander graphs [227] or randomization [251] The best known deterministic algorithm, given by Katz and Sharir [227] runs in O(n 4=3 log 3 n) time. Segment center. Given a set S of n points in R 2 and a line segment e, find a placement of e that minimizes the largest distance from the ....

P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri, Selecting distances in the plane, Algorithmica, 9 (1993), 495--514. Arrangements May 26, References 61

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P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. Algorithmica, 9:495--514, 1993.

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P. K. Agarwal, B. Aronov, M. Sharir, and S. Suri. Selecting distances in the plane. In Proc. 6th Annu. ACM Sympos. Comput. Geom., pages 321--331, 1990.

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