SHARIR, M., AND WELZL, E. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom. (1996), pp. 122--132.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and, if so, we compute the rectangular region where the center of R can be found. We check whether this region contains some point of Q. If the answer is positive, we are done, so assume that it does not. If P cannot be covered by a single square of area we apply the linear time algorithm of [23] (see also [6] to check whether P can be covered by two squares R 1 and R 2 of A. If not, then obviously we have no solution for our decision problem. Thus, we assume that there exist two squares (or even only one) of area that cover P , and that at least two discrete squares of area are ....

....to determine whether M contains a # Y Y # entry without having to construct the entire matrix. We inspect only O(n) entries in M advancing along a connected path within M [7] For each such entry, we must maintain dynamically whether P i,j is (1, Q) coverable, z r . Equivalently (see e.g. [23]) we can instead maintain dynamically the rectangle R = p#P i,j R(p) z r ) and ask whether 6 it contains a point of Q. This can be answered in O(log n) time by using a standard orthogonal range searching data structure (a range tree) of size O(n log n) 5] Thus the time and space ....

M. Sharir and E. Welzl, "Rectilinear and polygonal p-piercing and p-center problems", Proc. 12th ACM Symp. on Computational Geometry , 122--132, 1996. 25

....runtime was presented in Hwang et al. 23] for the planar discrete Euclidean k center problem. Recently, Agarwal and Procopiuc [1] extended and simpli ed the technique by Hwang et al. 24] to obtain an n O(k 1 1=d time algorithm for computing the Euclidean k center problem. Sharir and Welzl [32] explain a reduction from the rectilinear k center problem to the k piercing problem (under L1 metric) using a sorted matrices searching technique developed by Frederickson and Johnson [13] Ko et al. 26] proved the hardness of the planar version of the rectilinear k center and presented a O(n ....

Micha Sharir and Emo Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos.

....in Hwang et al. 23] for the planar discrete Euclidean k center problem. Recently, Agarwal and Procopiuc [1] extended and simpli ed the technique by Hwang et al. 24] to obtain an n O(k 1 1=d time algorithm for computing the Euclidean k center problem in d dimensions. Sharir and Welzl [32] explain a reduction from the rectilinear k center problem to the k piercing set problem (under L1 metric) using a sorted matrix searching technique developed by Frederickson and Johnson [13] Ko et al. 26] proved the hardness of the planar version of the rectilinear k center and presented an ....

Micha Sharir and Emo Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. on Computational Geometry, pages 122-132, 1996. 23

....the linear algorithm that is induced by the proof of Danzer and Grunbaum (described at the beginning of this section) into a linear algorithm for the 3 piercing problem for rectangles in the plane. Combining the approach of this algorithm with some dynamic data structures, Sharir and Welzl [SW96] have recently obtained O(n polylog(n) solutions to the 4 and 5 piercing problems for rectangles in the plane. 7 E F h s orientation o s 1 new B Case 2 (s s 6= s s 1 B s B Figure 1: Considering a pair of opposite orientations o and o in (B 1 ; B 2 ....

....the set of objects that are not pierced by p is 2 pierceable. We can maintain the 2 pierceability property of this complementary set under insertions and deletions of objects in logarithmic time, using a linear size dynamic data structure that is reminiscent of the data structures constructed in [SW96] for the 4 and 5 piercing problems for rectangles in the plane. We describe this data structure in the proof of Lemma 7 below. We thus obtain an O(n log n) algorithm. Theorem 6 Let S be a set of n homothetic triangles. Then it is possible to find a piercing triplet for S (if such exists) in O(n ....

[Article contains additional citation context not shown here]

M. Sharir and E. Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th ACM Symp. on Computational Geometry , pages 122--132, 1996. 15

....scaling of one coordinate axis. The resulting set of p squares will be called (a) minimal (rectilinear) p covering of P and can be seen as a natural generalization of the bounding box of P. If the input consists of points instead of polygons, the problem is known as rectilinear p center problem [4, 13]. So, another way to look at polygon covering is as a generalization of the rectilinear p center problem to in nite point sets. Since the latter problem cannot be approximated within a factor of less than two unless P = NP [10, 9] one cannot expect to do better for the more general problem of ....

....a factor of less than two unless P = NP [10, 9] one cannot expect to do better for the more general problem of covering polygons. Despite the apparent intractability, ecient algorithms have been developed for small values of p. The rectilinear 2 and 3 center problem can be solved in linear time [4, 8, 13], while for p = 4 already there is a lower bound of n log n) and algorithms of matching complexity for p 5 [3, 11, 12] The hope is that also the polygon covering problem can be solved eciently for a small number of covering squares. 2 Results While in principle techniques similar to the ....

Sharir, M., and Welzl, E. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom. (1996), pp. 122-132. 4

....input and d 2, or if d is part of the input and p 3 [78, 143] Ko et al. 120] showed that computing a solution set S with c(D; S) 2r is also NP Hard. The rectilinear 1 center problem is trivially solved in linear time. See [62, 118, 119, 143] for some earlier results. Sharir and Welzl [172] developed a linear time algorithm for the rectilinear 3 center problem, by showing that it is an LP type problem (as is the rectilinear 2 center problem) This is an instance of nonconvex programming that is LP type. Using the technique described in Section 2.3, Chan [36] proposed an O(n log n) ....

....2.3, Chan [36] proposed an O(n log n) expected time randomized algorithm for computing a rectilinear 5 center, which is optimal in the worstcase. Chan also presented an O(n log n) expected time algorithm for computing the smallest square that contains k of a given set of n points in the plane. See [114, 172] for additional related results. 6.2 Euclidean p line center Let D be a set of n points in R d and be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir showed that the problem ....

M. Sharir and E. Welzl, Rectilinear and polygonal p-piercing and p-center problems, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 122-132.

....sides solved in time O(n 2 ) by Jaromczyk et al. 14] Hershberger and Suri [13] and Glozman et al. 12] considered the problem of covering the set S by two axis parallel rectangles such that the size of the larger rectangle is minimized. They present an O(n log n) algorithm for this problem. In [22, 19, 20] several algorithms are presented that deal with a number of squares (rather than 2) that are not constrained. A recent paper of Katz et al. 17] presents three algorithms for various versions of the discrete two square problem. In the first version, the squares are axis parallel (O(n log 2 n) ....

Sharir M., Welzl E.: Rectilinear and polygonal p-piercing and p-center problems. Proc. 12th ACM Symp. on Comput. Geometry (1996) 122--132

....and p 3 [82, 149] Ko et al. 126] showed that computing a solution set S with c(D; S) 2r , where r is the size of an optimal solution, is also NP Complete. The rectilinear 1 center problem is trivially solved in linear time. See [66, 124, 125, 149] for some earlier results. Sharir and Welzl [182] developed a linear time algorithm for the rectilinear 3 center problem, by showing that it is an LP type problem (as is the rectilinear 2 center problem) This is an instance of nonconvex programming that is LP type. Using the technique described in Section 2.3, Chan [38] proposed an O(n log n) ....

....2.3, Chan [38] proposed an O(n log n) expected time randomized algorithm for computing a rectilinear 5 center, which is optimal in the worstcase. Chan also presented an O(n log n) expected time algorithm for computing the smallest square that contains k of a given set of n points in the plane. See [120, 182] for additional related results. 6.2 Euclidean p line center Let D be a set of n points in R d and ffi be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir [151] showed that the ....

M. Sharir and E. Welzl, Rectilinear and polygonal p-piercing and p-center problems, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 122--132.

....52] where demand arises from the continuous point sets along the edges in a graph. See [51] for results on the placement of k capacitated facilities serving a continuous demand on a one dimensional interval. Also, k center problems have been studied extensively in a geometric setting, see e.g. [1, 15, 24, 26, 27, 28, 29, 30, 31, 36, 38, 49, 50]. However, designing discrete algorithms for k center problems can generally be expected to be more immediate than for k median problems, since the set of demand points that determine a critical center location will usually form just a nite set of d 1 points in d dimensional space. Continuous ....

....on the methods of Megiddo. The geodesic 1 center of simple polygons has an O(n log n) algorithm [47] Exciting recent results of Sharir 3 et al. 49, 20, 8] have yielded nearly linear time algorithms for the planar two center problem. The more general p center problem has been studied recently by [50]. The results outlined in this abstract constitute much of the PhD thesis of Weinbrecht [54] most of the details necessarily omitted here are presented in depth in the thesis. 2 Preliminaries We will let Z = x; y) denote a candidate center point in F 2 . We concentrate on ....

M. Sharir and E. Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 122-132, 1996.

....sizes, then we have the weighted rectilinear p center problem, and if R is a set of arbitrary axisparallel rectangles (and the scaling points are also arbitrary) then we face the general rectilinear p center problem. Results for the problems defined above, for p = 2 and d = 2, were obtained by [3,4,11,12]. Hershberger and Suri [4] solve the following clustering problem: Given a planar set of points S, a rectangular measure acting on S and a pair of values 1 and 2 , does there exist a bipartition S = S 1 [S 2 satisfying (S i ) i for i = 1; 2) They present an algorithm which solves this ....

....an algorithm which solves this problem in O(n log n) time. Based on this algorithm and using the sorted matrix technique of Frederickson and Johnson [2] Glozman et al. 3] obtained an O(n log n) time algorithm that solves min maxbox problem in the plane. In a very recent paper Sharir and Welzl [12] using LP type framework and Helly type results obtained an O(n) expected time algorithm for the general rectilinear 2 center problem, where d = 2. The paper of Segal [10] solves the min max two box problem in three dimensions. The runtime of the algorithm in [10] is O(n 2 log n) In this paper ....

Micha Sharir and Emo Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 122--132, 1996. 9

....Abstract Rectangular p centers of a finite planar point set P are the centers of at most p axis parallel congruent squares of minimal size covering P . We give a simple linear time algorithm based on linear selection for the case p = 3. A linear algorithm for this problem is already known [7]. But it makes use of an LP type [6] formulation of the problem with high combinatorial dimension (roughly 40) which makes it unlikely to perform well in an actual implementation. The motivation for our algorithm is such an implementation. 1 Theoretical Results Let P be a set of points in the ....

....] x l x r ; y b y t . For the sake of simplicity let us furthermore assume that P is in general position, i.e. no two points have a common x or y coordinate nor the same jj Delta jj 1 distance to one of the corners of BP . We will first repeat a number of simple observations as listed in [7]. Observation 1 1. We can restrict ourselves to squares that are contained in BP . 2. Since each of the four line segments bounding BP contains at least one point from P, we have to place a square on each of them. Consequently, This work is partially supported by the ESPRIT IV LTR Projects No. ....

Sharir, M., and Welzl, E. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom. (1996), pp. 122--132.

....two planar n point sets under translations can be found in O(n 2 log n) expected time. This problem was studied by Chew and Kedem [20] who gave an O(n 2 log 2 n) time algorithm. ffl The two dimensional rectilinear 5 center problem can be solved in O(n log n) expected time. Recent works [46, 62, 66] gave an O(n log 2 n) time bound. ffl The two dimensional linear programming problem with k violated constraints can be solved in O(n log n) expected time in the feasible case for any 0 k n. If k is not too large, the time bound is actually linear. This type of problem was considered by ....

....literature [3] First, if the search space has linear size, then an ordinary binary search is sufficient. For many rectilinear problems, the search space forms a matrix with sorted rows columns, and one can use Frederickson and Johnson s selection algorithm [37] to carry out the binary search [20, 39, 66]; that algorithm relies heavily on repeated weighted median computations. In other instances, one can employ nontrivial explicit 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 ....

[Article contains additional citation context not shown here]

M. Sharir and E. Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th ACM Sympos. Comput. Geom., pages 122--132, 1996.

....2.1 The decision algorithm The decision problem is stated as follows: Given a set P of n points, are there two constrained axis parallel squares, each of a given area A, whose union covers P . We present an O(n log n) algorithm for solving the decision problem. We adopt the notation of [20] (see also [12, 17] Denote by R the set of axisparallel squares of area A centered at the points of P . R is p pierceable if there exists a set X of p points which intersects each of the squares in R. The set X is called a piercing set for R. Notice that X is a piercing set for R if and only if ....

....If R is not empty then R is 1 pierceable, and we check whether it is also 1 constrained pierceable by checking whether P has a point in R. If R is 1 constrained pierceable then we are done, so assume that it is not. If R was not found to be 1 pierceable, then we apply the linear time algorithm of [20] (see also [5] to check whether R is 2 pierceable. If R is neither 1 pierceable nor 2 pierceable, then obviously R is not 2 constrained pierceable and we are done. Assume therefore that R is 2 pierceable (or 1 pierceable) Assume R is 2 constrained pierceable, and let p 1 ; p 2 2 P be a pair of ....

M. Sharir and E. Welzl, "Rectilinear and polygonal p-piercing and p-center problems", Proc. 12th ACM Symp. on Computational Geometry, 122--132, 1996. This article was processed using the L a T E X macro package with LLNCS style

....imply that there is not much hope for a subquadratic solution for k = 3 or for any other value of k greater than 3. Obviously, if the set R is not k pierceable, then there is no solution. Therefore, we assume that R is k pierceable. We can check whether R is k pierceable, 1 k 2, in O(m) time [19]. When solving a problem, we first present a solution to the corresponding decision problem, and then apply the sorted matrices technique of Frederickson and Johnson [10] or the parametric searching technique of Megiddo [14] to obtain a solution to the original problem. That is, we first solve a ....

....bound of Omega Gamma n log n) Consider the GAP EXISTENCE problem: Given a set A of n real numbers A = fa 1 ; a n g, determine whether there exist two consecutive numbers in the sorted sequence obtained from A, such that the difference between them is greater than 1. Sharir and Welzl [19] observed that this problem has a lower bound of Omega Gamma n log n) We transform a i , i = 1; n, to the one dimensional rectangle [a i ; a i 1] thus obtaining a set R of n rectangles. We define R = min a i 2A a i ; max a i 2A a i ] It is clear that R is not covered by the ....

[Article contains additional citation context not shown here]

M. Sharir and E. Welzl "Rectilinear and polygonal p-piercing and p-center problems" In Proc. 12th ACM Symp. on Computational Geometry, pp. 122--132, 1996.

....sizes, then we have the weighted rectilinear p center problem, and if R is a set of arbitrary axisparallel rectangles (and the scaling points are also arbitrary) then we face the general rectilinear p center problem. Results for the above defined problems, for p = 2 and d = 2, were obtained by [4,3,12,11]. Hershberger and Suri [4] solve the following clustering problem: Given a planar set of points S, a rectangular measure acting on S and a pair of values 1 and 2 , does there exist a bipartition S = S 1 [S 2 satisfying (S i ) i for i = 1; 2) They present an algorithm which solves this ....

....an algorithm which solves this problem in O(n log n) time. Based on this algorithm and using the sorted matrix technique of Frederickson and Johnson [2] Glozman et al. 3] obtained an O(n log n) time algorithm that solves min maxbox problem in the plane. In a very recent paper Sharir and Welzl [12] using LP type framework and Helly type results obtained an O(n) expected time algorithm for the general rectilinear 2 center problem, where d = 2. The paper of Segal [10] solves the min max two box problem in three dimensions. The runtime of the algorithm in [10] is O(n 2 log n) In this paper ....

Micha Sharir and Emo Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 122--132, 1996.

....the linear algorithm that is induced by the proof of Danzer and Grunbaum (described at the beginning of this section) into a linear algorithm for the 3 piercing problem for rectangles in the plane. Combining the approach of this algorithm with some dynamic data structures, Sharir and Welzl [SW96] have recently obtained O(n polylog(n) solutions to the 4 and 5 piercing problems for rectangles in the plane. 3 Piercing Sets of c Oriented Polytopes with 2 Points Let H be a set of c halfspaces in E d , for some constant c. In this section, we consider the class CH of c oriented convex ....

....the set of objects that are not pierced by p is 2 pierceable. We can maintain the 2 pierceability property of this complementary set under insertions and deletions of objects in logarithmic time, using a linear size dynamic data structure that is reminiscent of the data structures constructed in [SW96] for the 4 and 5 piercing problems for rectangles in the plane. We describe this data structure in the proof of Lemma 7 below. We thus obtain an O(n log n) algorithm. Theorem 6 Let S be a set of n homothetic triangles. Then it is possible to find a piercing triplet for S (if such exists) in O(n ....

[Article contains additional citation context not shown here]

M. Sharir and E. Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th ACM Symp. on Computational Geometry , pages 122--132, 1996.

....rectangles was investigated; let us mention only the very recent papers. The 1 piercing problem was easily solved in linear time using the observation that 1 piercing problem for rectangles is equivalent to finding whether the intersection of rectangles empty or not. In Sharir and Welzl [7] 2 and 3 piercing problems in the plane are solved in linear time, while they reach only O(n log 3 n) bound for the 4 piercing problem and O(n log 4 n) bound for the 5 piercing problem. Katz and Nielsen [2] present a linear time algorithm for d dimensional boxes (d 2) when p = 2. In this ....

....and Nielsen [2] present a linear time algorithm for d dimensional boxes (d 2) when p = 2. In this paper we present a new technique which allows to obtain simple linear time algorithms for p = 1; 2; 3, and obtain an O(n log n) time solution for p = 4; 5, thus improving the previous results of [7]. We improve the existing algorithm of [7] for general (but fixed) p, and we extend our algorithms to higher dimensional space. We also consider the problem of piercing the set of rectangular rings. The boundary of a rectangular ring consists of two concentric rectangles, where the inner rectangle ....

[Article contains additional citation context not shown here]

M. Sharir, E. Welzl, "Rectilinear and polygonal p-piercing and p-center problems", In Proc. 12th ACM Symp. on Computational Geometry, 1996. This article was processed using the L a T E X macro package with LLNCS style

....al. requires only very basic properties of LP in order to work, and these properties are shared by many other problems, including nonlinear and even nonconvex optimization problems, for which in some cases the best known bounds are obtained using the uniform algorithm for the general problem class [19, 21]. The abstract class of problems amenable to this approach has been termed LP type problems. Similarly, Kalai s subexponential algorithm works in the more general setting of so called abstract objective functions (AOF) which he uses as a tool to derive extremely simple and beautiful proofs for ....

M. Sharir and E. Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 122--132, 1996.

....subset of a planar n point set [7, 22, 28] can be found in O(n log n) expected time for any 1 k n. ffl The minimum L1 Hausdorff distance between two planar n point sets under translations [16] can be found in O(n 2 log n) expected time. ffl The two dimensional rectilinear 5 center problem [50, 53] can be solved in O(n log n) expected time. ffl The two dimensional linear programming problem with k violations [41, 49] can be solved in O(n log n) expected time in the feasible case for any 0 k n. For a wide range of k, the time bound is actually linear. ffl Given an infeasible ....

....the extension of the problem to higher dimensions. Our technique again improves their running time by a logarithmic factor. Rectilinear p centers. A class of facility location problems known as p center problems has received much attention in the computational geometry literature (e.g. see [4, 24, 27, 51, 53]) We consider the case where p is a constant, the dimension is two, and the metric is L1 : given an n point set P IR 2 , find p congruent squares of the smallest size covering P . The decision problem reduces to a rectangular p piercing problem: given a set of n rectangles in the plane, ....

[Article contains additional citation context not shown here]

M. Sharir and E. Welzl. Rectilinear and polygonal p- piercing and p-center problems. In Proc. 12th ACM Sympos. Comput. Geom., pages 122--132, 1996.

....points, are there two axis aligned constrained squares, each of a given area A, whose union covers P . We present an O(n log n) algorithm for solving the decision problem. Applying the sorted matrices technique [10] we obtain an O(n log 2 n) time optimization algorithm. We adopt the notation of [22] (see also [14, 19] Denote by R the set of axis aligned squares of area A centered at the points of P . R is called p pierceable if there exists a set of p points which intersects every member in R. These points are called piercing points and the union of the axisaligned squares of area A ....

....If R is not empty then R is 1 pierceable, and we check whether it is also 1 constrained pierceable by checking whether P has a point in R. If R is 1 constrained pierceable then we are done, so assume that it is not. If R was not found to be 1 pierceable, then we apply the linear time algorithm of [22] (see also [6] to check whether R is 2 pierceable. If R is neither 1 pierceable nor 2 pierceable, then obviously R is not 2 constrained pierceable and we are done. Assume therefore that R is 2 pierceable (or 1 pierceable) Assume R is 2 constrained pierceable, and let p 1 ; p 2 2 P be a pair of ....

M. Sharir and E. Welzl, "Rectilinear and polygonal p-piercing and p-center problems", Proc. 12th ACM Symp. on Computational Geometry , 122--132, 1996. (iii) (iv) (ii) (i) (v)

....imply that there is not much hope for a subquadratic solution for k = 3 or for any other value of k greater than 3. Obviously, if the set R is not k pierceable, then there is no solution. Therefore, we assume that R is k pierceable. We can check whether R is k pierceable, 1 k 2, in O(m) time [18]. When solving a problem, we first present a solution to the corresponding decision problem, and then apply the sorted matrices technique of Frederickson and Johnson [10] or the parametric searching technique of Megiddo [14] to obtain a solution to the original problem. That is, we first solve a ....

....bound of Omega Gamma n log n) Consider the GAP EXISTENCE problem: Given a set A of n real numbers A = fa 1 ; a n g, determine whether there exist two consecutive numbers in the sorted sequence obtained from A, such that the difference between them is greater than 1. Sharir and Welzl [18] observed that this problem has a lower bound of Omega Gamma n log n) We transform a i , i = 1; n, to the one dimensional rectangle [a i ; a i 1] thus obtaining a set R of n rectangles. We define R = min a i 2A a i ; max a i 2A a i ] It is clear that R is not covered by the ....

[Article contains additional citation context not shown here]

M. Sharir and E. Welzl "Rectilinear and polygonal p-piercing and p-center problems" In Proc. 12th ACM Symp. on Computational Geometry, pp. 122--132, 1996.

....sizes, then we have the weighted rectilinear p center problem, and if R is a set of arbitrary axis parallel rectangles (and the scaling points are also arbitrary) then we face the general rectilinear p center problem. Results for the above defined problems, for p = 2 and d = 2, were obtained by [4, 1, 7]. Hershberger and Suri [4] solve the following clustering problem: Given a planar set of points S, a rectangular measure acting on S and a pair of values 1 and 2 , does there exist a bipartition S = S 1 [S 2 satisfying (S i ) i for i = 1; 2) They present an algorithm which solves this ....

....an algorithm which solves this problem in O(n log n) time. Based on this algorithm and using the sorted matrix technique of Frederickson and Johnson [2] Glozman et.al [1] obtained an O(n log n) time algorithm that solves min max box problem in the plane. In a very recent paper Sharir and Welzl [7] using LP type framework and Helly type results obtained an O(n) expected time algorithm for the general rectilinear 2 center problem, where d = 2. The paper of Segal [5] solves the same problem as this paper but only in the case where d = 3. The runtime of the algorithm in [5] is O(n 2 log n) ....

M. Sharir, E. Welzl "Rectilinear and polygonal p-piercing and p-center problems", In Proc. 12th ACM Symp. on Computational Geometry, 1996.

....for the rectilinear 2 center problem, even if d is unbounded, is given in [216] A linear time algorithm for the planar rectilinear 2 center problem is given by Drezner [94] see also [184] Ko and Lee [185] gave an O(n log n) time algorithm for the weighted case. Recently, Sharir and Welzl [257] have developed a linear time algorithm for the rectilinear 3 center problem, by showing that it is an LP type problem (as is the rectilinear 2 center problem) They have also obtained an O(n log n) time algorithm for computing a rectilinear 4 center, using the matrix searching technique by ....

....for computing a rectilinear 4 center, using the matrix searching technique by Frederickson and Johnson; they have shown that this algorithm is optimal in the worst case. Recently Chan [54] developed an O(n log n) expected time randomized algorithm for computing a rectilinear 5 center. See [177, 257] for additional related results. 7.2 Euclidean p line center and ffi be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir showed that the problem of determining whether w = 0 ....

M. Sharir and E. Welzl, Rectilinear and polygonal p-piercing and p-center problems, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 122--132.

....for the rectilinear 2 center problem, even if d is unbounded, is given in [186] A linear time algorithm for the planar rectilinear 2 center problem is given by Drezner [81] see also [157] Ko and Lee [158] gave an O(n log n) time algorithm for the weighted case. Recently, Sharir and Welzl [219] have developed a linear time algorithm for the rectilinear 3 center problem, by showing that it is an LP type problem (as is the rectilinear 2 center problem) They have also obtained an O(n log n) time algorithm for computing a rectilinear 4 center (and have shown that this algorithm is ....

....algorithm for computing a rectilinear 4 center (and have shown that this algorithm is worst case optimal) and an O(n log 5 n) time algorithm for computing a rectilinear 5 center. The algorithms for the 4 center and 5 center employ the Frederickson Johnson matrix searching technique. See [152, 219] for additional related results. 7.2 Euclidean p line center Let D be a set of n points in R d and ffi be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir showed that the problem of ....

M. Sharir and E. Welzl, Rectilinear and polygonal p-piercing and p-center problems, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 122--132.

....straight edges, the minimal angle between two adjacent edges in the triangulated polygon is maximized (combinatorial dimension 3) Rectilinear 3 centers in the plane. Given a set P of n points in the plane, find three points c 1 ; c 2 ; c 3 so that max p2P min i=1;2;3 kc i Gamma pk1 is minimal [27]. Spherical separability. Given n red and n blue segments in d space, find the smallest radius ball covering all red segments and disjoint from all blue segments. Holds also for many computationally simple objects instead of line segments [28] Width of thin point sets in the plane. The width ....

Sharir, M., Welzl, E.: Rectilinear and polygonal p-piercing and p-center problems. Manuscript, submitted (1995)

....for the rectilinear 2 center problem, even if d is unbounded, is given in [217] A linear time algorithm for the planar rectilinear 2 center problem is given by Drezner [95] see also [185] Ko and Lee [186] gave an O(n log n) time algorithm for the weighted case. Recently, Sharir and Welzl [259] have developed a linear time algorithm for the rectilinear 3 center problem, by showing that it is an LP type problem (as is the rectilinear 2 center problem) They have also obtained an O(n log n) time algorithm for computing a rectilinear 4 center, using the matrix searching technique by ....

....for computing a rectilinear 4 center, using the matrix searching technique by Frederickson and Johnson; they have shown that this algorithm is optimal in the worst case. Recently Chan [56] developed an O(n log n) expected time randomized algorithm for computing a rectilinear 5 center. See [178, 259] for additional related results. 7.2 Euclidean p line center Let D be a set of n points in R d and ffi be the Euclidean distance function. We wish to compute the smallest real value w so that D can be covered by the union of p strips of width w . Megiddo and Tamir showed that the problem ....

M. Sharir and E. Welzl, Rectilinear and polygonal p-piercing and p-center problems, Proc. 12th Annu. ACM Sympos. Comput. Geom., 1996, pp. 122--132.

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SHARIR, M., AND WELZL, E. Rectilinear and polygonal p-piercing and p-center problems. In Proc. 12th Annu. ACM Sympos. Comput. Geom. (1996), pp. 122--132.

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