P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 67--75, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... by several output sensitive algorithms, as surveyed in [12] The simplest to implement is perhaps Basch et al. s kinetic tournament tree method [6] for (expected) O(n) size output, the (expected) running time is O(n (n) log n) Theoretically, the randomized algorithms by Agarwal et al. [2, 12] and Har Peled [34] have faster expected running time, of O(n (n) log n) and O(n (n) log n) respectively. Brodal and Jacob s recent announcement regarding optimal kinetic heaps [7] implies the ultimate running time of O(n log n) Step 1: how to construct the concave chains second option. We ....

....concave chains second option. We now offer a different method for step 1 that also achieves O(n log n) expected time and has the advantage of being generalizable to 3 d (as we will see in Section 3) This second option is based on Matousek s shallow cutting lemma [37] Ramos [46] building on [2, 14]) has given a corresponding randomized construction in 3 d, which will be useful to us. We restate the 3 d result below in a form we find convenient. Lemma 2.1 Given n lower halfspaces in IR , the ( k) level can be covered by O(n=k) cells, each intersecting at most O(k) bounding planes. ....

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654-- 667, 1998.

....probability p. Alternatively, we say that R is an r sample, where r = pjXj, when we want to emphasize the expected size. Let n4 = jK(4)j = jS j4 j. The excess t 4 = t 4 (p; X) of 4 2 C with respect to p is pn4 . The following theorem is implicit in the work of de Berg et al. [14] and Agarwal et al. [2]. Previously, the corresponding result was known with property (3) instead of (3 ) Chazelle and Friedman [21] and Matou sek [46] We assume 1, and choose the value of p 0 so that the resulting factor C is O(1) independent from ; in fact C 4 and decreases to e as goes to in nity) ....

P. K. Agarwal, M. de Berg, J. Matousek and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27 (1998), 654-667.

....of bottom up and top down approaches is indispensable. A nal technicality is the proof of appropriate sampling bounds for the sizes of the chain con ict lists of our conformal decomposition: such bounds are known under locality or monotonicity properties that our decomposition does not satisfy [1, 9, 11, 22, 24]. Fortunately, we can prove appropriate bounds using the fact that, although the faces in the decomposition do not satisfy a locality property, they are chosen from a relatively small pool of candidates that satisfy a locality property. This paper is organized as follows. First, for comparison ....

....that for a chain trapezoid e 2 e T (K) L j e denotes the list of con icts of e in L, and that e n e = jL j e j. Unfortunately, we cannot prove the bound in Eqn. 2) Such a bound can be proved in the framework of con guration spaces, when certain locality [9, 24] or monotonicity [11, 1] properties hold for the decomposition induced by the sample (see [22] for a survey) but neither of these properties hold for our chain trapezoidation. Fortunately, we can prove a weaker bound that is only a factor O(f(log ) larger, and that suces to verify that our algorithm has expected ....

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654-667, 1998.

....below (any point in) OE. A clustering of A k (P ) consists of a partition of the facets into O( k =k) families, suchthat for each family Phi, its associated cluster S OE2 Phi P OE contains O(k) planes. The entire clustering can be stored using O( k =B) blocks. Lemma 6. 1 (Agarwal et al. [2]) Let P be a set of N planes. For any 1 k N , a clustering of A k (P ) can be constructed using O(Nk 5=3 ) expected I Os. We construct our data structure for P as follows. Let fi = cB log B n for some constant c 1. If the number of planes in P is less than fi,we simply store the planes in ....

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 67--75, 1994.

....case d = 3 however remains an open problem, even though a number of algorithms have been developed. A randomized algorithm of Mulmuley [52] for instance runs in expected time O(nk 2 log(n=k) which is just a logarithmic factor away from optimal. Agarwal, de Berg, Matousek, and Schwarzkopf [2] proposed a different randomized algorithm with expected running time O(n log 3 n nk 2 ) which is optimal for sufficiently large k. In Section 3, we settle the complexity of the three dimensional case completely by giving a randomized algorithm with an O(n log n nk 2 ) expected time bound; ....

....for k n=2 [32, 46] The k level in this case is related to the order k Voronoi diagram of n points in the Euclidean plane a natural extension to one of the most fundamental and useful geometric structures, the Voronoi diagram. 3 Some of the algorithmic results obtained by Agarwal et al. [2] on the k level can be improved by our techniques. Specifically, their expected time bound of O(n log 2 n nk 1=3 log 2=3 n) for the construction of k levels in the plane can be reduced to O(n log n nk 1=3 log 2=3 k) Furthermore, their O(n log 3 n nk log n) algorithm for order k ....

[Article contains additional citation context not shown here]

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654--667, 1998.

.... in computation of higher order Voronoi diagrams ( Aur91, Mul93] in orthogonal L 1 hyperplane fitting ( KM93] and in halfspace range searching ( CP86, AM95] The first output sensitive algorithm for enumerating k sets was given in [EW86] for R 2 ) and other such algorithms appeared in [Mul91, AM95, AMdS94]. While the above algorithms concentrate on time efficiency and require sophisticated data structures, we present here two output sensitive algorithms which are highly memory efficient. They are based on the reverse search technique introduced in [AF92, AF96] Except for being memory efficient, ....

Pankaj K. Agarwal, Jir'i Matousek, Mark de Berg, and Otfried Schwarzkopf. Constructing levels in arrangements and higher order voronoi diagrams. In Proceedings of the Tenth Annual Symposium on Computational Geometry, pages 67--75, Stony Brook, New York, June 1994.

....on the worst case size is known, namely O(nk) This leads to a trivial lower bound n log n nk) for constructing a k level in this particular case. Despite much work, an algorithm whose running time matches that lower bound has not been found. The most recent breakthroughs were by Agarwal et al. [1] and by Chan [11] complete references to previous work can be found in those papers) The rst paper gives an algorithm with expected running time O(n log 3 n nk log n) while the second one obtains expected running time O(n log n nk log k) using the rst algorithm as a black box) With a ....

....While in the rst approach the problem is subdivided and then each subproblem solved independently, the second maintains a global structure throughout the computation. Here, we use the incremental construction via gradations [26, 17] This is akin to the lazy randomized construction [7] used in [1]. For completeness and to be able to supply a complete analysis, we rst recall the approach and sketch its analysis. Then, we describe the algorithms for computing the k level and the cuttings needed in the data structure of Subsection 2.2.2. 3.1 Incremental Construction via Gradations The ....

[Article contains additional citation context not shown here]

P. K. Agarwal, M. de Berg, J. Matousek and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27 (1998), 654-667.

....p. Alternatively, we say that R is an r sample, where r = pjXj, when we want to emphasize the expected size. Let n4 = jK(4)j = jS j4 j. The excess t 4 = t 4 (p; X) of 4 2 C with respect to p is pn4 . The following theorem is implicit in the work of de Berg et al. 15] and Agarwal et al. [2]. Previously, the corresponding result was known with property (3) instead of (3 ) Chazelle and Friedman [23] and Matou sek [49] We assume 1, and choose the value of p 0 so that the resulting factor C is O(1) independent from ; in fact C 4 and decreases to e as goes to in nity) ....

P. K. Agarwal, M. de Berg, J. Matousek and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27 (1998), 654-667.

....on the worstcase size is known, namely O(nk) This leads to a trivial lower bound n log n nk) for constructing a k level in this particular case. Despite much work, an algorithm whose running time matches that lower bound has not been found. The most recent breakthroughs were by Agarwal et al. [1] and by Chan [9] complete references to previous work can be found in those papers) The rst paper gives an algorithm with expected running time O(n log 3 n nk log n) while the second one obtains expected running time O(n log n nk log k) using the rst algorithm as a black box) With a ....

....While in the rst approach the problem is subdivided and then each subproblem solved independently, the second maintains a global structure throughout the computation. Here, we use the incremental construction via gradations [22, 15] This is akin to the lazy randomized construction [5] used in [1]. For completeness and to be able to supply a complete analysis, we rst recall the approach and sketch its analysis. Then, we describe the algorithms for computing the k level and the cuttings needed in the data structure of Subsection 2.2.2. 3.1 Incremental Construction via Gradations The ....

[Article contains additional citation context not shown here]

P. K. Agarwal, M. de Berg, J. Matousek and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27 (1998), 654-667.

....case. We give an algorithm that guarantees O(n log n nk 1=3 ) expected time. 1 Introduction The notion of k levels [2, 20, 26, 34] has proved to be an important one in computational geometry, exploited directly or indirectly in various algorithms for computing higher order Voronoi diagrams [1, 11, 14] (used, for instance, in finding clusters of points [3, 23] designing data structures for halfspace range searching [16, 17] and solving hyperplane partitioning problems (such as hamsandwich cuts [32] and weak line separators [25] The concept is also fundamental in the combinatorics of ....

.... a more traditional randomized incremental algorithm produces essentially the same expected time bound, except for a mere ff(n) factor (possibly an artifact of our analysis) O( n m)ff(n) 2 log n) Actually, this algorithm is an already known one by Agarwal, de Berg, Matousek, and Schwarzkopf [1] from 1994. They looked at the problem in both the planar and the three dimensional case, and examined not only the k level but also a related structure known as the ( k) level. Although they did not give an output sensitive analysis for the planar k level case, they provided basically all the ....

[Article contains additional citation context not shown here]

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654--667, 1998.

....case d = 3 however remains an open problem, even though a number of algorithms have been developed. A randomized algorithm of Mulmuley [47] for instance runs in expected time O(nk 2 log(n=k) which is just a logarithmic factor away from optimal. Agarwal, de Berg, Matousek, and Schwarzkopf [2] proposed a different randomized algorithm with expected running time O(n log 3 n nk 2 ) which is optimal for sufficiently large k. In Section 3, we settle the complexity of the three dimensional case completely by giving a randomized algorithm with an O(n log n nk 2 ) expected time bound; ....

....for k n=2 [29, 41] The k level in this case is related to the order k Voronoi diagram of n points in the Euclidean plane a natural extension to one of the most fundamental and useful geometric structures, the Voronoi diagram. Some of the algorithmic results obtained by Agarwal et al. [2] on the k level can be improved by our techniques. Specifically, their expected time bound of O(n log 2 n nk 1=3 log 2=3 n) for the construction of k levels in the plane can be reduced to O(n log n nk 1=3 log 2=3 k) Furthermore, their O(n log 3 n nk log n) algorithm for order k ....

[Article contains additional citation context not shown here]

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654--667, 1998.

....to O(n log m m log 1 n) deterministically. The algorithm uses O(n) space. Very recently, Har Peled [22] announced a marginally faster O( n m)ff(n) log n) expected time bound via a randomized approach. Actually, as the author observed [10] an earlier randomized algorithm by Agarwal et al. [1] gives almost the same result. However, it is unclear whether the space complexity of either randomized algorithm can be made O(n) ffl The order k Voronoi diagram of n points in the plane is another well known structure in computational geometry, with various applications of its own. An early ....

....some complicated machinery (namely, shallow cuttings [9] in conjunction with this new result, we can construct the diagram in time O(nk log 1 k Delta (log n= log k) O(1) which is currently the best determinstic bound. Simpler and slightly faster randomized algorithms are known though [1, 9, 33]. ffl Basch, Guibas, and Ramkumar [5] examined a natural version of the segment intersection problem: given a connected family R of n red line segments and a connected family B of n blue line segments in the plane, report all intersecting pairs from R Theta B. They (with an observation ....

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654--667, 1998.

.... others in computation of higher order Voronoi diagrams ( Aur91, Mul93] in orthogonal L 1 hyperplane fitting ( KM93] and in halfspace range searching ( CP86, AM95] The first outputsensitive algorithm for enumerating k sets was given in [EW86] for R 2 ) and other such algorithms appeared in [Mul91, AM95, AMdS94]. While the above algorithms concentrate on time efficiency and require sophisticated data structures, we present here two output sensitive algorithms which are highly memory efficient. They are based on the reverse search technique introduced in [AF92, AF96] Except for being memory efficient, ....

Pankaj K. Agarwal, Jir'i Matousek, Mark de Berg, and Otfried Schwarzkopf. Constructing levels in arrangements and higher order voronoi diagrams. In Proceedings of the Tenth Annual Symposium on Computational Geometry, pages 67--75, Stony Brook, New York, June 1994.

....in O(n 0 ) time. It is easy to prove that the ( k) level of an arrangement of fc 1 ; Delta Delta Delta ; c n 0 g is precisely Lk (H 0 ; h ) The remaining work is to compute the ( k) level of the arrangement. For doing this, we use the following previous result due to Agarwal et al. [1]. Lemma 2 [1] The ( k) level of an arrangement of n x monotone Jordan curves in the plane such that any pair intersects in at most s points can be computed in expected O(k 2 s (n=k) min( s (n) log 2 n; k 2 s (n=k) log n) time by a lazy randomized incremental algorithm. Here, s (m) is ....

....time. It is easy to prove that the ( k) level of an arrangement of fc 1 ; Delta Delta Delta ; c n 0 g is precisely Lk (H 0 ; h ) The remaining work is to compute the ( k) level of the arrangement. For doing this, we use the following previous result due to Agarwal et al. [1] Lemma 2 [1] The ( k) level of an arrangement of n x monotone Jordan curves in the plane such that any pair intersects in at most s points can be computed in expected O(k 2 s (n=k) min( s (n) log 2 n; k 2 s (n=k) log n) time by a lazy randomized incremental algorithm. Here, s (m) is the maximum ....

Pankaj K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 67--75, 1994.

.... time [EW86] Cha95] where f is the combinatorial complexity of the k level, and the ( k) level in worst case optimal time O(n log n kn) ERvK93] Algorithms for computing the ( k) level in an arrangement of Jordan arcs and the ( k) level in an arrangement of planes in R 3 are described in [AdBMS94] For computing the k level in an arrangement of hyperplanes in R d see [Cha95] MANY CELLS There are efficient algorithms (deterministic and randomized) for computing a set of selected faces in arrangements of lines or segments in the plane. These algorithms are nearly worst case optimal ....

P.K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 67--75, 1994.

....case d = 3 however remains an open problem, even though a number of algorithms have been developed. A randomized algorithm of Mulmuley [42] for instance runs in expected time O(nk 2 log(n=k) which is just a logarithmic factor away from optimal. Agarwal, de Berg, Matousek, and Schwarzkopf [2] proposed a different randomized algorithm with expected running time O(n log 3 n nk 2 ) which is optimal for sufficiently large k. In Section 3, we settle the complexity of the three dimensional case completely by giving a randomized algorithm with an O(n log n nk 2 ) expected time ....

....for k n=2 [25, 37] The k level in this case is related to the order k Voronoi diagram of n points in the Euclidean plane a natural extension to one of the most fundamental and useful geometric structures, the Voronoi diagram. Some of the algorithmic results obtained by Agarwal et al. [2] on the k level can be improved by our techniques. Specifically, their expected time bound of O(n log 2 n nk 1=3 log 2=3 n) for the construction of k levels in the plane can be reduced to O(n log n nk 1=3 log 2=3 k) Furthermore, their O(n log 3 n nk log n) algorithm for order k ....

[Article contains additional citation context not shown here]

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput.,

....in the induced ranking. Now, the ranking on S 0 is a uniform random variable on all permutations of S 0 , and thus the probability that s i ; s j are parent and child is 2= ffi S (v) 2) ut Levels in arrangements of lines and segments are a well studied topic in computational geometry [2, 25]. Although estimating the exact number of vertices at level has proven difficult, a simple bound on the number of vertices of level at most can be obtained using standard random sampling techniques [12] Lemma 4.4. Let S be a set of s segments in the plane. Denote by t the number of ....

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 67--75, 1994.

No context found.

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654-667, 1998.

....[Cha99a] presented a faster algorithm for the dynamic maintenance of the intersection of halfplanes, requiring O(log n) amortized time for each operation. Thus, one can compute the level l in O(n log n jE l j log deterministic time. Chan [Cha99b] also showed that by using the algorithm of [AdBMS98] one can compute the level l in O(n jE l jff(n) log n) randomized expected time. We note, however, that our algorithm is still faster and simpler than those two algorithms. 9.4.2 Other Applications In this subsection, we provide some additional applications of CompZoneOnline and CompLevel. ....

P.K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654--667, 1998.

....direction. The expected running time of the algorithm is O( n= 3 ) log 3 n) Our algorithm computes levels in an arrangement of weighted lines (See Section 2 for a precise definition of levels in weighted arrangements) Although much work has been done on computing levels of unweighted lines [1, 8, 10, 11, 13, 6, 7, 16], little is known about computing levels in arrangement of weighted lines. No sub quadratic bound is known on the complexity of a level in an arrangement of n lines with weights on the plane. In fact an Omega Gamma n 5=3 ) bound is known on the complexity of the median level in weighted ....

....of the k level. A recent result by Chan [6] improves this bound to O(n log n b log 1 ffi n) for any ffi 0. Har Peled used a different technique to obtain an O( n b)ff(n) log n) expected time algorithm for the same problem, where ff(n) is the inverse Ackermann function. Agarwal et al. [1] gave a randomized algorithm for computing the k level in expected time O(nk 1=3 n log n) A recent analysis by Chan [7] shows that the running time of this algorithm is O( n b)ff 2 (n) log n) The problem of computing approximate levels in arrangement of lines have also been considered by ....

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf, Constructing levels in arrangements and higher order Voronoi diagrams, SIAM J. Comput., 27 (1998), 654--667.

.... ( Gamma i ) where Gamma i is the set of arcs added so far, the expected running time of the algorithm is O( s 2 (n)k log(n=k) see, e.g. 275] Everett et al. 170] showed that if Gamma is a set of n lines, the expected running time can be improved to O(n log n nk) Recently Agarwal et al. [13] Arrangements May 26, Computing Substructures in Arrangements 49 gave another randomized incremental algorithm that can compute Ak ( Gamma) in expected time O( s 2 (n) k log n) In higher dimensions, little is known about computing Ak ( Gamma) for collections Gamma of surface patches. ....

....surface patches. For d = 3, Mulmuley [275] gave a randomized incremental algorithm for computing the k level in an arrangement of n planes whose expected running time is O(nk 2 log(n=k) The expected running time can be improved to O(n log 3 n nk 2 ) using the algorithm by Agarwal et al. [13]. There are, however, several technical difficulties in extending this approach to computing levels in arrangements of surface patches. Using the random sampling technique, Agarwal et al. 14] developed an O(n 2 k) expected time algorithm for computing Ak ( Gamma) for a collection Gamma of ....

[Article contains additional citation context not shown here]

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf, Constructing levels in arrangements and higher order Voronoi diagrams, SIAM J. Comput., 27 (1998), 654--667.

....lie below (any point in) OE. A clustering of A k (P ) consists of a partition of the facets into O( k =k) families, such that for each family Phi, its associated cluster S OE2 Phi P OE contains O(k) planes. The entire clustering can be stored using O( k =B) blocks. Lemma 6. 1 (Agarwal et al. [2]) Let P be a set of N planes. For any 1 k N , a clustering of A k (P ) can be constructed using O(Nk 5=3 ) expected I Os. We construct our data structure for P as follows. Let fi = cB log B n for some constant c 1. If the number of planes in P is less than fi, we simply store the planes ....

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 67--75, 1994.

No context found.

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 67--75, 1994.

No context found.

Agarwal, P. K., de Berg, M., Matou sek, J., and Schwarzkopf, O. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27 (1998), 654--667.

No context found.

Agarwal, P. K., de Berg, M., Matou sek, J., and Schwarzkopf, O. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27 (1998), 654--667.

No context found.

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654-667, 1998.

No context found.

P. K. Agarwal, M. de Berg, J. Matousek, and O. Schwarzkopf. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput., 27:654-667, 1998.

No context found.

Agarwal, P. K., de Berg, M., Matou sek, J., and Schwarzkopf, O. Constructing levels in arrangements and higher order Voronoi diagrams. SIAM J. Comput. 27 (1998), 654--667.

No context found.

P. K. Agarwal, M. de Berg, J. Matousek and O. Schwarzkopf, "Constructing Levels in Arrangements and Higher Order Voronoi Diagrams," Proc. 10th Annual ACM Symp. Comput. Geometry, 1994, 67-75.

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