D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287--299, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....n) lower bound, every convex hull algorithm must require (n log n) time for some inputs. Despite these matching upper and lower bounds, and probably because of the many applications of convex hulls, a number of other planar convex hull algorithms have been published since Graham s algorithm [1, 2, 4, 6, 11, 17, 21, 28, 29, 36]. Of particular note is the Ultimate( algorithm of Kirkpatrick and Seidel [21] that computes the convex hull of a set of n points in the plane in O(n log h) time, where h is the number of vertices of the convex hull. Later, the same result was obtained by Chan using a much simpler algorithm ....

.... upper and lower bounds, and probably because of the many applications of convex hulls, a number of other planar convex hull algorithms have been published since Graham s algorithm [1, 2, 4, 6, 11, 17, 21, 28, 29, 36] Of particular note is the Ultimate( algorithm of Kirkpatrick and Seidel [21] that computes the convex hull of a set of n points in the plane in O(n log h) time, where h is the number of vertices of the convex hull. Later, the same result was obtained by Chan using a much simpler algorithm [3] The same authors show that, on algebraic decision trees of any xed ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287-299, 1986.

....log n) lower bound, every convex hull algorithm must require (n log n) time for some inputs. Despite these matching upper and lower bounds, and probably because of the many applications of convex hulls, a number of other planar convex hull algorithms have been published since Graham s algorithm [1,2,4,6,11,17,27,28,21,35]. Of particular note is the Ultimate( algorithm of Kirkpatrick and Seidel [21] that computes the convex hull of a set of n points in the plane in O(n log h) time, where h is the number of vertices of the convex hull. The same authors show that, on algebraic decision trees of any xed order, ....

.... matching upper and lower bounds, and probably because of the many applications of convex hulls, a number of other planar convex hull algorithms have been published since Graham s algorithm [1,2,4,6,11,17,27,28,21,35] Of particular note is the Ultimate( algorithm of Kirkpatrick and Seidel [21] that computes the convex hull of a set of n points in the plane in O(n log h) time, where h is the number of vertices of the convex hull. The same authors show that, on algebraic decision trees of any xed order, n log h) is a lower bound for computing convex hulls of sets of n points having ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287-299, 1986.

....are referred to as output sensitive algorithms. Readers familiar with the literature on output sensitive convex hull algorithms may recognize the expression O(n log k) as the running time of optimal algorithms for computing convex hulls of n point sets with k extreme points, in 2 or 3 dimensions [2, 4, 5, 12, 18]. This is no coincidence. Given a set of n points in , we can color them all red and add three blue points at infinity (see Figure 2) In this set, the only points that contribute to the nearest neighbour decision boundary are the three blue points and the red points on the convex hull of the ....

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287--299, 1986.

....[6] can be used to find the convex hull of S in O(jSj log jSj) time. However, observe that we need to find only the edges of the convex hull that intersect the median line. This can be formulated as a linear programming problem. Inspired by linear programming algorithms, Kirkpatrick and Seidel [7] provide a linear algorithm to find such edges, which they call bridges. Compared to the Graham scan, the algorithm is quite complex, and its implementation uses finite precision floating point arithmetics and is complicated. Therefore our implementation uses a bridge finding algorithm based on ....

P. G. Kirkpatrick and R. Seidel. The Ultimate Planar Convex Hull Algorithm? SIAM Journal on Computing 15(1): 287-- 299, 1986.

....are referred to as output sensitive algorithms. Readers familiar with the literature on output sensitive convex hull algorithms may recognize the expression O(n log k) as the running time of optimal algorithms for computing convex hulls of n point sets with k extreme points, in 2 or 3 dimensions [2, 4, 5, 12, 18]. This is no coincidence. Given a set of n points in R , we can color them all red and add three blue points at infinity (see Figure 2) In this set, the only points that contribute to the nearest neighbour decision boundary are the three blue points and the red points on the convex hull of the ....

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287--299, 1986.

....form the set of non trivial cones CV = fc1 ; c2 ; crg. In effect, the above algorithm determines the optimal scheduling hyperplanes of all non trivial cones: they coincide with the inner facets. For more on this computational geometry method the interested reader may refer to [5] and [11]. 4.2 Calculation of the Trivial Cones For the calculation of the trivial cones, the computation of the outer facets suffices. To each outer facet corresponds a trivial cone. The number of the vectors that form the trivial cone depends on the specific facet. The following algorithm determines ....

D. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15:287--299, 1986.

....a convex set, there exists a smallest convex set containing any given set. This smallest convex set is called the convex hull of the input set. Algorithms for computing the Euclidean convex hull of a finite set of points of the Euclidean plane can be found in [10] 19, chaps. 4 5] 31] 18] and [16]. An extended bibliography on this topic is given in [20] In image analysis, convex hulls are at the basis of useful shape indices such as the concavity index, i.e. the ratio between the surface area of a 2 D connected bounded set and that of its convex hull. Alternatively, the difference ....

D. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM J. Cornput., 15(1):287-299, February 1986.

....hull (or halfspace intersection) algorithms have been analyzed not only in terms of n, the size of the input, but also in terms of f , the number of faces of the output polytope. The only known lower bound in terms of n and f is Omega Gamma n log f f) time. In 1986 Kirkpatrick and Seidel [22] published a paper entitled The Ultimate Planar Convex Hull Algorithm which computes the convex hull of n points in the plane in O(n log f) time. This algorithm is, therefore, output sensitive and worst case optimal. The ultimate algorithm is a divide and conquer algorithm based on a ....

....scheme for dividing a problem into subproblems. In the next section we set up notation and point out some preliminary facts about the construction of convex hulls. Section 3 gives a simple O(n log f) convex hull algorithm in the plane and compares it with Kirkpatrick and Seidel s algorithm [22]. Section 4 details and analyzes the algorithm in four dimensions. We briefly describe the application to Voronoi diagrams in Section 5. An extension of our 4 d algorithm to higher dimensions is given in Section 6. Note: A preliminary version of this paper, written for the problem of computing ....

[Article contains additional citation context not shown here]

D. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15:287--299, 1986.

....order and then eliminates vertices that are not on the convex hull. There is a solution by Preparata and Hong [PH77] that achieves also O(n log n) time. It is a divide and combine algorithm based on a recursive bridge finding between vertically separated convex hulls. Later Kirkpatrick and Seidel [KS86] settled the asymptotic computational complexity of the problem by giving a recursive bridge finding algorithm that runs in O(n log h) time. This algorithm introduces the marriage before conquer paradigm. They also show a matching lower bound. Finally Chan [Cha96] gave a much simpler algorithm ....

.... result is strengthened by van Emde Boas [vEB80] and Preparata and Hong [PH77] They consider the task of identifying the vertices of the convex hull (without making their order explicit) Even this simpler problem requires time# n log n) This lower bound is tightened by Kirkpatrick and Seidel [KS86] to # n log h) matching their algorithm. Here the parameter h denotes the number of vertices on the convex hull. This immediately leads to that n insert and query operations require a total of # n log n) time. The above lower bounds for the static problem imply some lower bound on the dynamic ....

D. G. Kirkpatrick and R. Seidel, The ultimate planar convex hull algorithm ?, SIAM J. Comput. 15 (1986), no. 1, 287--299.

....filtering: A technique based on the work of Tamassia and Vitter [35] that allows I O optimal on line queries in fractional cascaded data structures based on balanced binary trees. cx tcral marriagc before coqucst: an externalmemory analog to the well known technique of Kirkpatrick and Seidel [22] for performing output sensitive hull constructions. We apply these techniques to derive optimal external memory algorithms for the following fundamental problems in computational geometry: computing the pairwise intersection of N orthogonal segments, answering K range queries on N points, ....

.... two dimensional case we show how to beat this lower bound for the case when the output size T is much smaller than N (in the extreme case, T = O(1) We develop an output sensitive algorithm based upon an external memory version of the marriage before conquest paradigm of Kirkpatrick and Seidel [22]. Our 3 d convex hull is somewhat esoteric, so we also describe a simplified version that, although not optimal asymptotically, is simpler to implement and is faster for the vast majority of practical cases. 6.1 A worst case optimal two dimensional For the two dimensional case, a number of main ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick and R. Seidel, "The Ultimate Pla- nar Convex Hull Algorithm?," SIAM J. Cornput. 15 (1986), 287 299.

....of Driscoll el al. batch filtering: a general method for performing K simultaneous external memory searches in data structures that can be modeled as a planar layered dags; external marriage before conquest: an externalmemory analog to the well known technique of Kirkpatrick and Seidel [22] for performing output sensitive hull constructions. We apply these techniques to derive optimal external memory algorithms for the following fundamental problems in computational geometry: computing the pairwise intersection of N orthogonal segments, answering Ix2 range queries on N points, ....

.... two dimensional case we show how to beat this lower bound for the case when the output size T is much smaller than N (in the extreme case, T O(1) We develop an output sensitive algorithm based upon an external memory version of the marriage before conquest paradigm of Kirkpatrick and Seidel [22]. Our 3 d convex hull is somewhat esoteric, so we also describe a simplified version which, although not optimal asymptotically, is simpler to implement, and will be faster for the vast majority of practical cases. 5.1 A worst case optimal two dimensional For the two dimensional case, a number of ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick & R. Seidel, "The ultimate planar convex hull algorithm?," SlAM J. Coalput. 15 (1986), 287 299.

....to all of the data points to the left of L 2 . LB 1 is the lower edge, which intersects L 1 , of the convex hull of the points compared with L 1 . We can find all of the LB s and UB s in O(nlogb) time, using a modified version of the linear time bridge finding algorithm of Kirkpatrick and Siedel [11]. Since the LH s are a subset of the LB s (and the UH s are a subset of the UB s) we can then compute the UH i s and LH i s in O(b 2 ) additional time (in fact O(blogb) time is sufficient) Once we have computed the UH i s and LH i s, we can then check if each of the query points lies in the ....

Kirkpatrick, D., Seidel, R. The ultimate planar convex hull algorithm? SIAM Journal of Computing, 15:287-299, 1986.

....when n p 3 # (only O(n p) data is sent and received by each processor) The algorithm is quite complicated, however, and it is unclear what the constants in the work are. Edelsbrunner and Shi [12] present a 3D convex hull algorithm based on the 2D algorithm of Kirkpatrick and Seidel [18]. The algorithm divides the problem by first using linear programming to find a facet of the 3D convex hull above a splitting point, then using projection onto vertical planes and 2D convex hulls to find two paths of convex hull edges. These paths are then used to divide the problem into four ....

....triplets. 2D Convex Hull. The 2D convex hull is central to our algorithm, and is the most expensive component. We considered three candidates for the convex hull algorithm: 1) Overmars and van Leeuwens [23] which is O(n) work for sorted points. 2) Kirkpatrick and Seidel s O(n log h) algorithm [18], and its much simplified form as presented by Chan et al. 21] 3) A simple worst case O(n 2 ) quickhull algorithm, as in [25] and [26] All of these algorithms can be naturally parallelized. Using the algorithm of Overmars and van Leeuwen [23] the convex hull of presorted points can be ....

[Article contains additional citation context not shown here]

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287--299, February 1986.

No context found.

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287--299, 1986.

No context found.

Kirkpatrick, D. G. and Seidel, R., "The ultimate planar convex hull algorithm?" SIAM Journal on Computing, vol. 15, No. 1, February 1986, pp. 287-299.

No context found.

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287-299, 1986.

No context found.

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287-299, 1986.

No context found.

P. G. Kirkpatrick and R. Seidel. The Ultimate Planar Convex Hull Algorithm? SIAM Journal on Computing 15(1): 287-- 299, 1986.

No context found.

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287--299, 1986.

No context found.

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM J. Comput., 15:287-299, 1986.

No context found.

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287--299, 1986.

No context found.

D. G. Kirkpatrick, R. Seidel, The ultimate planar convex hull algorithm?, SIAM Journal on Computing 15 (1) (1986) 287--299.

No context found.

D. G. Kirkpatrick and R. Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287--299, 1986.

No context found.

D. G. Kirkpatrick, R. Seidel, The ultimate planar convex hull algorithm?, SIAM J. Comput. 15 (1) (1986) 287--299.

No context found.

D. G. Kirkpatrick, R. Seidel, The Ultimate Planar Convex Hull Algorithm?, SIAM J. Computing vol. 15, No. 1, (1986) pp. 287{ 299.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC