W.F. Eddy, \A new convex hull algorithm for planar sets," ACM Trans. Math. Software, Vol. 3, No. 4, 1977, pp. 398403 and 411-412.  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 ....

W. Eddy. A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software, 3(4):398-403, 1977.

....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, ....

....1 3 and authors 4 6) discovered they were both working on in place computational geometry algorithms and decided to merge their results. Some of these results have been omitted due to space constraints. These include in place or in situ implementations of Eddy s algorithm (also known as quickhull) [11], Kirkpatrick and Seidel s algorithm [21] Seidel s randomized linear programming algorithm [31] and Megiddo s deterministic linear programming algorithm [24] The ideas presented in this paper also apply to other problems. The maximal elements problem is that of determining all elements S[i] ....

W. Eddy. A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software, 3(4):398-403, 1977.

....the points furthest in a direction perpendicular to the line through y max and x max . In this way we always search in new directions and we guarantee that points are global rather than local extrema. This latter algorithm has been independently discovered by several researchers (see for example [18], 19] As with algorithm CH2 it is more dicult than CHl to prove correct but it is extremely fast. In fact, it was proven in [19] and [20] that in many situations it executes in time proportional to n, the number of input points. 3 The Diameter Problem The diameter of a set of points P = fp 1 ....

W.F. Eddy, \A new convex hull algorithm for planar sets," ACM Trans. Math. Software, Vol. 3, No. 4, 1977, pp. 398403 and 411-412.

....the points furthest in a direction perpendicular to the line through y max and x max . In this way we always search in new directions and we guarantee that points are global rather than local extrema. This latter algorithm has been independently discovered by several researchers (see for example [18], 19] As with algorithm CH2 it is more dicult than CHl to prove correct but it is extremely fast. In fact, it was proven in [19] and [20] that in many situations it executes in time proportional to n, the number of input points. 3 The Diameter Problem The diameter of a set of points P = fp 1 ....

W.F. Eddy, "A new convex hull algorithm for planar sets," ACM Trans. Math. Software, Vol. 3, No. 4, 1977, pp. 398403 and 411-412.

....Remember that, depending on the goodness of the pivot chosen, the new subproblems may have rather di erent weights. Figure 13 presents the speed ups on a digital Alpha Server, an Origin 2000, an IBM SP2, a CRAY T3E and a CRAY T3D. The size of the problem was 1M integers. The QuickHull algorithm [5] constitutes our fourth example. The algorithm shares a few similarities with its namesake, QuickSort: QuickHull is also recursive and each recursive step partitions data into several groups. Figure 14 presents the speedups for di erent platforms: A digital Alpha Server, a Hitachi, an Origin ....

Eddy, W.: A New Convex Hull Algorithm for Planar Sets, ACM Transactions on Mathematical Software 3(4), (1977) 398-403.

....year, R. Jarvis gave an algorithm whose running time depends on the output size [14] Jarvis s algorithm runs in O(nh) time where h is the number of points in the convex hull. The next ten years saw many other algorithms for finding convex hulls in the plane most of which run in O(n log n) time [1, 4, 11, 13, 16]. Some very simple algorithms were proposed which have O(n) expected running time for many distributions of points in the plane (such as points with normal density) 10, 3] During this period, Avis [2] and Yao [20] proved lower bounds of Omega Gamma n log n) on the time to find a convex hull, ....

Eddy, W. F. A new convex hull algorithm for planar sets. ACM Trans. Math. Softw. 3 (1977), 398--403 and 411--412.

....facet s outside set only if it is above the facet. Like Clarkson and Shor s algorithm, an unprocessed point is in exactly one outside set. Our variation is to process the furthest point of an outside set instead of a random point. In R 2 , this is the well known Quickhull Algorithm [Bykat 1978] [Eddy 1977] [Floyd 1976] Green and Silverman 1979] Other variations of the Clarkson and Shor algorithm do not maintain conflict graphs or outside sets. Instead, they retain old facets of the convex hull with links to the new facets that replaced them. This hierarchy begins with an initial simplex formed ....

Eddy, W. 1977. A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software 3, 4, 398--403.

....of interest in these matters. One of the first efforts in the OR literature was a sophisticated application of experimental design principles by Lin and Rardin (1980) Golden and Stewart (1985) and more recently Amini and Racer (1992) used rigorous statistical methods to analyze test results. Eddy (1977) and Hart (1983) used less elaborate analyses. Barton (1987) discussed experimental design for comparing optimization procedures, and McGeoch (1992) variance reduction techniques. Crowder, Dembo and Mulvey (1978) as well as Hoaglin and Andrews (1975) addressed the issue of computational reporting. ....

Eddy, W. F. 1977. A new convex hull algorithm for planar sets, ACM Transactions on Mathematical Software 3 398-402.

....S. The computational problem when d = 2 is to find the subset of S comprising the vertices of the hull, as well as their radial order as seen from an interior point. We use the unit cost RAM model to measure algorithmic complexity. One of the earliest, and most attractive methods is due to Eddy [2], a divide andconquer algorithm called QUICKHULL, in analogy to Quicksort (see also O Rourke [8] As described in [8] it finds A and B, the two points with smallest and largest x coordinates, respectively. To find the upper chain, hull vertices above the segment AB, a procedure UPPER is called. ....

....U : P above AC and R = P = x, y) # U : P above CB 4. CALL UPPER(A,C,L) and UPPER(C,B,R) Figure 1. GENERIC QUICKHULL # Supported in part by NSF DIMATIA Kontakt 055 96 and by DIMACS. Supported in part by NSF DIMATIA Kontakt 337 99 1 The lower chain is obtained in a similar way. In [2], Eddy chose C in Step 2 as the support point for the upper tangent line to U with the same slope as line AB (e.g. the point of max distance above AB) obtained in linear time. In Step 3, the points in L are in the vertical strip between A and C, and on or above the segment AC; those in R are ....

W. Eddy. A new convex hull algorithm for planar sets, ACM Trans. Math. Software 3 (1977) 398-403 & 411-412.

....also be seen as a generalized and automated version of the taxometric map (Carmichael and Sneath 1969) which showed the clusters but not the objects. Clusplots might be extended in several ways, e.g. by replacing each ellipse by the convex hull of all points in the cluster using the algorithm of Eddy (1977), or by the bagplot (Rousseeuw and Ruts 1997) of the cluster. Another possibility is to draw ellipses based on the Minimum Covariance Determinant estimator, which is easily obtained with the function cov.mcd (Rousseeuw and Van Driessen 1997) built into S PLUS 4.0. A natural question is whether ....

Eddy, W.F. (1977), "A New Convex Hull Algorithm for Planar Sets," ACM Trans. Math.

....is uniform on the unit circle, then on the average O(n) points will be left after steps 1 and 2, and much depends upon the algorithm used in step 3. We do not wish to specify an algorithm in step 3 because steps 1 and 2 should be considered as preprocessing steps in all generality. Remark 1: Eddy [10] has proposed an algorithm that uses an idea similar to that of steps 1 and 2 but it repeats these steps by finding extrema in different directions instead of proceeding to step 3. Furthermore, after having initially found the X i s in the x direction (X min and X max ) they search for ....

W.F. Eddy. A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software, 3:398--403, 411--412, 1977.

....is in a facet s outside set only if it is above the facet. Like Clarkson and Shor s algorithm, an unprocessed point is in exactly one outside set. Our variation is to process the furthest point of an outside set instead of a random point. In R 2 , this is the well known Quickhull Algorithm [10] [20] [22] 26] Other variations of the Clarkson and Shor algorithm do not maintain conflict graphs or outside sets. Instead, they retain old facets of the convex hull with links to the new facets that replaced them. This hierarchy begins with an initial simplex formed from d 1 of the input points. ....

W. Eddy. A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software, 1977.

....given for d = 3 with the same complexity. 4] Like the earlier one, this one also can be derandomized for d 3 to obtain a deterministic algorithm with the same asymptotic complexity. 3] This result is perhaps best interpreted as a rough theoretical justification for algorithms like Quickhull[5, 10], which find extreme points on their way to computing the hull. As further evidence of the interest of this approach, x5 gives an output sensitive algorithm for the problem of thinning a point set for nearest neighbor (NN) classification. Here we are given a set S of sites (points) in R d , and ....

W. F. Eddy. A new convex hull algorithm for planar sets. ACM Trans. Math. Softw., 3:398-- 403 and 411--412, 1977.

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W.F. Eddy, \A new convex hull algorithm for planar sets," ACM Trans. Math. Software, Vol. 3, No. 4, 1977, pp. 398403 and 411-412.

No context found.

W.F. Eddy, "A new convex hull algorithm for planar sets," ACM Trans. Math. Software, Vol. 3, No. 4, 1977, pp. 398403 and 411-412.

No context found.

W. Eddy. A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software, 3(4):398-403, 1977.

No context found.

W. Eddy. A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software, 3(4):398-403, 1977.

No context found.

W.F. Eddy. A new convex hull algorithm for planar sets. ACM Transactions on Mathematical Software, 3:398--403, 411--412, 1977.

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W. Eddy, A new convex hull algorithm for planar sets, ACM Transactions on Mathematical Software 3 (4) (1977) 398--403. 17

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Eddy W.: A New Convex Hull Algorithm for Planar Sets, ACM Trans. Math. Software 3 (4), 1977.

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