F.P. Preparata and S.J. Hong, "Convex hulls of finite sets of points in two and three dimensions", Comm.ACM 2, pp. 87--93.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....convex hull, and intersect it with v = 0. Since the Minkowski sum of two convex polygons can be computed in linear time [7] we spend O(mn ) time in computing the polygons G i;j . Their convex hull can be computed in O(mn log n) time, using the divide and conquer algorithm of [10] (which has now only O(log n) recursive levels, because we start with the already available polygons G i;j ) Hence, the total running time is O(mn log n) This completes the proof of part (b) Note that in practical terms, the implementation of this algorithm is a straightforward setup ....

F. Preparata and S. Hong, Convex hulls of finite sets of points in two and three dimensions, Commun. ACM 20 (1977), 87--93. 10

....pose hypotheses remains linear in the number of corners. d) With an algorithm described in [2] bitangents on pairs of image regions can be computed in an expected time of O(log (n) where n is the number of vertices on the convex hull of the regions. This improves on a previous algorithm [4] that has a complexity of O(n) A focus feature consists of two bitangents with four invariant points (again a least squares solution is used for pose computation) Figure 5 illustrates feature types a,b and d. A corner feature is used in the lower part of figure 3. P1 P2 P3 dist = max ....

F.P. Preparata and S.J. Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 20(2):87--93, February 1977.

....the trees T i and Voronoi diagrams of the sets V i. Now (i) and (ii) follow from Theorem 3, and (iii) follows from Theorem 3 and from Kirkpatrick [6] The next example is the set problem compute the convex hull of a set of n points in three dimensional space . As was shown by Preparata and Hong [8], this problem is 0(n) order decomposable. A third example is the set problem find the intersection of a set of n halfspaces in three dimensions . Using a result of Brown [3] it can be shown that this problem is 0(n) order decomposable. Finally, according to Edelsbrunner, Overmars and Wood [4] ....

F.P.preparata and S.J. Hong, Convex Hulls of Finite Sets of Points in Two and Three Dimensions, Corem.of the ACM 20 (1977), pp.87-93.

....briefly addressed in an appendix. Examples of 3 dimensional channel constructions are shown in figures 28 and 29. Since all of the points in the hull set are initially embedded in faces and edges, it is possible to construct all of these convex hulls in O(n) time for n edges (Preparata and Hong, [20]) Of course in two dimensions, the construction time is constant because there are only 2 vertices per edge. 4.5. A Constant Time Constructor for the 2 Dimensional Case The channel constructor can be expressed as a very simple algorithm in two dimensions. We can construct the region between A ....

Prcparata, F. and Hong, S. "Convex Hulls of Finite Sets of Points in Two and Three Dimensions," Communications of the ACM 23, 3 (1977).

....search. Then there is Graham s scan [Gra72] using time O(n log n) Andrew s vertical sweep line variant of Graham s scan [And79] This algorithm first sorts the points in lexicographical 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 ....

....example by Shamos in his PhD thesis [Sha78] or in the textbook by Preparata and Shamos [PS85] This establishes together with Ben Or s [BO83] result a # n log n) lower bound on the real RAM for computing the convex hull. This 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 ....

F. P. Preparata and S. J. Hong, Convex hulls of finite sets of points in two and three dimensions, Comm. ACM 20 (1977), no. 2, 87--93.

....constant c 0, this implies that the total number of I Os is O(ulog which is optimal for any value of T. 5. 3 Three dimensional convex hulls Even in main memory, sweep plane algorithms fail to solve the 3 d convex hull problem, and we must resort to more advanced divide and conquer approaches [29]. One idea is to use a plane to partition the points into equally sized sets, recursively construct the convex hull for each set, and then stitch the recursive solutions together in linear time. Unfortunately, we know no way of implementing an algorithm of this type in secondary memory; the ....

F. P. Preparata & S. J. Hong, "Convex hulls of finite sets of points in two and three dimensions," Comm. ACM 20 (1977), 87 93.

....using the above mentioned next v algorithm or it can be found in form of a table in [27] There are plenty of algorithms for point set problems which are based on computing the orientation of a sequence of points. Prime examples are algorithms for the construction of convex hulls; see for example [66, 72, 73] or [21,54, 67] References to other applications can be found in [27] which also extends the concept of orientation to points with homogeneous coordinates in arbitrary dimensions. Sphere test. The crucial primitive operation for constructing Delaunay triangulations is to check if a triangle is ....

F P Preparata and S J Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 20(2):87 93, 1977.

....Clearly, M can be obtained as the upper envelope of M 1 and M 2 , and M Gamma can be obtained as the lower envelope of M Gamma 1 and M Gamma 2 . It is well known that one can compute the union and the intersection of two convex polygons of at most n sides in time O(n) [PH], S] Thus, if we know convP i ( convP i ( Gamma ) M i ; and M Gamma i for i = 1; 2, then in linear time we can determine convP i (i = 1; 2) M and M Gamma . If any of the conditions of the Claim is not satisfied, we conclude that P does not admit a perfect ....

F.P. Preparata and S.J. Hong, Convex hulls of finite sets of points in two and three dimensions, Communications ACM 20 (1977), 87--93.

....original set of sites. This correspondence is true for arbitrary dimension d. Guibas and Stolfi [17] note that their divide and conquer algorithm for computing the Deluanay triangulation can be viewed as a variant of Preparata and Hong s algorithm for computing three dimensional convex hulls [21]. Others have also used this approach. Recently Barber [2] has developed a Delaunay Triangulation algorithm based on a convex hull algorithm that he calls Quickhull. He combines the 2 dimensional divide and conquer Quickhull algorithm with the general dimension Beneath beyond algorithm to obtain ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....Overmars. Key words. computational geometry, convex polytopes, lower bounds, decision trees, adversary arguments AMS subject classifications. 68Q25, 68U05, 52B55, 52B05 1. Introduction. The construction of convex hulls is one of the most basic and well studied problems in computational geometry [2,3,5,10,11,12,13,15,17,18, 29,34,35,38,39,47,41,45,43,44,47,48]. Over twentyyears ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [29] The same running time was first achieved in three dimensions by Preparata and Hong [38] Yao [48] proved a lower bound of Omega n log n) on the complexity of ....

.... geometry [2,3,5,10,11,12,13,15,17,18, 29,34,35,38,39,47,41,45,43,44,47,48] Over twentyyears ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [29] The same running time was first achieved in three dimensions by Preparata and Hong [38]. Yao [48] proved a lower bound of Omega n log n) on the complexity of identifying the convex hull vertices, in the quadratic decision tree model. This lower bound was later generalized to the algebraic decision tree and algebraic computation tree models by Ben Or [7] It follows that both ....

F. P. Preparata and S. J. Hong, Convex hulls of finite sets of points in two and three dimensions, Commun. ACM, 20 (1977), pp. 87--93.

....linear array, mesh connected computer or hypercube. 5.2 Convex Hull This section considers the problem of constructing the convex hull from a finite set S of points in the two dimensional real space IR Theta IR. The algorithm given here is mainly an adaptation of a sequential one presented in [PH77] with major changes to fit the massively parallel paradigm. Preliminaries and Operational Specifications Given a set S = fs 1 ; s 2 ; s 2n g of points in the plane, the convex hull of S is the smallest convex polygon P , for which each point in S is either on the boundary of P or in its ....

.... (a; b) x = a max x p q = q ; if p:x q :x p; otherwise min x p q = p; if p:x q :x q ; otherwise (p = q) p:x = q :x ) p:y = q :y) a; b) y = b max y p q = q ; if p:y q :y p; otherwise min y p q = p; if p:y q :y q ; otherwise The DC method of constructing UH (S ) given in [PH77] is as follows: Let S be a sequence of 2n points in the plane such that s 1 :x s 2 :x : s 2n :x where n is a power of 2. If n 1, then S itself is an upper hull of S (primitive case) Otherwise, we subdivide S into two subsequences S 1 = s 1 ; s 2 ; s n ] and S 2 = s n 1 ; ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of The ACM, 20:88--93, 1977.

....a fundamental approach to solid modelling applications. Many other geometric problems can be solved by D C methods, especially in discrete computational geometry. Examples include the construction of Voronoi diagrams and Delaunay triangulation (Shamos 1977) and the determination of convex hulls (Preparata Hong 1977). It thus appears that the generic nature of D C, and of geometric decomposition trees in particular, makes GEL applicable far beyond its initial aims. On the other hand, the limitation to D C means that not all possible algorithms on the types provided are currently included in GEL; for ....

Preparata, F. P. & Hong, S. J. (1977), `Convex Hulls of a Finite Set of Points in Two and Three Dimensions', Communications of the ACM 20(2), 87--93.

....the region are closer to or farther away from the corresponding point than from any other point in the set. Similarly, the largest inscribed circle can be searched on the closest point Voronoi diagram. The algorithm for generating two dimensional convex hulls from a set of points can be found in [32]. Hopp and Reeve [20] have presented an algorithm that solves the smallest circumscribed sphere in arbitrary dimensions. As with two sided minimax fitting, one sided minimax fitting is very sensitive to data outliers. 12 3.2 Algorithms For Calculating Minimum Tolerance Zones ANSI Y14.5M 1982 ....

Preparata, F.P. and Hong, S.J., "Convex Hulls of Finite Sets of Points in Two and Three Dimensions," Comm. of the ACM, Vol. 20, No. 2, Feb. 1977, 87-93.

....plane, Shamos [Sha78] established an Omega Gamma n log n) lower bound for the time for computing a planar triangulation and provided an algorithm reaching this bound, thus optimal in the worst case sense. Other optimal algorithms have been proposed by Lee and Schacter [LS80] Preparata and Hong [PH77] Guibas and Stolfi [GS85] and Fortune [For87] In higher dimensional spaces, Klee [Kle80] has shown that the Delaunay triangulation may have Omega n d d 2 e d d 2 e simplices. See also [Pas82, Sei82] It is well known that the problem of computing the Delaunay triangulation of ....

F.P. Preparata and S.J. Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 2(20):87--93, 1977.

.... that d dimensional convex hulls can have Omega (n bd=2c ) facets, the previously best lower bound for either of the problems we consider is only Omega n log n) 1 Introduction The construction of convex hulls is one of the most basic and well studied problems in computational geometry [4,5, 6, 7, 9, 10, 15, 18,20,21,22, 29, 24, 26, 27]. Over twentyyears ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [15] The same running time was first achieved in three dimensions by Preparata and Hong [21] Yao [30]proved a lower bound of Omega (n log n)onthe complexity of ....

.... in computational geometry [4,5, 6, 7, 9, 10, 15, 18,20,21,22, 29, 24, 26, 27] Over twentyyears ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [15] The same running time was first achieved in three dimensions by Preparata and Hong [21]. Yao [30]proved a lower bound of Omega (n log n)onthe complexity of identifying the convex hull vertices, in the quadratic decision tree model. This lower bound was This researchwas partially supported by a GAANN Fellowship. later generalized to the algebraic decision tree and algebraic ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....There are a few discrete attributes that are the object of this thesis. The convex hull diagram and the closest pair have been introduced in the previous chapter. Both can be computed in O(n log n) time for a set of n items positioned in the plane using a variety of different techniques [57, 82, 83, 84]. Data structures exist to perform efficient updates of these attributes upon item insertion and or deletion [25, 29, 55, 64, 71, 81, 84] Another classical structure is the Voronoi diagram, which encodes the nearest item to each point in the plane. The Voronoi diagram is a planar map that can ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....to compute P Q. 5. Merging Convex Hulls A typical divide and conquer approach to finding the convex hull of a set of n points on the plane consists of sorting the points along the x axis and subsequently merging bigger and bigger convex polygons until one final convex polygon is obtained [9]. Performing the merge in linear time will guarantee an O(n log n) upper bound on the complexity of the entire process. Merging two convex polygons P, Q consists of essentially finding two pairs of vertices p i , p j and q k , q l such that the new edges p i q k and q l p j , together with the two ....

....q k , q k 1 , q l and p j , p j 1 , p i form the convex hull of P Q. An edge such as p i q k is called a bridge and the vertices making up a bridge (such as p i and q k ) are referred to as bridge points. While O(n) algorithms exist for finding the bridges of two disjoint convex polygons [9], we show here that the bridges can also be computed very simply with the rotating calipers. p i p i 2 q i q j f j Q P q j 1 q j 1 p i 1 Fig. 5 6 The following theorem leads to the desired algorithm. Theorem 5.1: Two vertices p i e P and q j e Q are bridge points if, and only if, ....

F.P. Preparata and S. Hong, "Convex hulls of finite sets of points in two and three dimensions ", Comm. ACM, Vol. 20, 1977, pp. 87-93.

....later. Given a subset R ae H , the ( 0) level in the arrangement of R (also called the lower envelope of R) is a convex polyhedron. Computing this polyhedron is equivalent to constructing the convex hull [37, 53, 55] by duality. For d = 3, several O(jRj log jRj) time algorithms are known [32, 54], among the simplest of which are based on the randomized incremental paradigm. Let CT 0 (R) denote the collection of (closed) full dimensional simplices in the canonical triangulation of the ( 0) level, as defined for instance by Clarkson [31] also called the bottom vertex triangulation) The ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....bucket. This implies that the total number of I Os is O( log ) which is optimal for any value of T . 6. 3 Three dimensional convex hulls Even in main memory, space sweeping algorithms fail to solve the 3 d convex hull problem, and we must resort to more advanced divide and conquer approaches [29]. One idea is to use a plane to partition the points into equally sized sets, recursively construct the convex hull for each set, and then merge the recursive solutions together in linear time. Unfortunately, we know no way of implementing an algorithm of this type in secondary memory; the problem ....

F. P. Preparata and S. J. Hong, "Convex Hulls of Finite Sets of Points in Two and Three Dimensions," Comm. ACM 20 (1977), 87--93.

....ordering of the vertices of the hull. In three dimensions it can be the cyclic ordering of the hull edges around each hull vertex. The problem of constructing convex hulls has attracted a great deal of attention from the inception of computational geometry. In three dimensions, Preparata and Hong [PH77], described the first O(n log n) time algorithm. Clarkson and Shor [CS89] presented the first (randomized) optimal output sensitive algorithm which ran in O(n log h) expected time, where h is the number of hull vertices. Their algorithm was subsequently derandomized optimally by Chazelle and ....

F P Preparata and S J Hong. Convex hulls of finite sets of points in two and three dimensions. Comm. ACM, 20:87--93, 1977.

....is defined as the smallest convex set containing P . The convex hull problem has received considerable attention in computational geometry [11, 21, 23, 25] In E 2 , an algorithm known as Graham s scan [15] achieves O(n log n) running time, and in E 3 , an algorithm by Preparata and Hong [24] has the same complexity. These algorithms are optimal in the worst case, but if h, the number of hull vertices, is small, then it is possible to obtain better time bounds. For example, in E 2 , a simple algorithm called Jarvis s march [19] can construct the convex hull in O(nh) time. This bound ....

....time. We can use the same grouping idea from the previous section to improve the time complexity to optimal O(n log h) while maintaining linear space. The calls to Graham s scan (line 3 of Hull2D(P; m; H) are now replaced by calls to Preparata and Hong s three dimensional convex hull algorithm [24], which has the same complexity. To make line 8 work in E 3 , we need to calculate tangents or supporting planes of convex polyhedra through a given line (or, in dual space, intersect convex polyhedra with a ray) If we use the hierarchical representation of Dobkin and Kirkpatrick [9, 10] to ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....practical performance of these methods. This will be the main objective of this paper. To our knowledge, the only known practical result for computing 3D convex hull (hence 2D Voronoi Diagram) on a coarse grained machine is quite restrictive [Day91] It does parallelize Preparata Hong s algorithm [PH77] however, at the cost of storing a copy of the n input points on each of the processors. This is kind of forcing the coarse grained machine (which is share nothing ) to share everything ; moreover, when the number of processors P is increased it is unlikely that we can afford to have a copy ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....its orientation is the opposite of the orientation of (p i 0 ; p i 1 ; p i d ) Simulation of Simplicity 8 There are plenty of algorithms for point set problems which are based on computing the orientation of a sequence of points. Prime examples are the construction of convex hulls (see [PH77], PS85] Se81] Se86] or [Ed87] computing matrices as discussed in [GP83] and [Ed87] and finding convex subsets (see [CK80] EG89] and [Ed87] The remainder of this section considers the primitive operations required by the three dimensional convex hull algorithm of Preparata and Hong ....

.... or [Ed87] computing matrices as discussed in [GP83] and [Ed87] and finding convex subsets (see [CK80] EG89] and [Ed87] The remainder of this section considers the primitive operations required by the three dimensional convex hull algorithm of Preparata and Hong which is described in [PH77], PS85] and [Ed87] The first step of the algorithm sorts the points in x 1 direction. To perform this step, it needs to compare the x 1 coordinates of two points, which can be done by computing the orientation of their orthogonal projections onto the x 1 axis. Second, it constructs the ....

F. P. Preparata and S. J. Hong. Convex Hulls of Finite Sets of Points in Two and Three Dimensions. Communications of the ACM, 20(2):87--93, February 1977.

....mind that a cavity may have more than one cap if the cavity is a tunnel or network of tunnels. The dihedral angle examination can be used to determine if each void or cavity is convex nonconvex. An O(n log n) time convex hull algorithm for n points in E 3 has been described by Preparata and Hong [78] and implemented by Day [18] Day writes that he found the task . definitely not a trivial exercise . due to degeneracies and special cases. However, if an O(n 2 ) algorithm [27, section 8.4] is satisfactory, a much simpler implementation is possible. Even an O(n log n) implementation can ....

PREPARATA, F. P. and HONG, S. J. Convex Hulls of Finite Sets of Points in Two and Three Dimensions. Comm. ACM 20 (1977), 87--93.

....Kirkpatrick and Seidel obtained a deterministic algorithm for planar convex hulls with the same time bound [31] We also give a Las Vegas incremental algorithm requiring O(n log n) expected time for d = 3 and O(n bd=2c ) expected time for d 3. This improves known results for odd dimensions [36, 40, 41, 20]. For independently identically distributed points, the algorithm requires O(n) P 1rn f(r) r 2 expected time, where f(r) is the expected size of the convex hull of r such points. Here f(r) must be nondecreasing. The algorithm is not complicated. Spherical intersections and diametral ....

....complexity [29] Like polytopes, spherical intersections have duals, which were introduced as ff hulls in [21] Unfortunately, spherical intersections do not share with convex polytopes some properties helpful for algorithms. In particular, the divide and conquer technique of Preparata and Hong[36] does not seem to lead to a fast algorithm for computing spherical intersections. Our simple algorithm for spherical intersections, requiring O(n log n) expected time, is asymptotically faster than any previous algorithm. The spherical intersection problem arises in a classic problem of ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 20:87--93, 1977.

....approach due to the ease with which it can be implemented. Proposition 1 The bounding k dop of a set S of n triangles can be computed in time O(minfkn; n log n k log hg) where h = jch(S)j is the complexity of the convex hull of S. Proof. We compute ch(S) in time O(n log n) e.g. using [80]) We then compute the Dobkin Kirkpatrick hierarchy ( 30] for ch(S) in time O(h) after which extremal queries in each of the k directions can be answered in time O(log h) per query. ut In thinking of k dops as intervals along a set of k=2 directions, our (conservative) test for intersection ....

F.P. Preparata and S.J. Hong. Convex hulls of finite sets of points in two and three dimensions. Comm. ACM 20:87--93, 1977.

....algorithms (as well as variants on previous algorithms) were proposed in E 2 . We mention here two divide and conquer algorithms [PS85] MergeHull and QuickHull modeled after the sorting algorithms MergeSort and QuickSort. The former divide andconquer algorithm, due to Preparata and Hong [PH77], runs in O(n log n) time. The latter algorithm, discovered independently by several researchers around the late 1970s, runs in O(nh) time in the worst case, but is usually faster in practice (as its name suggests) At the time, no asymptotic improvement to the original bounds by Graham and ....

....algorithm and providing a matching lower bound. It would appear that Kirkpatrick and Seidel s optimal algorithm is thus the ultimate convex hull algorithm for d = 2 or is it The convex hull problem in its three dimensional setting (d = 3) has also been studied intensively. Preparata and Hong [PH77] presented the first O(n log n) time algorithm in E 3 , based on divide and conquer. The first output sensitive algorithm is the giftwrapping method of Chand and Kapur [CK70] which works in arbitrary dimensions and is a generalization of Jarvis s march in two dimensions (although historically, ....

[Article contains additional citation context not shown here]

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the Association for Computing Machinery, 20:87--93, 1977.

....Let (S 1 ; S 2 ) be a partition of S such that all the points in S 1 are to the left of those in S 2 . Given the convex hulls of S 1 and S 2 , the convex hull of S can be computed in O(log n) time in two dimensions [124] the bridging of the previous section) and in O(n) time in three dimensions [132]. This property allows one to apply the dynamization technique for order decomposable problems. Hence, planar convex hulls can be dynamically maintained using O(n) space, O(log 2 n) update time, O(log n) time for a find query, and O(k) for a report query [121] For three dimensional convex ....

F.P. Preparata and S.J. Hong, "Convex Hulls of Finite Sets of Points in Two and Three Dimensions," Comm. ACM 2(20)(1977), 87--93.

....convex hull, and intersect it with v = 0. Since the Minkowski sum of two convex polygons can be computed in linear time [10] we spend O(mn 2 ) time in computing the polygons G i;j . Their convex hull can be computed in O(mn 2 log n) time, using the divide and conquer algorithm of [17] (which has now only O(logn) recursive levels, because we start with the already available polygons G i;j ) Hence, the total running time is O(mn 2 log n) 2 Note that in practical terms, the implementation of this algorithm is a straightforward setup followed by a convex hull computation, ....

F. Preparata and S. Hong, Convex hulls of finite sets of points in two and three dimensions, Commun. ACM 20 (1977), 87--93.

....we consider is only Omega Gamma n log n) This research was partially supported by a GAANN Fellowship. New Lower Bounds for Convex Hull Problems in Odd Dimensions 1 1 Introduction The construction of convex hulls is one of the most basic and well studied problems in computational geometry [4, 5, 6, 7, 9, 10, 14, 17, 19, 20, 21, 28, 23, 25, 26]. Over twenty years ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time. The same running time was achieved in three dimensions by Preparata and Hong [20] Yao [29] proved a lower bound of Omega Gamma n log n) on the complexity of ....

....problems in computational geometry [4, 5, 6, 7, 9, 10, 14, 17, 19, 20, 21, 28, 23, 25, 26] Over twenty years ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time. The same running time was achieved in three dimensions by Preparata and Hong [20]. Yao [29] proved a lower bound of Omega Gamma n log n) on the complexity of identifying the convex hull vertices, in the quadratic decision tree model. This lower bound was later generalized to the algebraic decision tree and algebraic computation tree models by Ben Or [3] It follows that both ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....(An animation of traveling salesman heuristics, alluded to in the video, was omitted for reasons of length. Convex hulls: The first animation shows Graham s incremental algorithm [17] while the second animation shows Preparata and Hong s divide and conquer method [27]. Proximity problems: We present a plane sweep for computing the closest pair [19] and a plane sweep for computing for each given point a nearest neighbor to theleft [20] Experimenting with the XYZ GeoBench: The final demonstration shows how the GeoBench can be used interactively to approximate ....

F. Preparata and S. Hong. Convex hulls of finite sets of points in two and three dimensions. Comm. ACM, 20(2):87--93, 1977.

....convex hull, and intersect it with v = 0. Since the Minkowski sum of two convex polygons can be computed in linear time [7] we spend O(mn 2 ) time in computing the polygons G i;j . Their convex hull can be computed in O(mn 2 log n) time, using the divide and conquer algorithm of [10] (which has now only O(log n) recursive levels, because we start with the already available polygons G i;j ) Hence, the total running time is O(mn 2 log n) This completes the proof of part (b) Note that in practical terms, the implementation of this algorithm is a straightforward setup ....

F. Preparata and S. Hong, Convex hulls of finite sets of points in two and three dimensions, Commun. ACM 20 (1977), 87--93.

....representation is shown in figure 4. j H 1 1 2 3 3 6 4 8 5 10 i VERTEX edge NEXT 1 1 a 2 2 1 b 1 3 2 b 4 4 2 f 5 5 2 c 3 6 3 c 7 7 3 d 6 8 4 d 9 9 4 e 8 10 5 a 11 11 5 e 12 12 5 f 10 This representation of a planar graph is precisely that one obtained by the algorithm of Preparata and Hong ( PH77] which constructs the convex hull of a set of points in three dimensions. Indeed, the surface of a convex polyhedron is topologically a planar graph. It is clear that filling the vertex to edge representation can be done in O(n) time by inserting each edge with its vertices (indices of ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 20:87--93, 1977.

....y x L n 2 2 L n 2 1 L n 2 R n 2 R n 2 1 R n 2 2 R 1 R 2 R 3 Fig. 3 E E pairs include (L i L i 1 , R j R j 1 ) For i, j 1, n 2 1 Algorithm 3. 1 Width in Three Dimensions: Begin Step 1: Construct the convex hull in O(n log n) time using Preparata and Hong s algorithm [10]. Step 2: Transform the polyhedron into two planar subdivisions in O(n) time as described in [3] Step 3: Compute the overlay of the two planar subdivisions using Guibas and Seidel s algorithm [8] in O(n I) time. Step 4: Examine all vertices and all parallel rays of unbounded regions of the ....

F. P. Preparata and S. J. Hong, "Convex hulls of finite sets of points in two and three dimensions, " Communications of the ACM, vol. 20, pp. 87-93, 1977.

.... i i i i i i h h h h h h q q q q q A program for finding the convex hull of a set of points can be written using a divide andconquer algorithm, using the fact that it is relatively easy to combine two non overlapping convex polygons into a single convex polygon which encloses the original two[15]. This involves much less work than combining two overlapping convex polygons. We can take advantage of this if we divide the original set of points into non overlapping subsets. An easy way to do this is to order the points in order of their X coordinates (and if two points have equal ....

F.P. Preparata and S.J. Hong, Convex Hulls of Finite Sets of Points in Two and Three Dimensions, Comm. ACM, 20(2) (1977) 87--93.

.... d dimensional convex hulls can have Omega Gamma n bd=2c ) facets, the previously best lower bound for either of the problems we consider is only Omega Gamma n log n) 1 Introduction The construction of convex hulls is one of the most basic and well studied problems in computational geometry [4, 5, 6, 7, 9, 10, 15, 18, 20, 21, 22, 29, 24, 26, 27]. Over twenty years ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [15] The same running time was first achieved in three dimensions by Preparata and Hong [21] Yao [30] proved a lower bound of Omega Gamma n log n) on the complexity ....

.... computational geometry [4, 5, 6, 7, 9, 10, 15, 18, 20, 21, 22, 29, 24, 26, 27] Over twenty years ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [15] The same running time was first achieved in three dimensions by Preparata and Hong [21]. Yao [30] proved a lower bound of Omega Gamma n log n) on the complexity of identifying the convex hull vertices, in the quadratic decision tree model. This lower bound was This research was partially supported by a GAANN Fellowship. later generalized to the algebraic decision tree and ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....time. Hence, this method requires O(n log n) preprocessing time, O(n) space, and O(log n) query time. The same preprocessing time, space, and query time can be obtained in three dimensions: in the preprocessing, compute the polytope T H by the dual of Preparata and Hong s convex hull algorithm [35] and construct its Dobkin Kirkpatrick hierarchical representation [15] then use the query algorithm from [15] Our first observation is that a preprocessing time query time tradeoff is possible using a standard grouping technique. Using this observation, we can perform q queries in O(n log q) ....

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

....Overmars. Key words. computational geometry, convex polytopes, lower bounds, decision trees, adversary arguments AMS subject classifications. 68Q25, 68U05, 52B55, 52B05 1. Introduction. The construction of convex hulls is one of the most basic and well studied problems in computational geometry [2, 3, 5, 10, 11, 12, 13, 15, 17, 18, 29, 34, 35, 38, 39, 47, 41, 45, 43, 44, 47, 48]. Over twenty years ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [29] The same running time was first achieved in three dimensions by Preparata and Hong [38] Yao [48] proved a lower bound of Omega Gamma n log n) on the ....

.... 5, 10, 11, 12, 13, 15, 17, 18, 29, 34, 35, 38, 39, 47, 41, 45, 43, 44, 47, 48] Over twenty years ago, Graham described an algorithm that constructs the convex hull of n points in the plane in O(n log n) time [29] The same running time was first achieved in three dimensions by Preparata and Hong [38]. Yao [48] proved a lower bound of Omega Gamma n log n) on the complexity of identifying the convex hull vertices, in the quadratic decision tree model. This lower bound was later generalized to the algebraic decision tree and algebraic computation tree models by Ben Or [7] It follows that both ....

F. P. Preparata and S. J. Hong, Convex hulls of finite sets of points in two and three dimensions, Commun. ACM, 20 (1977), pp. 87--93.

....convex hulls. The sequential complexity of computing the convex hull of a set S of n points in d , d 2, is known to be Omega Gamma n log n) see, e.g. PS85] Although there exist several optimal serial algorithms for this problem when d = 2 (refer, e.g. to [Gra72, And79] and d = 3 [PH77] optimal parallel algorithms are known only for d = 2 [AG86, ACG 88, AG88, MS88, Che90] In general, a parallel algorithm is said to be optimal if it runs in time O(log c n) for some constant c (often referred to as polylogarithmic, or polylog time) and the product of the time and the ....

....phase of the sequential three dimensional convex hull algorithm. parallel algorithms for this problem use at least O(log 2 n) time. 1 All traditional parallel algorithms for the three dimensional convex hull problem are based on the serial divide and conquer algorithm of Preparata and Hong [PH77] Recent exceptions are the randomized algorithm of Reif and Sen [RS92] and its derandomized version due to Goodrich [Goo93] Let CH(S) denote the convex hull of the point set S. The serial algorithm for computing the convex hull of a point set S is outlined in Algorithm 1.1. Algorithm 1.1 ....

[Article contains additional citation context not shown here]

F. P. Preparata and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

No context found.

F.P. Preparata and S.J. Hong, "Convex hulls of finite sets of points in two and three dimensions", Comm.ACM 2, pp. 87--93.

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F. Preparata and S. Hong, Convex hulls of finite sets of points in two and three dimensions, Commun. ACM 20 (1977), 87--93. 10

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F. P. Preparata, and S. J. Hong. Convex hulls of finite sets of points in two and three dimensions. Commun. ACM, 20:87--93, 1977.

No context found.

Preparata, F. and S. Hong, Convex Hulls of Finite Sets of Points in Two and Three dimensions, Communications of the ACM, 2,20, pp. 87-93, 1977.

No context found.

F P Preparata and S J Hong. Convex hulls of finite sets of points in two and three dimensions. Comm. ACM, 20:87--93, 1977.

No context found.

F. P. Preparata and S J Hong. Convex hulls of finite sets of points in two and three dimensions. Comm. ACM, 20:87--93, 1977.

No context found.

Preparata, F. and S. Hong, Convex hulls of finite sets of points in two and three dimensions, Communications of the ACM, 2,20, p. 87-93, 1977.

No context found.

F. P. Preparata and S. J. Hong, Convex hulls of finite sets of points in two and three dimensions, Commun. ACM, 20, 2 (1977), pp. 87--93.

No context found.

Preparata, F. and Hong,S., Convex hulls of finite sets of points in two and three dimensions, CACM 20 (1977), 87-93.

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