M. I. Shamos, "Geometric complexity," Proc. 7th ACM Symposium on the Theory of Computing, pp. 224-233, May 1975. f  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Historically, the first efficient algorithm for specifically computing the nearest neighbour decision boundary is due to Dasarathy and White [7] and runs in O(n ) time. The first O(n log n) time algorithm for computing the Vorono diagram of a set of n points in the plane is due to Shamos [14]. Figure 2: The relationship between convex hulls and decision boundaries. Each vertex of the convex hull of R contributes to the decision boundary. 2 A 1 Dimensional Algorithm In the 1 dimensional version of the nearest neighbour decision boundary problem, the input set S consists of n real ....

M. I. Shamos. Geometric complexity. In Proceedings of the 7th ACM Symposium on the Theory of Computing (STOC 1975.

....Historically, the first efficient algorithm for specifically computing the nearest neighbour decision boundary is due to Dasarathy and White [7] and runs in O(n ) time. The first O(n log n) time algorithm for computing the Vorono diagram of a set of n points in the plane is due to Shamos [14]. Figure 2: The relationship between convex hulls and decision boundaries. Each vertex of the convex hull of R contributes to the decision boundary. 2 A 1 Dimensional Algorithm In the 1 dimensional version of the nearest neighbour decision boundary problem, the input set S consists of n real ....

M. I. Shamos. Geometric complexity. In Proceedings of the 7th ACM Symposium on the Theory of Computing (STOC 1975.

....solving practical problems and creating beautiful mathematics [SH75, SH76, SB77] Computational geometry has always concerned itself with the design of efficient data structures and algorithms for solving problems which involve geometric objects. Two decades after the landmark paper by M. I. Shamos in 1975 [Sha75] and only ten years after the publication of the first text book [PS85] entirely devoted to the topic, computational geometry has established itself as an important research field in computer science. Numerous papers have been produced by researchers all over the world. Several ....

....problems and creating beautiful mathematics [SH75, SH76, SB77] Computational geometry has always concerned itself with the design of efficient data structures and algorithms for solving problems which involve geometric objects. Two decades after the landmark paper by M. I. Shamos in 1975 [Sha75] and only ten years after the publication of the first text book [PS85] entirely devoted to the topic, computational geometry has established itself as an important research field in computer science. Numerous papers have been produced by researchers all over the world. Several conferences and ....

M. I. Shamos. Geometric complexity. In Proc. 7th Annu. ACM Sympos. Theory Comput., pages 224--233, 1975.

....of M, and so has constant time complexity. Computing the radius function has O(Ivertices( M)l ) time complexity. The radius function is the solution to the problem of determining the minimum free half plane which the robot M can occupy. The radius function is related to the support function [17] of a convex polygon, and to the shi[t distance of edges in the computation of the grown obstacles. To see the concept of shift distance, let s slide robot J1 along the wall w, keeping M at fixed orientation 0. The reference point O draw a line parallel to the wall, distant from the wall by R(O, ....

Shamos, MI. "Geometric complexity," 7th Annual ACM SFmposium on Theory of Computing (1975).

....correctness, performance guarantees, and worst case analysis of algorithms. On the application side, however, the eld is quite broad, and interacts with a large number of applied elds that deal with geometric problems. Historically, most people consider the publication of Michael Shamos paper [40] as the birthday of computational geometry. While a number of isolated results existed before this publication, Shamos paper hit a kind of critical mass in that it ignited a urry of research and also attracted several brilliant graduate students to the nascent eld. In the past quarter century, ....

Michael I. Shamos. Geometric complexity. In Proceedings of the 16th ACM Symposium on Theory of Computing (STOC), pages 224-233, 1975. 115

....made by several authors (such as Duda and Hart [1] and Johnson [2] that d max (S 1 , S 2 ) diam(S) is not always true. Were this so, there would be no need for this paper since the diameter of S 1 S 2 could be found in O(n log n) time with either the convex hull approach of Shamos [3, 4], or the furthest point Voronoi diagram method of [5] Although the idea of using the furthest point Voronoi diagram to find the diameter of a set is originally due to Shamos [4] his algorithm is based on another invalid claim that the diameter is an edge in the dual of the Voronoi diagram. A ....

....2j ) and output it as d max (S 1 , S 2 ) end Theorem 3.2: Algorithm MAXDIST 2 computes the maximum distance between S 1 and S 2 . Proof: The correctness of Step 1 follows from Lemma 2.1. Because the diameter of a set is equal to the diameter of the convex hull of the set (see [3 4]) Steps 3 and 4 correctly compute the diameter of CH(S 1i ) CH(S 2j ) Finally, the correctness of Steps 2 and 5 follows directly from Theorem 2.1. Q.E.D. 2 1 2 2 p i n n n log max i j , Worst Case Analysis Step 1 can be computed in O(n log n) time [6] Step 2 is clearly an O(n) ....

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M. I. Shamos, "Geometric complexity," Proc. 7th ACM Symposium on the Theory of

....The closest pair problem: The simplest version of this problem is stated as follows. Problem 1. 1 Given a set S of n points in IR D , find a closest pair of S, i.e. two points P; Q 2 S, such that d(P; Q) minfd(p; q) p; q 2 S; p 6= qg: This problem appeared for the first time in Shamos [111], one of the first papers in computational geometry. Since there are i n 2 j pairs of distinct points, and since the distance between two points can be computed in O(D) O(1) time, Problem 1.1 can trivially be solved in O(n 2 ) time. In the early years of computational geometry, ....

....to D Delta cffi= cD c) ffi. This is clearly a contradiction. 2.2 Algorithms that are optimal in the algebraic computation tree model 2.2. 1 An algorithm based on the Voronoi diagram The first optimal algorithm for solving the planar version of the closest pair problem is due to Shamos [111] and Shamos and Hoey [112] Their algorithm is as follows. In O(n log n) time, compute the Voronoi diagram of S. Then, for each edge e of this diagram, compute the distance between the two points whose Voronoi regions share e. This takes only linear time. The smallest distance found in this way ....

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M.I. Shamos. Geometric complexity. Proceedings 7th Annual ACM Symposium on the Theory of Computing, 1975, pp. 224--233.

....[Gr2] In 1974, Dobkin and Lipton [DL1, DL2] gave the 3 first algorithms for searching in spatial subdivisions. Their work arose from an open problem given in Knuth [Kn] asking how to preprocess a set of points to be able to find nearest neighbors efficiently. In 1975, Shamos and Hoey [S, SH1] proposed efficient algorithms for finding a closest pair from a set of points. This work became part of Shamos thesis which initiated the field of computational geometry. Although it wasn t realized at the time, finding convex hulls, building and searching spatial subdivisions and finding ....

Shamos, M.I., "Geometric complexity", Proceedings of the 7th ACM Symposium on the Theory of Computing, 224-233, 1975.

.... algorithms for the construction of the Voronoi diagram of a set of line segments are given by Kirkpatrick [14] Lee and Drysdale [15] and Yap [33] The algorithms in [14, 33] run in O(n log n) time, which is optimal since a lower bound of Omega Gamma n log n) is known for this problem [27]. The run time of the algorithm in [15] is O(nlog 2 n) We will repeatedly refer to Yap s algorithm in the coming sections, since it lends itself to efficient parallelization, whereas the other two techniques do not. Goodrich et al. 9] give a CREW PRAM algorithm for Voronoi diagram construction ....

M. I. Shamos. Geometric Complexity. In Proc. 7th ACM Symp. on Theory of Computing, pages 224--233, 1975.

....of minimum perimeter, or a simple circuit that bounds the minimum area, with no increase in computational complexity. 1 Introduction It is always possible to construct a simple polygon passing through every point of a planar point set, and this task can be accomplished in Theta(n log n) time [12]. However, for a set of line segments, it is not always possible to obtain a simple circuit passing through every line segment. An example is shown in Fig. 1. If we can find a simple circuit that passes through every segment of a set of line segments, then we say that the set admits a simple ....

M.I. Shamos. Geometric complexity. In Proc. 7th ACM Annu. Symp. Theory Comput., pages 224--233, 1975.

....(S 1 ; S 2 ) problem into 81 diameter problems on the sets of the form (S 1u [ S 2v ) In this note we present a very simple algorithm based on searching a generalization of the notion of antipodal pairs of points. We assume that the reader is familiar with the O(n) diameter algorithm of Shamos [10]. 2 Preliminary Results Let CH(S i ) i = 1; 2 denote the set of points of S i which are vertices of the convex hull of S i . Denote the diameter of S i by diam(S i ) i.e. diam(S 1 ) max i;j fd(p i ; p j )g; i; j = 1; 2: n: We now review and establish some results which will form ....

....; S 2 ) diam(S) It should be noted here that the claim made by several authors (such as Duda and Hart [3] and Johnson [6] that d max (S 1 ; S 2 ) diam(S) is not always true. Were this so the diameter of S 1 [ S 2 could be found in O(n log n) time with either the convex hull approach of Shamos [10, 11] of the furthest point Voronoi diagram methods of [13] and [7] We now define some new terms and establish some additional results. Let LS i (a) denote the directed line of support of set S i through point a 2 S i such that no points of S i lie to the left of LS i (a) Definition 2.1 Given two ....

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M.I. Shamos. Geometric complexity. In Proc. 7th ACM Symposium on the Theory of Computing, pages 224--233, May 1975.

....Set Distance and the Closest Pair The closest pair problem consists of determining the pair of points in a set S that are closest together in the sense of minimizing the euclidean distance. This problem has received attention recently in the literature on computational geometry [3] 4] Shamos [5] has shown that W(n log n) is a lower bound for this problem, and in [3] presents a divide and conquer algorithm that runs in O(n log n) time, and is thus optimal. When the input is a convex n vertex polygon rather than an arbitrary set of points the W(n log n) lower bound does not hold. Lee and ....

M. I. Shamos, "Geometric complexity," Proc. 7th ACM Symposium on the Theory of Computing, May 1975, pp. 224-233.

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M. I. Shamos, "Geometric complexity," Proc. 7th ACM Symposium on the Theory of Computing, pp. 224-233, May 1975. f

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M. I. Shamos. Geometric complexity. In Proceedings of the 7th ACM Symposium on the Theory of Computing (STOC 1975.

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Michael Shamos. Geometric complexity. In Proceedings of the Seventh ACM Symposium on the Theory of Computing, pages 224-253, 1975.

No context found.

M. I. Shamos. Geometric complexity. In Proceedings of the 7th ACM Symposium on the Theory of Computing (STOC 1975.

No context found.

M. I. Shamos. Geometric complexity. In Proceedings of the 7th ACM Symposium on the Theory of Computing (STOC 1975), pages 224--253, 1975.

No context found.

Michael Shamos. Geometric complexity. In Proceedings of the Seventh ACM Symposium on the Theory of Computing, pages 224-253, 1975.

No context found.

M. I. Shamos. Geometric complexity. In Proceedings of the 7th ACM Symposium on the Theory of Computing (STOC 1975), pages 224--253, 1975.

No context found.

M. I. Shamos. Geometric complexity. In Proc. 7th Annu. ACM Sympos. Theory Comput., pages 224--233, 1975.

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