H. Edelsbrunner, J. O'Rourke and R. Seidel, "Constructing arrangements of lines and hyperplanes with applications", SIAM J. Computing 15(2), pp 341-363, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that may be used to extend the path. We will show that this can be done in O(n) time, after some preprocessing of the point set which takes ) time and storage. In the preprocessing phase we compute and store the circular order of the edges from each point to all other points of S in O(n time [10]. At this stage, we can already eliminate edges which do not lie insideP . The next edge which is added to a partial path must ful ll Properties 2 (pointedness) and 3 (empty area) of De nition 2, the non empty opposite wedge condition of Lemma 4, and it must not cross the previous edges of the ....

H. Edelsbrunner and J. O'Rourke and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. In SIAM J. Comput., 15:341-363, 1986.

....i and we already know that A inherits all directions for B with only one more intersection. Thus, the directions for A are contained in the union. As a corollary of Lemma 3, the depth of all points with respect to a set of hyperplanes can be computed by constructing the arrangement of hyperplanes [9, 10] and labeling cells in a breadthfirst search. The unbounded cells are labeled with their depth zero. Then, for i = 1, 2, all cells with label i 1 cause their adjacent, unlabeled cells to be labeled i. Finally, lower dimensional cells can be labeled according to Lemma 1. Corollary 4 For ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput., 15:341--363, 1986.

....7. Together with the regions R ff (fB i g) this will give us all the discs C(p; q) that we have to consider. This takes O(km) time. Next we compute the set L of lines tangent to two polygons; it is straightforward to do this in O(km) time. We then construct the arrangement A(L) in O(k ) time [5]. So the first stage of the algorithm takes O(k km) time in total. The second stage of the algorithm is as follows. For every cell c in A(L) we compute its visibility cycle V c . This can easily be done in O(k log k m) time as follows. Pick some point x in the cell, replace every B i by a ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. In Proc. 24th Annu. IEEE Sympos. Found. Comput. Sci., pages 93--91, 1983.

.... well known relationship between higher order VoronoY diagrams in d dimensions and arrangements of hyperplanes in d 1 dimensions can be used for the design of an algorithm that constructs the order tz 1 VoronoY diagrams in time and storage O(tz ) as shown by Edelsbrunner, O Rourke and Seidel [13]. In d 2 dimensions, Clarkson [10] has shown that the size of the order k VoronoY diagrams is O(k[ nc ) Recently, Mulmuley [11,14] has obtained a randomized algorithm whose expected complexity meets this bound for d 2 and whose complexity is O(nk 2 nlog n) for d = 2. This algorithm ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM Journal on Computing, 15:341 363, 1986.

....developed two versions of a better algorithm for large k. The first one has a O(n log n k(n Gamma k) log n) time complexity, with a O(k(n Gamma k) storage, while the second one runs in O(n k(n Gamma k) log n) time with O(n ) storage. H. Edelsbrunner, J. O Rourke and R. Seidel [EOS86] use duality (Section 1.4) to construct all orders n Gamma 1 diagrams in O(n ) time and space complexity. 1.2.3 Voronoi diagram of line segments S now denotes a set of objects : points or line segments, in Euclidean space IE . The Voronoi diagram of S is defined in the same ways as the ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM Journal on Computing, 15:341--363, 1986.

....neighbors between lines and points: a) primal, nearest point neighbor to a line# (b) dual, vertical ray shooting in a line arrangement. minimum area triangle with s is the nearest vertical neighbor of l. This observation was used to develop O(n ) algorithms for the minimum triangle problem [7, 16, 15]. We can tighten this characterization as follows. Let 4pqr be the minimum area triangle, and assume that the vertical projection of r is between those of p and q. Then as before r is the nearest neighbor of line pq, but the vertical segment connecting r and line pq actually touches segment pq. ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput., 15:341--363, 1986.

....geometric optimization problems of finding a k gon minimizing or maximizing a certain objective function. A celebrated result in this area is that a minimum area triangle can be found in time O(n by using geometric duality to transform the problem into one of searching a line arrangement [7, 8]. Algorithms are also known for optimizing other functions including minimum perimeter [1, 5, 9] and maximum perimeter and area [2, 4] For some time it remained open whether the minimum area triangle result could be generalized to finding minimum area k gons. There are actually four reasonable ....

....the area of the triangle is c d(x, l) where c is half the length of the horizontal projection of s. Therefore the point in P forming the minimum area triangle with s is the nearest vertical neighbor of l. This observation was used to develop O(n algorithms for the minimum triangle problem [7, 8]. We can tighten this characterization as follows. Let xyz be the minimum area triangle, and assume that the horizontal projection of y is between those of x and z. Then as before y is the nearest neighbor of line xz, but the vertical segment connecting y and line xz actually touches segment xz. ....

H. Edelsbrunner, J. O'Rourke and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput. 15 (1986) 341--363.

....said to be simple (or uniform) and there are clearly points of intersection. Arrangements of lines have been studied extensively in a variety of contexts and there is a large literature. The text by Edelsbrunner [1] provides a good introduction to the area; other related references include [2] and [5] A (line) arrangement graph (denoted as G(L) is the abstract graph obtained from A(L) where the vertex set of G(L) is the intersection points of A(L) and two vertices v 1 and v 2 are adjacent in G(L) if and only if the intersection points corresponding to v 1 and v 2 share a common ....

....into connected pieces of dimensions 0, 1, d. If the hyperplanes are in general position, that is, no d 1 subset of hyperplanes intersect at a common point, then the arrangement is called simple. Edelsbrunner [1] provides a good introduction to the area; other related references include [2] and [5] Recall that a polygon P is an inducing polygon of a line arrangement A(L) provided it has the following properties: 1. P is a simple polygon, 2. every vertex of P is a vertex of A(L) 3. every edge of P lies on a line of L, 4. each line in L is de ned by 2 consecutive vertices of P ....

H. Edelsbrunner, J. O'Rourke, R. Seidel, Constructing Arrangements of Lines and Hyperplanes with Applications, SIAM J. Computing 15 (1986), 341-363.

.... arrangement of lines in the plane; this last theorem states that the average value of the square of the number of vertices of a face of the arrangement is a O(1) this is a well known consequence of the linear bound on the complexity of the so called zone of a line in an arrangement of lines; see [10, 17, 13, 15] and [21, 4] for an higher dimensional analogue. To state our sum of squares theorem we need to introduce new operators. First we extend the de nition of the operator to the case where the set of obstacles is augmented with a set of bitangent obstacles : for H a set of pairwise disjoint ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput., 15:341-363, 1986.

....on . ffl for every intersection point we have a radially sorted list of all lines intersecting the point. An arrangement of n lines may be constructed in Theta(n 2 ) time and space. This result was first obtained by Chazelle, Guibas and Lee [CGL85] and by Edelsbrunner, O Rourke and Seidel [EOS86] The proof of this result is described well in [O R95] The same algorithm may be used to construct an arrangement of line segments. A nice application of arrangements is for sorting all points about every point in a data set in O(n 2 ) time [LC85, Ede87] This may be used in algorithms ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput., 15(2):341-- 363, 1986.

....the generators are in general position. The theorem immediately implies that f 0 (Z) O(n d 1 ) and thus, the number of extreme points is polynomially bounded as d is fixed. The polynomial solvability of the FRC 01QP problem follows 5 directly from the existence of an e#cient algorithm (see [9, 8]) to generate all extreme points of Z. Again, the original algorithm [9] is designed for arrangements and needs to be dualized for our purpose. Theorem 3.2 For d # 3, there is an O(n d 1 ) time algorithm to generate all extreme points of a d dimensional zonotope given by n rational ....

....f 0 (Z) O(n d 1 ) and thus, the number of extreme points is polynomially bounded as d is fixed. The polynomial solvability of the FRC 01QP problem follows 5 directly from the existence of an e#cient algorithm (see [9, 8] to generate all extreme points of Z. Again, the original algorithm [9] is designed for arrangements and needs to be dualized for our purpose. Theorem 3.2 For d # 3, there is an O(n d 1 ) time algorithm to generate all extreme points of a d dimensional zonotope given by n rational generators. Corollary 3.3 The FRC 01QP problem is polynomially solvable. ....

[Article contains additional citation context not shown here]

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput., 15:341--363, 1986.

.... the overall time to O(n 2 ) 5, 6] The dual of the point (x; y) is the line v = xu y, and the dual of the line y = mx b is the point ( m; b) The duality preserves incidence and the above below relationships [3] The arrangement of n lines in a plane can be constructed in O(n 2 ) time [4]. After the arrangement of the n dual lines is constructed, we can traverse the dual line of p to obtain the ordering of the intersections with the other n 1 dual lines in O(n) time. This ordering corresponds to the angular ordering of the n 1 primal points around p. We can then proceed to count ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel, \Constructing Arrangements of Lines and Hyperplanes with Applications", SIAM J. Comput. 15 (1986), 341-363.

....space. 4 G OMEZ, HURTADO, SELLAR ES AND TOUSSAINT s r r t p p s c t FIG. 1. Non regular segment projection Proof. Let S be the set of n segments that are the projection of the segments of S. It suces to determine the forbidden situations using the technique of [16] that computes the arrangement of segments A(S ) in O(n 2 ) time and space. A point that cannot be the projection center of a regular perspective projection of S is called a forbidden point. Let us to analyze the di erent situations in which forbidden points can arise. If a point lies on a ....

....of S, we can choose a point c covered by the minimum number of such regions. Let P be the set of the O(n 2 ) planes determined by the segments of S taken two by two. Let A(P ) be the arrangement corresponding to the planes of P . It suces to compute the arrangement A(P ) using the algorithm of [16], and during the construction, for each cell of the arrangement to compute the T (s; s 0 ) regions that contain it. The entire procedure can be done in O(n 6 ) time and space. By doing depth rst search on the dual graph of A(P ) we can compute in O(n 6 ) time the cell of A(P ) covered by ....

[Article contains additional citation context not shown here]

H. Edelsbrunner, J. O'Rourke and R. Seidel, Constructing arrangements of lines and hyperplanes with applications, SIAM J. Computing 15 (2), pp 341-363, 1986.

....b) 7 y = ax b. It is well known that there are three collinear points in the primal space if and only if there are vertices of degree six or more in the arrangement of lines in the dual space. Constructing such an arrangement and checking for those vertices takes O(n 2 ) time and space (see [6]) 2 The decision problem admits a reduction from the 3 collinear problem. We recall the latter problem belongs to the so called 3 SUM hard class (see [10] Theorem 3.2 Given a set of n distinct points, to decide whether a given center produces a non collinear projection is 3 SUM hard. Proof. ....

....a point if, and only if, the corresponding points are non collinear and, 2) four planes are concurrent if and only if the corresponding points are cocircular. By constructing the arrangement of planes, we can determine if there are four or more concurrent planes in O(n 3 ) time and space (see [6]) c Figure 4: Cocircular projection 2 4.1 Existence of non cocircular perspective projections A center of projection is called a forbidden center if it produces a cocircular projection. As before, let us call P = fp 1 ; Delta Delta Delta ; p n g the set of projected points and, for i; ....

[Article contains additional citation context not shown here]

H. Edelsbrunner, J. O'Rourke and R. Seidel, "Constructing arrangements of lines and hyperplanes with applications", SIAM J. Computing 15(2), pp 341-363, 1986.

....p i j 1 are collinear this is one plus the number for p i j 1 p i , and otherwise it is zero. Next, we consider the time and space complexity: Step (a) takes O(n 2 ) time and space and Step (b) takes O(n log n) time and linear space. Applying the results of Edelsbrunner, O Rourke, and Seidel [10], Step (c) can be performed using only quadratic time and space. Finally, Step (d) consists of two nested for loops, and each substep in the loop is a simple addition or subtraction. Hence, Step (d) costs at most O(n 2 ) time and space, too. Thus we have proved the following theorem: Theorem 2.1 ....

....xy in time and space O(n) each. This gives the claimed complexity. # 9 If the points are not in general position, the minimum k point set may be four points in a line, or a triangle with a point on one of the sides. Such a set can be found analogously to the minimum area triangle algorithm [9, 10]. We compute the line arrangement dual to the point set; then four collinear points correspond to four coincident lines. A minimum area triangle with a point on one side corresponds to three coincident lines and the nearest line directly above or below their point of intersection. These ....

H. Edelsbrunner, J. O'Rourke and R. Seidel, Constructing arrangements of lines and hyperplanes with applications, SIAM J. Computing 15 (1986), 341--363.

....funnel in a coastal face. Therefore, we can compensate for any overlooked edges by counting all edges of all coastal faces of all funnels. We obtain a bound on this number General Collections of Line Segments 8 by separately considering A # i and A ## i , as before. By the zone theorem in [EOS, ESS], the combinatorial complexity of all coastal faces in A # i is O(n i ) The combinatorial complexity of all coastal faces in A ## i is O(k i #(k i ) because by removing the sides of f i we get al..l coastal faces as part of one unbounded face in an arrangement of k i line segments. By the ....

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Computing, 15 (1986), 341--363.

No context found.

H. Edelsbrunner, J. O'Rourke and R. Seidel, "Constructing arrangements of lines and hyperplanes with applications", SIAM J. Computing 15(2), pp 341-363, 1986.

No context found.

H. Edelsbrunner, J. O'Rourke, and R. Seidel, "Constructing arrangements of lines and hyperplanes with applications," Society for Industrial and Applied Mathematics Journal on Computing, vol. 15, pp. 341--363, 1986.

No context found.

H. Edelsbrunner, J. O'Rourke, and R. Seidel, "Constructing arrangements of lines and hyperplanes with applications," Society for Industrial and Applied Mathematics Journal on Computing, vol. 15, pp. 341--363, 1986.

No context found.

H. Edelsbrunner, J. O'Rourke, and R. Seidel, "Constructing arrangements of lines and hyperplanes with applications," Society for Industrial and Applied Mathematics Journal on Computing, vol. 15, pp. 341--363, 1986.

No context found.

H. Edelsbrunner, J. O'Rourke, and R. Seidel, "Constructing arrangements of lines and hyperplanes with applications," Society for Industrial and Applied Mathematics Journal on Computing, vol. 15, pp. 341--363, 1986.

No context found.

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM Journal on Computing, 15:341-363, 1986.

No context found.

H. Edelsbrunner, J. O'Rourke, and R. Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM J. Comput., 15:341363, 1986.

No context found.

H. Edelsbrunner, J. O'Rourke and R. Seidel, Constructing arrangements of lines and hyperplanes with applications, SIAM J. Comput., 15:2 (1986), pp. 341-363.

No context found.

Edelsbrunner, H., J. and Seidel, R., Constructing arrangements of lines and hyperplanes with applications, in: Proceedings 24th Symposium on Foundations of Computer Science (1983) 83-91.

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