L. Guibas and M. Sharir. Combinatorics and algorithms of arrangements. In J. Pach, editor, New Trends in Discrete and Computational Geometry, pages 9--36. Springer-Verlag, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of surfaces in d space is the partitioning of d space induced by a collection of surfaces. Arrangements play a central role in computational geometry, and the analysis of many geometric algorithms relies on the complexity of an arrangement or of portions of an arrangement (see, e.g. 10] [12]) The complexity of an arrangement of surfaces in 3 space, for example, is the overall number of faces of dimensions 0; 1; 2 and 3 in the partitioning of space induced by these surfaces. We obtain the following result which we believe to be of independent interest (Proposition 2.3) Given a ....

L. Guibas and M. Sharir, Combinatorics and algorithms of arrangements, in New Trends in Discrete and Computational Geometry (J. Pach, Ed.), Springer-Verlag, 1993, pp. 9--36.

....be simply connected in an arrangement of segments. Line and segment arrangements have been extensively studied in computational geometry (as well as in some other areas) as a wide variety of computational geometry problems can be formulated in terms of computing such arrangements or their parts [11, 14]. Given a set L of n lines and a set P of m points in the plane, we define A(L; P ) to be the collection of all cells of A(L) containing at least one point of P . The combinatorial complexity of a cell C, denoted by jCj, in A(L) is the number of edges of C. Let (L; P ) C2A(L;P ) jCj denote ....

L. Guibas and M. Sharir, Combinatorics and algorithms of arrangements, in New Trends in Discrete and Computational Geometry (J. Pach, ed.), Springer-Verlag, New York-BerlinHeidelberg, 1993, 9--36.

....l q intersect at most twice. Lemma 4 The lower envelope L consists of O(n) vertices. Proof: We will show that the names of the points that correspond to the edges of L, when we traverse L from left to right, form a Davenport Schinzel sequence of order two. This will prove the claim. See e.g. [8]. Hence, we must show that for any pair p and q of distinct points of S, this sequence of names does not contain a subsequence of the form p : q : p : q. But this follows from the fact that l p and l q intersect at most twice, and from the restrictions on the endpoint of these ....

....Let R be the anchored ray that makes an angle of with the positive x axis. Then ffi = min p2S d(p; R ) It is easy to see that R = R . Next we analyze the running time of our algorithm. Step 1 takes O(n) time. The lower envelope L can be computed by a divide and conquer algorithm. See e.g. [8]. Since L has linear size, this algorithm, and hence Step 2, takes O(n log n) time. Step 3 takes O(n) time, and Step 4 takes O(1) time. We have proved the following result. Theorem 1 Let S be a set of n points in the plane, and let each point p of S have a positive weight w(p) In O(n log n) ....

L. Guibas and M. Sharir. Combinatorics and algorithms of arrangements. In: New Trends in Discrete and Computational Geometry, Ed. J. Pach. Springer-Verlag, Berlin, 1993, pp. 9--36.

.... p and l q do not intersect. Lemma 4 The lower envelope L consists of O(n) vertices. Proof: We will show that the names of the points that correspond to the edges of L, when we traverse L from left to right, form a Davenport Schinzel sequence of order two. This will prove the claim. See e.g. [6]. Hence, we must show that for any pair p and q of distinct points of S, this sequence of names does not contain a subsequence of the form p . q . p . q. But this follows from the fact that l p and l q intersect at most twice, and from the restrictions on the endpoint of these ....

....Let R # be the anchored ray that makes an angle of # with the positive x axis. Then # = min p#S d(p, R # ) It is easy to see that R = R # . Next we analyze the running time of our algorithm. Step 1 takes O(n) time. The lower envelope L can be computed by a divide and conquer algorithm. See e.g. [6]. Since L has linear size, this algorithm, and hence Step 2, takes O(n log n) time. Step 3 takes O(n) time, and Step 4 takes O(1) time. We have proved the following result. Theorem 1 Let S be a set of n points in the plane, and let each point p of S have a positive weight w(p) In O(n log n) ....

L. Guibas and M. Sharir. Combinatorics and algorithms of arrangements. In: New Trends in Discrete and Computational Geometry, Ed. J. Pach. Springer-Verlag, Berlin, 1993, pp. 9--36.

....later) Bibliography and remarks. A survey on randomized algorithms is Karp [Kar91] a recent book is Motwani and Raghavan [MR95] Randomized algorithms in computational geometry (mainly incremental ones) are treated extensively in Mulmuley [Mul94] other good sources are Guibas and Sharir [GS93] Seidel [Sei93] Agarwal [Aga91] Clarkson [Cla92] As was mentioned above, most algorithms in computational geometry are designed under the assumption that the space dimension is a (small) constant. One problem where the dependence of the running time on the dimension has been studied ....

....algorithm (based on a shallow cutting construction and vertex accounting; see remarks to section 5.1) All algorithms in this paragraph are in the EREW PRAM model. Diameter in IR 3 . The first derandomization of the Clarkson Shor 3 dimensional diameter algorithm is due to Chazelle et al. CEGS93] and has O(n 1 ffi ) running time for any fixed Delta 0. This was improved by Matousek and Schwarzkopf [MS96] to O(n log const n) by Ramos [Ram94] to O(n log 5 n) by Amato et al. AGR94] to 9 O(n log 3 n) and finally by Ramos [Ram97b] to the current best time O(n log 2 n) ....

L. Guibas and M. Sharir. Combinatorics and algorithms of arrangements. In J. Pach, editor, New Trends in Discrete and Computational Geometry, volume 10 of Algorithms and Combinatorics, pages 9--36. Springer-Verlag, 1993.

....be simply connected in an arrangement of segments. Line and segment arrangements have been extensively studied in computational geometry (as well as in some other areas) as a wide range of computational geometry problems can be formulated in terms of computing such arrangements or their parts [11, 14]. Given a set S of n lines and a set P of m points in the plane, we define A(S, P ) to be the collection of all cells of A(S) that contain at least one point of P . The combinatorial complexity of a cell C in A(S) denoted by C , is the number of edges of C. Let #(S, P ) P C#A(S,P ) ....

L. GUIBAS AND M. SHARIR, Combinatorics and algorithms of arrangements, in New Trends in Discrete and Computational Geometry, J. Pach, ed., Springer-Verlag, Heidelberg, 1993, pp. 9--36.

....be algebraic of bounded maximum degree, and bounded by algebraic surfaces of maximum constant degree. See [6] for detailed discussions on arrangements of lines and of hyperplanes in higher dimensional spaces. For discussion of arrangements of curves and surfaces (not necessarily linear) see, e.g. [7, 10]; in particular, the thesis [10] focuses on arrangements of surfaces in 3 space that are induced by motion planning problems and are closely related to the arrangements discussed in Section 4 of this paper. We will often refer to two quantities regarding arrangements of m surfaces in d space: The ....

....The OE upper envelope of M , E OE (M) is defined to be the union of G( M) over all lines whose angle with the positive x direction is OE, whenever G( M) is defined. If OE is 90 ffi , then our definition of E OE (M) is similar to the standard definition of upper envelopes (see, e.g. [7]) Figure 2(d) shows the OE upper envelope of the polygon in Figure 2(b) for OE = 90 ffi . Corresponding to each of the two envelopes, we now define a shadow. The central shadow of M , SC (M ) is the collection of points in the plane such that the line connecting them to the origin intersects ....

L. Guibas and M. Sharir. Combinatorics and algorithms of arrangements. In J. Pach, editor, New Trends in Discrete and Computational Geometry, pages 9--36. Springer, 1993.

....parts) we conclude that S 2 consists of O(n 2 ) critical surfaces. The surfaces (or more precisely, surface patches) in S 1 and S 2 are clearly algebraic of bounded degree. It is well known that the maximum number of cells in a 3D arrangement induced by m such surfaces is O(m 3 ) see, e.g. [4, 5, 11]) Since there are O(n 2 ) surfaces in each of S 1 and S 2 , the maximum number of cells in the subdivision of (x; y; OE) space is O(n 6 ) The algorithm requires that we visit all cells in the arrangement, at each step moving from a cell to one of its neighbors. This can be done in time very ....

L. Guibas and M. Sharir. Combinatorics and algorithms of arrangements. In J. Pach, editor, New Trends in Discrete and Computational Geometry, pages 9--36. Springer, 1993.

....O(k) time is also presented, where k n is an inputdependent parameter indicating the maximum times an edge projected on the projection plane can be intersected by the projections of other edges. The visibility problem in the plane can be solved by using a sequential computer in O(n log n) time [7] and hidden surface and line elimination problems can be solved in O(n 2 ) worst case optimal time [19] Hidden line and surface elimination problems can also be solved in a manner dependent on k, where k is the number of intersections between projected edges [6, 24, 8, 26] Besides objectspace ....

.... stating that the complexity of the lower envelope of n curve segments any two of which may intersect at most in s points (i.e. the total number nLE of visible segments produced by n curve segments) is s 2 (n) where s (n) is the maximum length of an (n; s) Davenport Schinzel sequence [7]. In our case, s = 1 and it is known that 3 (n) O(nff(n) where ff(n) is the extremely slowly growing functional inverse of Ackermann s function. In fact, ff(n) 4 for most practical n. Thus, by adding to Procedure Find Lower Envelope one more step of sending visible segments from the ....

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L. Guibas and M. Sharir. Combinatorics and algorithms of arrangements. In J. Pack, editor, New Trends in Discrete and Computational Geometry, pages 10--36. Springer, 1993.

....be simply connected in an arrangement of segments. Line and segment arrangements have been extensively studied in computational geometry (as well as in some other areas) as a wide variety of computational geometry problems can be formulated in terms of computing such arrangements or their parts [11, 14]. Given a set L of n lines and a set P of m points in the plane, we define A(L; P ) to be the collection of all cells of A(L) containing at least one point of P . The combinatorial complexity of a cell C, denoted by jCj, in A(L) is the number of edges of C. Let (L; P ) P C2A(L;P ) jCj denote ....

L. Guibas and M. Sharir, Combinatorics and algorithms of arrangements, in New Trends in Discrete and Computational Geometry (J. Pach, ed.), Springer-Verlag, New York-BerlinHeidelberg, 1993, 9--36.

....be simply connected in an arrangement of segments. Line and segment arrangements have been extensively studied in computational geometry (as well as in some other areas) as a wide variety of computational geometry problems can be formulated in terms of computing such arrangements or their parts [11, 14]. A part of this work was done while the first and third authors were visiting Charles University and while the first author was visiting Utrecht University. The first author has been supported by National Science Foundation Grant CCR 93 01259 and an NYI award. The second author has been ....

L. Guibas and M. Sharir, Combinatorics and algorithms of arrangements, in New Trends in Discrete and Computational Geometry (J. Pach, ed.), Springer-Verlag, New York-BerlinHeidelberg, 1993, 9--36.

....will be explained in more detail below. We will also describe below more applications of higher dimensional arrangements to problems in visibility, in geometric optimization, and involving generalized Voronoi diagrams. For some basic terminology related to arrangements, the reader is referred to [30, 39, 45, 65]. This survey describes many recent advances in the study of combinatorial, topological, and algorithmic problems involving arrangements of algebraic surfaces in higher dimensions. In these studies there are three main relevant parameters: the number n of surfaces, their maximum algebraic degree ....

L. Guibas and M. Sharir, Combinatorics and algorithms of arrangements, in New Trends in Discrete and Computational Geometry, (J. Pach, Ed.), Springer-Verlag, 1993, 9--36.

.... three degrees of freedom, including several restricted cases of the polygon motion planning problem that we consider here, where the shape of B and or the shape of V is further restricted; these latter bounds are also all close to quadratic, and are reported in [7] See also two recent surveys [5, 6] for more details concerning motion planning problems and arrangements of surfaces. In this paper we exploit the new bounds derived in [8] introduce a special cell decomposition scheme for the cell arising in our motion planning problem, and obtain an efficient algorithm for constructing such a ....

L. Guibas and M. Sharir, Combinatorics and algorithms of arrangements, in New Trends in Discrete and Computational Geometry, (J. Pach, Ed.), Springer-Verlag, 1993, 9--36.

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L. Guibas and M. Sharir. Combinatorics and algorithms of arrangements. In J. Pach, editor, New Trends in Discrete and Computational Geometry, pages 9--36. Springer-Verlag, 1993.

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