N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 26th Annual ACM Sympos. Theory Comput., 672-682, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is performed separately, but simultaneously, over the red, blue, and purple arrangements, in a manner that processes only a small number of red blue intersections. See [69, 134] for more details. Hence, the overall running time of the algorithm is O( s 2 (n) log 2 n) Recently, Goodrich et al. [15] have succeeded in derandomizing the algorithm by Chazelle et al. 32] described above, for a set of segments. The worst case running time of their algorithm is O(nff 2 (n) log n) It is still an open problem whether a single face in an arrangement of Jordan arcs, each pair of which intersects ....

....running time of their algorithm is O(nff 2 (n) log n) It is still an open problem whether a single face in an arrangement of Jordan arcs, each pair of which intersects in at most s points, can be computed in deterministic O( s 2 (n) log n) time. Hence, we can conclude: Theorem 6. 6 ([15, 69]) Given a collection Gamma of n Jordan arcs, each pair of which intersect in at most s points, and a point x not lying on any arc, the face of A( Gamma) containing x can be computed by a deterministic algorithm in time O( s 2 (n) log 2 n) in an appropriate Davenport Schinzel Sequences ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos, Computing faces in segment and simplex arrangements, Proc. 27th Symp. Theory of Comput., 1995, pp. 672--682.

....Our bibliography is by no means complete here; the reader may consult some of the recent papers mentioned here for references to other important works in this area. The k wise independence methods were applied by Berger et al. BRS94] and later by Goodrich [Goo93] Goo96] by Amato et al. AGR95] and by Mahajan et al. MRS97] for parallelization of derandomized geometric algorithms. Mulmuley [Mul96] extends earlier work of Karloff and Raghavan [KR93] on limiting random resources using bounded independence distributions and demonstrates that a polylogarithmic number of random bits is ....

.... (Goodrich [Goo93] The observation that partition trees can produce approximations in geometric situations quickly is also in [Mat92a] together with some applications; more applications can be found in Chazelle and Matousek [CM94] Matousek and Schwarzkopf [MS96] and Amato et al. AGR94] AGR95] The linearization produced by assigning a new coordinate to each monomial is well known in algebraic geometry (the so called Veronese map, see e.g. Har92] It has been used by Yao and Yao [YY85] to show that various geometrically defined set systems can be embedded into the set systems ....

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N. M. Amato, M. T. Goodrich, and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 27th Annu. ACM Sympos. Theory Comput., pages 672-- 682, 1995.

....that adds certain globality to the plain divide and conquer approach. We apply this approach to the segments problem and obtain a second algorithm that constructs a structure of optimal size O(n k) in contrast to O(n log log n k) for the rst algorithm in this paper and a previous one [8] (this previous algorithm also had the drawback of handling only line segments) The second algorithm also results in an ecient deterministic algorithm for computing a cutting of optimal size for a set of curve segments. These algorithms are easily parallelizable and can be made deterministic ....

....are easily parallelizable and can be made deterministic via derandomization techniques. The second algorithm can be implemented in the EREW PRAM model so that it uses optimal work and time O(log 3=2 n) in contrast to O(log 2 n) for the rst algorithm in this paper and the previous one [8]. The approach of divide and conquer with partial clean up also simpli es other previous algorithms (3 d convex hulls, 2 d abstract Voronoi diagrams, 3 d diameter, single face in an arrangement of segments [7, 8, 10] and also leads to the same time speed up for the corresponding parallel ....

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N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 26th Annual ACM Sympos. Theory Comput., 672-682, 1995.

.... is known that finds the intersections in O(N log N K) optimal time [7] There are also much simpler randomized algorithms [10, 16] All of these algorithms actually compute T (S) or some variation of it) There are also algorithms that use space O(N ) and output K(S) in time O(N log N K) [2, 4, 10]. In the external memory model, optimality means that the total number of I Os required to compute the K intersections or the trapezoidal decomposition of S, is Theta(n log m n k) where n = N=B, m = M=B and k = K=B. In fact, the first term derives from the optimal I O cost for sorting the N ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 27th Annu. ACM Sympos. Theory Comput., 672--682, 1995.

....v denote the edges in v s visibility polygon, let E 0 = v2V E v , and let n 0 = jE 0 j. Then, the common intersection of the V (v) for all v 2 V , can be obtained in O(n 0 log n 0 k 0 ) time, where k 0 is the number of intersection points, using one of a number of deterministic [2, 6] or randomized [8, 17] algorithms for line segment intersection. Thus, the total time required to determine the visibility sectors is O(n 2 n 0 log n 0 k 0 ) In many practical situations, n 0 = O(n) and so the total construction time would actually be O(n 2 ) We remark that a ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 27th Annu. ACM Sympos. Theory Comput., pages 672--682, 1995.

....segments in the plane. They both use a divide and conquer approach based on derandomized geometric sampling and achieve the optimal running time O(n log n k) where n is the number of segments and k is the number of intersections. The first algorithm, a simplified version of one presented in [1], generates a structure of size O(n log log n k) and its parallel implementation runs in time O(log 2 n) The second algorithm is better in that the decomposition of the arrangement constructed has optimal size O(n k) and it has a parallel implementation in the EREW PRAM model that runs in ....

....on random sampling. The approach extends previous work by Dehne et al. 7] Deng and Zhu [8] and Kuhn [9] that use small separators for planar graphs in the design of randomized geometric algorithms for coarse grained multicomputers. The approach simplifies other previous geometric algorithms [1, 2], and also has the potential of providing efficient deterministic algorithms for the external memory model. 1 Problem and Previous Work We consider a classical problem in computational geometry: computing the arrangement determined by a set of curve segments in the plane. There has been a ....

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N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 26th Annual ACM Sympos. Theory Comput., 672--682, 1995.

....can be computed in polynomial time. Through the use of a (1=r) approximation, the time can be reduced to O(nr C ) and for linearizable configuration spaces, for r n ffl , to O(n log r) In parallel, k wise independence can only guarantee part 2 of the theorem for j = O(k) and not part 1) [3, 4]. Modeling the sampling with leveled DFAs, and fooling them with relative error, we can construct in parallel a sample as guaranteed by the sampling theorem, except for a constant multiplicative factor. Relative error is needed because of the exponential weighting that makes even small probability ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 27th Annu. ACM Sympos. Theory Comput., 1995, 672--682.

....the complex K T (R) consisting of those that intersect the stated levels. 3. Compute the con ict lists for K with respect to H . This is translated into a problem of locating the points dual to the planes in H in the arrangement of planes dual to the vertices of the simplices in K. See e.g. [6, 24]. The total time is O(n log n) 4. Recurse on each 2 K, and piece together the results into the k level for H . The bound on the size of K in Step 3 follows from Claim 2. By de nition of (1=r) approximation, K is guaranteed to cover the k level of H and its a (1=r) cutting for H . The ....

....with storage O(n d = log bd=2c n) and with query time O(log n) Proof. We use a variation of the data structure by Chazelle [13] that uses O(log log n) levels augmented at each node with a data structure for ray shooting in a convex polytope. This is similar to a data structure in [6]. The data structure is a tree with a node v associated with a simplex v and its con ict list H v of size n v ( n v ) At the i th level of the tree n v n=r i . where r 0 = 1 and r i =r i 1 = n=r i 1 ) for a constant appropriately small. Note that n=r i = n (1 ) i . Consider a ....

N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. STOC'95, 672-682.

....are easily parallelizable and can be made deterministic via derandomization techniques. The second algorithm can be implemented in the EREW PRAM model so that it uses optimal work and time O(log 3=2 n) in contrast to O(log 2 n) for the rst algorithm in this paper and the previous one [8]. The approach of divide and conquer with partial clean up also simpli es other previous algorithms (3 d convex hulls, 2 d abstract Voronoi diagrams, 3 d diameter) and leads to the same time speed up for some of the corresponding parallel algorithms. This is reported elsewhere. Texas A M ....

....(a point where many segments intersect is counted only once) as shown by Seidel [69] and Burnikel [18] On the other hand, unlike its randomized counterparts in [27, 56] the deterministic algorithm in [21] can only handle line segments. An alternative deterministic algorithm by Amato et al. [8], which follows a divide and conquer approach based on derandomization of geometric sampling, has the advantage of being parallelizable, achieving a running time O(log 2 n) in the EREW PRAM model (previous parallel algorithms did not achieve work optimality in general [33, 67] However, it can ....

[Article contains additional citation context not shown here]

N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 26th Annual ACM Sympos. Theory Comput., 672-682, 1995.

No context found.

N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. STOC'85, 672--682.

....with storage O(n d = log bd=2c n) and with query time O(log n) Proof. We use a variation of the data structure by Chazelle [11] that uses O(log log n) levels augmented at each node with a data structure for ray shooting in a convex polytope. This is similar to a data structure in [4]. The data structure is a tree with a node v associated with a simplex v and its con ict list H v of size n v ( n v ) At the i th level of the tree n v n=r i , where r 0 = 1 and r i =r i 1 = n=r i 1 ) for a constant appropriately small. Note that n=r i = n (1 ) i . Consider a ....

N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. STOC'95, 672-682.

....can be reduced to O(nr c ) For a so called linearizable configuration space, for which the range space (X; fK(oe) oe 2 C(X)g) is linearizable in IR l for some l, there is an ffl = ffl(l) 0 ffl 1, so that for r n ffl , a good sample can be computed in O(n log r) time. In parallel [5, 6], k wise independence can guarantee (ii) of the theorem for j = O(k) but not (i) a weaker version, jK(oe)j C 0 dn r r ffi , follows from (ii) Modelling the sampling with levelled RFAs, and fooling them with relative error, we can construct in parallel a sample as guaranteed by the ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 27th Annu. ACM Sympos. Theory Comput., 1995, 672--682.

....log r) if r n ffl . In EREW PRAM the construction can be performed in times O(logn) and O(logn log r) respectively; in CRCW PRAM the construction can be performed in times O(log log n) and O(log log n log r) respectively (the output is padded, or add time O(logn) for not padded output) See [9, 33, 24, 4]. Only the CRCW construction has not been noted before, it uses the recent fast construction of approximation in [25, 27] Briefly, for later reference, a cutting is computed by first (i) obtaining a (1=2r) approximation A for (Q; Q E ) then (ii) computing a (1=2r) cutting T for (Y; A) and ....

....fast O(n log r) construction, for any constant k 0, by choosing ffl sufficiently small, we have that jAj k and jT j k are O(jQj) thus a polynomial amount of work in jAj and jT j is allowed. For our 1 d lower envelope algorithm, we need sparse cuttings, introduced in [9] and later used in [40, 3, 4]. In the next section, we state the specific constructions needed. 3 1 d lower envelopes 3.1 Statement of problem Let F be the class of 1 d algebraic functions, that is, functions f : IR IR such that f is defined in each of a constant number of intervals as an algebraic function of constant ....

[Article contains additional citation context not shown here]

N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 26th Annual ACM Sympos. Theory Comput., 672--682, 1995.

....O(1) Cutting. Under the assumptions above, the following geometric sampling result holds, and is the 11 Note that it is possible to have more than one component, for example, in the Voronoi diagram of line segments, or in the intersection of unit balls in R 3 . basis for the algorithm. See [3]. Lemma 4 For 0 ffi 1, there are constants C,r 0 ,ffl 0 such that for 0 ffl ffl 0 and r 0 r n ffl , a sample R S with the following properties can be constructed deterministically in time O(n log r) i) jjRj Gamma 3r=2j r, ii) max 2T (R) jS j j (n=r)r ffi , iii) jT (R)j ....

N. M. Amato, M. T. Goodrich and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 26th Annual ACM Sympos. Theory Comput., 672-- 682, 1995.

....and efficient deterministic methods for constructing small sized ffi relative ffl approximations in parallel and applying these methods to fixed dimensional linear programming. Our methods have other applications as well, including fixeddimensional convex hull and geometric partition construction [6, 7], but these are beyond the scope of this paper. 1.1 Previous work on derandomizing geometric algorithms Before we describe our results, however, let us review some related previous work. The study of random sampling in the design of efficient computational geometry methods really began in ....

....on an EREW PRAM, for fixed d. 6 Conclusion We have given a general scheme for derandomizing random sampling efficiently in parallel, and have shown how it can be used to solve the fixed dimensional linear programming problem efficiently in parallel. Interestingly, Amato, Goodrich, and Ramos [6, 7] have shown how to use such methods to derive efficient parallel algorithms for d dimensional convex hull construction, planar segment intersection computation, 1=r) cutting construction, and d dimensional point location. We suspect that there may be other applications, as well. ....

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 27th Annu. ACM Sympos. Theory Comput., 672--682, 1995.

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N. M. Amato, M. T. Goodrich, and E. A. Ramos. Computing faces in segment and simplex arrangements. In Proc. 27th Annu. ACM Sympos. Theory Comput., 672--682, 1995.

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