H. Edelsbrunner, L. Guibas and M. Sharir, The complexity of many faces in arrangements of lines and of segments, Discrete and Computational Geometry 5 (1990), 161--196.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Durham, NC 27708 0129, USA. Department of Applied Mathematics, Charles University, Malostransk e n am. 25, 118 00 Praha 1, Czech Republic. Department of Computer Science, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, the Netherlands. the marked cells of A(L) Edelsbrunner et al. [12] presented a randomized algorithm, based on the random sampling technique [16] for computing A(L; P ) whose expected running time was log n n log n log m) for any fixed 0. A deterministic algorithm with running O(1) n n log n m log n) was given by Agarwal [1] These ....

....combinatorial complexity of A(S; P ) over all sets S of n segments and sets P of m points in the plane. Aronov et al. 2] proved that j(n; m) O n log m n ff(n) A randomized algorithm with expected running time log n nff(n) log n log m) is described by Edelsbrunner et al. [12], and a slightly faster deterministic algorithm is presented by Agarwal [1] Following the same strategy as for the case of lines, we first develop a randomized algorithm with O( m n log m n ff(n) log n) expected running time. Let us remark that the above upper bound for j(n; m) is not known ....

H. Edelsbrunner, L. Guibas and M. Sharir, The complexity of many faces in arrangements of lines and of segments, Discrete and Computational Geometry 5 (1990), 161--196.

..... The storage is bounded by O(m n) 14 Proof. As for the storage, O(n m) space is used in one level of the algorithm and from one level to the next we need to store only one candidate triangle for each ray. The time bound follows from solving equation (7) this solution follows a schema in [EGS88] Fix ffi and choose r = r(ffi) to be sufficiently large (how large will be determined later in the proof) If m n 4 ffl then T (m; n) am log 2 n satisfies the bound assuming A a. Suppose m n 4 ffl . In this case m = m 4=5 Gammaffi m 1=5 ffi m 4=5 Gammaffi n 4=5 4ffi ffl 0 ....

....T in 3dimensional space, there is a data structure D(T ) that uses O(n 4 ffl ) storage and reports the first triangle hit by any ray ae in O(log n) time. D(T ) can be built in O(n 4 ffl ) expected time. 5. 1 Off line ray shooting on intersecting triangles The batching technique of [GOS89, EGS88] allows us to to divide recursively the problem into subproblems in which every line in the set of lines M oe , of size m oe , intersect all the triangles in the set N oe of size n oe . We solve the ray shooting problem restricted to N oe and M oe according to the following strategy. 1. If m oe ....

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H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity of many faces in arrangements of lines and segments. In Proceedings of the 4th ACM Symposium on Computational Geometry, pages 44--55, 1988.

....using a randomized algorithm whose expected running time is bounded by [Dm 5=6 Gammaffi n 5=6 5ffi log 2 n Am log 2 n Bn log n log m] for any ffi 0, where the coefficients A; B and D depend on ffi . The storage is bounded by O(m n) Proof. The proof is follows a schema in [EGS88b] Fix ffi and choose r = r(ffi) to be sufficiently large (how large will be determined later in the proof) If m n 5 then T (m; n) am log 2 n satisfies the bound assuming A a. Suppose m n 5 . In this case m = m 5=6 Gammaffi m 1=6 ffi m 5=6 Gammaffi n 5=6 5ffi First notice ....

H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity of many faces in arrangements of lines and segments. In Proceedings of the 4th ACM Symposium on Computational Geometry, pages 44--55, 1988.

.... V j I = f0; 1g k g: This quantity plays a important role in certain areas of statistics, in particular in the theory of empirical processes [Dud78,Vap82,GZ84,Dud84,Pol84,Tal87a,Tal87b,Tal88,Pol90] It has also been used recently in the fields of computational geometry [HW87] Wel88] MSW90] [EGS88] [CF88] CW89] and machine learning [BEHW89,HP88,RHW89,FC90,VW91] Let jV j denote the cardinality of V . The following result is well known, and was independently discovered by several people, including Sauer [Sau72] and Vapnik and Chervonenkis (see [Ass83] for a review, and also [Dud84] Lemma ....

H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity of many faces in arrangements of lines and of segments. In Proc. 4th Ann. ACM Symp. on Computational Geometry, pages 44--55, 1988.

....efficiently in an implicit way and therefore allows us to solve the query problem. Once we have the data structure D(L) we can produce its batched version in a quite general way which we call nested batching technique. This technique is a generalization of an approach to batching computations in [EGS88] This technique has been used in several recent papers (e.g. Pel90, AS91b] where the underlying predicate has only one conjunct. In [Pel93] the technique has been used for a formula with two conjuncts. Here we generalize it in an abstract setting for any constant number of conjuncts. 14 4.1 ....

....following is the main theorem of this section: Theorem 8 For any ffl and ffi ffl=d: T j (m; n) D j n d= d 1) ffl m d= d 1) B j m 1 ffi A j n log (1 j) m where D j ; B j ; A j depend on ffl and ffi. Proof. The proof is an induction on j and n inspired by similar argument is in [EGS88] For j = 0, C 0 = TRUE is satisfied by nm pairs. The product is computed in constant time. We consider j 0 and we use the inductive hypothesis on T j Gamma1 . Equation (8) becomes: T j (m; n) X T j (m ; n ) O(mr O(1) M(r) D j Gamma1 n d= d 1) ffl m d= d 1) B j Gamma1 m ....

H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity of many faces in arrangements of lines and segments. In Proceedings of the 4th ACM Symposium on Computational Geometry, pages 44--55, 1988.

....Edelsbrunner and Welzl [13] In this paper we study the problem of computing A(S, P ) that is, for each cell C # A(S, P ) we want to return the vertices of C in, say, clockwise order. We will refer to the cells of A(S, P ) as the marked cells of A(S) Edelsbrunner, Guibas, and Sharir [12] presented a randomized algorithm, based on the random sampling technique [16] for computing A(S, P ) whose expected running time was O(m 2 3 # n 2 3 2# log n n log n log m) for any fixed # 0. A deterministic algorithm with running time O(m 2 3 n 2 3 log O(1) n n log 3 n ....

....n segments and over all sets P of m points in the plane. Aronov et al. 2] proved that #(n, m) O m 2 3 n 2 3 n log m n#(n) A randomized algorithm with expected running time O(m 2 3 # n 2 3 2# log n n#(n) log 2 n log m) is described by Edelsbrunner, Guibas, and Sharir [12], and a slightly faster deterministic algorithm is presented by Agarwal [1] See [20] for results on computing a single cell in arrangements of segments. COMPUTING MANY FACES IN ARRANGEMENTS 493 Following the same strategy as for the case of lines, we first develop a randomized algorithm with ....

H. EDELSBRUNNER, L. GUIBAS, AND M. SHARIR, The complexity of many faces in arrangements of lines and of segments, Discrete Comput. Geom., 5 (1990), pp. 161--196.

....Durham, NC 27708 0129, USA. z Department of Applied Mathematics, Charles University, Malostransk e n am. 25, 118 00 Praha 1, Czech Republic. x Department of Computer Science, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, the Netherlands. the marked cells of A(L) Edelsbrunner et al. [12] presented a randomized algorithm, based on the random sampling technique [16] for computing A(L; P ) whose expected running time was O(m 2=3 Gamma n 2=3 2 log n n log n log m) for any fixed 0. A deterministic algorithm with running time O(m 2=3 n 2=3 log O(1) n n log 3 n ....

....sets S of n segments and sets P of m points in the plane. Aronov et al. 2] proved that j(n; m) O i m 2=3 n 2=3 n log m n ff(n) j : A randomized algorithm with expected running time O(m 2=3 Gamma n 2=3 2 log n nff(n) log 2 n log m) is described by Edelsbrunner et al. [12], and a slightly faster deterministic algorithm is presented by Agarwal [1] Following the same strategy as for the case of lines, we first develop a randomized algorithm with O( m 2 n log m n ff(n) log n) expected running time. Let us remark that the above upper bound for j(n; m) is not known ....

H. Edelsbrunner, L. Guibas and M. Sharir, The complexity of many faces in arrangements of lines and of segments, Discrete and Computational Geometry 5 (1990), 161--196.

....and Welzl [13] Szemer edi and Trotter [20] In this paper we study the problem of computing A(L; P ) that is, for each cell C 2 A(L; P ) we want return the vertices of C in, say, clockwise order. We will refer to the cells of A(L; P ) as the marked cells of A(L) Edelsbrunner et al. [12] presented a randomized algorithm, based on the random sampling technique [16] for computing A(L; P ) whose expected running time was O(m 2=3 Gamma n 2=3 2 log n n log n log m) for any fixed 0. A deterministic algorithm with running time O(m 2=3 n 2=3 log O(1) n n log 3 n ....

.... proved that j(n; m) O i m 2=3 n 2=3 n log m n ff(n) j : A randomized algorithm with expected running time O(m 2=3 Gamma n 2=3 2 log n nff(n) log 2 n log m) Computing Many Faces April 13, A generalization of Chazelle Friedman lemma 3 is described by Edelsbrunner et al. [12], and a slightly faster deterministic algorithm is presented by Agarwal [1] Following the same strategy as for the case of lines, we first develop a randomized algorithm with O( m 2 n log m n ff(n) log n) expected running time. Let us remark that the above upper bound for j(n; m) is not known ....

H. Edelsbrunner, L. Guibas and M. Sharir, The complexity of many faces in arrangements of lines and of segments, Discrete and Computational Geometry 5 (1990), 161--196.

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