## Computing faces in segment and simplex arrangements (1995)

Venue: | In Proc. 27th Annu. ACM Sympos. Theory Comput |

Citations: | 16 - 10 self |

### BibTeX

@INPROCEEDINGS{Amato95computingfaces,

author = {Nancy M. Amato and Michael T. Goodrich and Edgar A. Ramos},

title = {Computing faces in segment and simplex arrangements},

booktitle = {In Proc. 27th Annu. ACM Sympos. Theory Comput},

year = {1995},

pages = {672--682}

}

### Years of Citing Articles

### OpenURL

### Abstract

For a set S of n line segments in the plane, we give the first work-optimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log 2 n) time using O(n log n + k) work in the EREW PRAM model, where k is the number of intersecting line segment pairs, and provides a fairly simple divide-and-conquer alternative to the optimal sequential “plane-sweep ” algorithm of Chazelle and Edelsbrunner. Moreover, our method can be used to output all k intersecting pairs while using only O(n) working space, which solves an open problem posed by Chazelle and Edelsbrunner. We also describe a sequential algorithm for computing a single face in an arrangement of n line segments that runs in O(n 2 (n) log n) time, which improves on a previous O(n log 2 n) time algorithm. For collections of simplices in IR d, we give methods for constructing a set of m = O(n d,1 log c n+k) cells of constant descriptive complexity that covers their arrangement, where c> 1 is a constant and k is the number of faces in the arrangement. The construction is performed sequentially in O(m) time, or in O(log n) time using O(m) work in the EREW PRAM model. The covering can be augmented to answer point location queries in O(log n) time. In addition to supplying the first parallel methods for these problems, we improve on the previous best sequential methods by reducing the query times (from O(log 2 n) in IR 3 and O(log 3 n) in IR d, d>3), and also the size and construction cost of the covering (from O(n d,1+ + k)). 1

### Citations

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Citation Context ...intersections and constructing their arrangement that runs in O(n log n + k) time and uses a number of beautiful techniques, including plane sweeping and topological sweeping. A number of researchers =-=[12, 17, 33, 34]-=-, have given elegant randomized methods that run in O(n log n + k) expected time. In fact, if k n log 1+ n for some constant >0, then these methods run in this bound with high probability 2 [31]. In t... |

289 |
Computational Geometry: An Introduction Through Randomized Algorithms
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Citation Context ...intersections and constructing their arrangement that runs in O(n log n + k) time and uses a number of beautiful techniques, including plane sweeping and topological sweeping. A number of researchers =-=[12, 17, 33, 34]-=-, have given elegant randomized methods that run in O(n log n + k) expected time. In fact, if k n log 1+ n for some constant >0, then these methods run in this bound with high probability 2 [31]. In t... |

168 | An optimal algorithm for intersecting line segments in the plane
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- 1992
(Show Context)
Citation Context ...eing that of constructing a triangulation of the arrangement of the segments in S. This problem has been studied extensively in the computational geometry literature [6, 8]. Chazelle and Edelsbrunner =-=[11]-=- gave an optimal method for computing segment intersections and constructing their arrangement that runs in O(n log n + k) time and uses a number of beautiful techniques, including plane sweeping and ... |

150 |
editor. Synthesis of Parallel Algorithms
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Citation Context ...). Using randomization, the time is O(log n log log n) with n-polynomial probability. All steps in stage i can be parallelized in a straight forward manner using known parallel techniques (see, e.g., =-=[4, 37]-=-). In particular, stage i runs in O(log 2 ri) time using optimal work. Also, at each round it suffices to perform one single global processor allocation call. If a model that allows non-global process... |

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Citation Context ...ollection of all the hyperplanes bounding the linear cells ( ) for 2T(R). jHjis O(r D ). We can construct a point location data structure D for H to achieve an O(log n) query time with size O(jHj D ) =-=[9]-=-, but we can also use the following less-efficient construction: For each leaf l of D (its number is O(jHj D )), determine the list L(l) of each cell 2T(R)for which ( ) contains the cell in the arrang... |

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Citation Context ...intersections and constructing their arrangement that runs in O(n log n + k) time and uses a number of beautiful techniques, including plane sweeping and topological sweeping. A number of researchers =-=[12, 17, 33, 34]-=-, have given elegant randomized methods that run in O(n log n + k) expected time. In fact, if k n log 1+ n for some constant >0, then these methods run in this bound with high probability 2 [31]. In t... |

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Citation Context ...lgorithms are known for bothproblems on segments we are considering. Here, we point out that for both, algorithms with equal asymptotic performance can be obtained using the approach used by Chazelle =-=[10]-=- and Brönnimman et al [7] for half-space intersection. The sampling is global, that is, at a given stage a single global sample determines the sample in each subproblem, rather than an independent sam... |

76 | Efficient partition trees - Matoušek - 1992 |

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Citation Context ...et Tp(R) be the trapezoidal diagram T (R) restricted to fp(R). The complexity of fp(S) is O(n (n)) where is the very slowly-growing inverse of Ackerman’s function [24, 36]; this bound can be achieved =-=[40]-=-. The same bound applies to Tp(S). For Tp(R) monotonicity but not locality holds. Theorem 2.4 directly implies the following. Theorem 2.7 For 0 < <1, there are constants C; r0; such that for r0 r n , ... |

55 |
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Citation Context ...vertices and edges on its boundary. Let Tp(R) be the trapezoidal diagram T (R) restricted to fp(R). The complexity of fp(S) is O(n (n)) where is the very slowly-growing inverse of Ackerman’s function =-=[24, 36]-=-; this bound can be achieved [40]. The same bound applies to Tp(S). For Tp(R) monotonicity but not locality holds. Theorem 2.4 directly implies the following. Theorem 2.7 For 0 < <1, there are constan... |

48 | Product range spaces, sensitive sampling, and derandomization - Brönnimann, Chazelle, et al. - 1993 |

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Citation Context ...heorem below. We state it in some generality, and then specialize it to our particular problem. For completeness, we give a detailed proof. It uses standard techniques in geometric sampling (see e.g. =-=[1, 4, 17, 21, 29]-=-). For sequential computation, the r factor in property (ii) of the theorem can be improved to log r using derandomization by the method of conditional probabilities. However, we prefer our form for t... |

38 |
Efficient partition trees
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Citation Context ...2. Efficient construction For our purposes, we need a faster construction of a sample R. This can be achieved using -approximations which can be constructed efficiently using a technique of Matouˇsek =-=[30]-=-.Approximations. Apair(X;F), whereX Xis as before and F is a family of subsets of IR d defines a range space (X;R(X;F)) where R(X;F) =fT X: T =fx2X: \x6=;g for 2Fg. A X is a (1=r)-approximation for (... |

33 | On lazy randomized incremental construction
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Citation Context ...nverse of Ackerman’s function. The best previously known deterministic algorithm is an O(n log 2 n) method due to Mitchell [32], yet there are randomized ones that run in expected time O(n (n) log n) =-=[12, 18]-=-. The known lower bound is Ω(n log n). We describe an almost-optimal deterministic algorithm, in that it runs in time O(n 2 (n) log n). The algorithm uses a divide-and-conquer approach based on determ... |

31 |
Geometric partitioning made easier, even in parallel
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- 1993
(Show Context)
Citation Context ...heorem below. We state it in some generality, and then specialize it to our particular problem. For completeness, we give a detailed proof. It uses standard techniques in geometric sampling (see e.g. =-=[1, 4, 17, 21, 29]-=-). For sequential computation, the r factor in property (ii) of the theorem can be improved to log r using derandomization by the method of conditional probabilities. However, we prefer our form for t... |

29 | Separating two simple polygons by a sequence of translations - Pollack, Sharir, et al. - 1987 |

26 | A deterministic algorithm for the threedimensional diameter problem
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(Show Context)
Citation Context ...heorem below. We state it in some generality, and then specialize it to our particular problem. For completeness, we give a detailed proof. It uses standard techniques in geometric sampling (see e.g. =-=[1, 4, 17, 21, 29]-=-). For sequential computation, the r factor in property (ii) of the theorem can be improved to log r using derandomization by the method of conditional probabilities. However, we prefer our form for t... |

25 |
Partitioning arrangements of lines
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Citation Context ...by n = O(n + Theorem 2.4. By construction P 2T( ) k (r =n )). For each , step 2.1 is trivially performed in time O(n ), step 2.2 is performed in time O(n log n ) using well-known techniques (see e.g. =-=[2, 20]-=-), step 2.3 takes time as indicated above. Thus, steps 1-2 for taketime O((n +k (r =n )) log r + k = ). Using the invariant in the previous stage, this is O(((n + k= )+k= ) logri) where ri = max r and... |

25 | Randomized parallel algorithms for trapezoidal diagrams
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(Show Context)
Citation Context ...n in O(n log n + k) expected time. In fact, if k n log 1+ n for some constant >0, then these methods run in this bound with high probability 2 [31]. In the parallel domain, Clarkson, Cole, and Tarjan =-=[16, 15]-=- show that one can construct a segment arrangement in parallel in O(log n) time and O(n log n + k) expected work in the CRCW PRAM model. 3 There is no previous deterministic optimal-work parallel algo... |

24 | Optimal randomized parallel algorithms for computational geometry
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Citation Context ...bility when locality does not apply; and finally the method of limited independence is computationally simpler. We need the following tail estimate for random variables with limited independence (see =-=[38, 34]-=-). Lemma 2.1 Let I = P m j=1 Im be the sum of m 2K-wise independent (K fixed), identical 0-1 random variables (i.e. each variable is 1 with probability p > 0 and0otherwise). Let = E[I] = mp. Then, for... |

21 |
Reporting and counting segment intersections
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(Show Context)
Citation Context ... slightly more difficult variant being that of constructing a triangulation of the arrangement of the segments in S. This problem has been studied extensively in the computational geometry literature =-=[6, 8]-=-. Chazelle and Edelsbrunner [11] gave an optimal method for computing segment intersections and constructing their arrangement that runs in O(n log n + k) time and uses a number of beautiful technique... |

19 |
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(Show Context)
Citation Context ... some special cases [20, 22, 23, 39]. We show how to solve the problem of computing a segment arrangement in O(log 2 n) time and O(n log n+k) work 1 For background material on geometric sampling, see =-=[3, 14, 28, 34]-=-. 2 We say that an event parameterized by n holds “with high probability” if its probability is at least 1 , 1=n for some constant >0. 3 The CRCW PRAM is the synchronous shared-memory model that allow... |

18 | Epsilon-nets and computational geometry - Matoušek - 1993 |

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Citation Context ... 3 There is no previous deterministic optimal-work parallel algorithm for the general segment intersection problem, however. The best previous methods for the general problem are a method of Goodrich =-=[20]-=-, which runs in O(log n) time and O(n log 2 n + k log n) work in the CREW PRAM model and a method of Rüb, which runs in O(logn log log n) time using O((n + k) log n loglog n) work in the same parallel... |

14 | Vertical decompositions for triangles in 3-space
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Citation Context ... ,foranyfixedd3, e.g., triangles in IR 3 , or tetrahedra in IR 4 . When the simplices in S may intersect, the complexity k of A(S) can vary between n and nd . For d = 3, de Berg, Guibas, and Halperin =-=[19]-=- build a vertical decomposition D of A(S) of size O(n2+ + k) in O(n2+ + jDj logn) time, for any constant >0; it supports point location queries in O(log 2 n) time. For d 3, Pellegrini [35] constructs ... |

13 |
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Citation Context ...thod of Rüb, which runs in O(logn log log n) time using O((n + k) log n loglog n) work in the same parallel model. One can achieve an optimal O(n logn + k) work bound, however, for some special cases =-=[20, 22, 23, 39]-=-. We show how to solve the problem of computing a segment arrangement in O(log 2 n) time and O(n log n+k) work 1 For background material on geometric sampling, see [3, 14, 28, 34]. 2 We say that an ev... |

12 |
Computing a face in an arrangement of line segments
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Citation Context |

12 | An optimal randomized logarithmic time connectivity algorithm for the erew pram - Halperin, Zwick - 1994 |

12 |
Derandomization in computational geometry
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Citation Context ... some special cases [20, 22, 23, 39]. We show how to solve the problem of computing a segment arrangement in O(log 2 n) time and O(n log n+k) work 1 For background material on geometric sampling, see =-=[3, 14, 28, 34]-=-. 2 We say that an event parameterized by n holds “with high probability” if its probability is at least 1 , 1=n for some constant >0. 3 The CRCW PRAM is the synchronous shared-memory model that allow... |

8 |
Erratum: Randomized parallel algorithms for trapezoidaldiagrams
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(Show Context)
Citation Context ...n in O(n log n + k) expected time. In fact, if k n log 1+ n for some constant >0, then these methods run in this bound with high probability 2 [31]. In the parallel domain, Clarkson, Cole, and Tarjan =-=[16, 15]-=- show that one can construct a segment arrangement in parallel in O(log n) time and O(n log n + k) expected work in the CRCW PRAM model. 3 There is no previous deterministic optimal-work parallel algo... |

7 | On point location and motion planning among simplices
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Citation Context ...nd Halperin [19] build a vertical decomposition D of A(S) of size O(n2+ + k) in O(n2+ + jDj logn) time, for any constant >0; it supports point location queries in O(log 2 n) time. For d 3, Pellegrini =-=[35]-=- constructs a covering5 for A(S) of size O(nd,1+ + k) in O(nd,1+ +k) time, for any constant >0. Using the same space, but with O(nd,1+ + k log n) work, the covering can be augmented to support point l... |

6 |
Computing intersections and arrangements for red-blue curve segments in parallel
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(Show Context)
Citation Context ...thod of Rüb, which runs in O(logn log log n) time using O((n + k) log n loglog n) work in the same parallel model. One can achieve an optimal O(n logn + k) work bound, however, for some special cases =-=[20, 22, 23, 39]-=-. We show how to solve the problem of computing a segment arrangement in O(log 2 n) time and O(n log n+k) work 1 For background material on geometric sampling, see [3, 14, 28, 34]. 2 We say that an ev... |

4 | Tail estimates for the efficiency of randomized incremental algorithms for line segment intersection
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- 1993
(Show Context)
Citation Context ...7, 33, 34], have given elegant randomized methods that run in O(n log n + k) expected time. In fact, if k n log 1+ n for some constant >0, then these methods run in this bound with high probability 2 =-=[31]-=-. In the parallel domain, Clarkson, Cole, and Tarjan [16, 15] show that one can construct a segment arrangement in parallel in O(log n) time and O(n log n + k) expected work in the CRCW PRAM model. 3 ... |

3 | Cutting hyperplanearrangements - Matoušek - 1991 |

2 |
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- Bentley, Ottmann
- 1979
(Show Context)
Citation Context ... slightly more difficult variant being that of constructing a triangulation of the arrangement of the segments in S. This problem has been studied extensively in the computational geometry literature =-=[6, 8]-=-. Chazelle and Edelsbrunner [11] gave an optimal method for computing segment intersections and constructing their arrangement that runs in O(n log n + k) time and uses a number of beautiful technique... |

2 | Lam.Finding connected components in o(logn log logn) time on the EREW PRAM - Chong, W - 1993 |

2 | Addendum to "parallel methods for visibility and shortest path problems in simple polygons - Goodrich, Shauck, et al. - 1993 |

2 | On a set of almost deterministic k-independentrandomvariables - Joffe - 1974 |

2 |
On computing a single face in an arrangementof line segments
- Mitchell
- 1990
(Show Context)
Citation Context ...omplexity of a single face is O(n (n)) where (n) is the very slowly-growing inverse of Ackerman’s function. The best previously known deterministic algorithm is an O(n log 2 n) method due to Mitchell =-=[32]-=-, yet there are randomized ones that run in expected time O(n (n) log n) [12, 18]. The known lower bound is Ω(n log n). We describe an almost-optimal deterministic algorithm, in that it runs in time O... |

2 | A general approach toD-dimensionalgeometric queries - Yao, Yao - 1985 |

2 |
Computingmany faces in arrangements of lines and segments
- Agarwal, Matoušek, et al.
- 1994
(Show Context)
Citation Context ...for the complete arrangement. In some applications where one is interested only in a subset of the arrangement, for example a single face in an arrangement, locality fails. However, it has been noted =-=[1, 18]-=- that in this case a property that we call monotonicity holds: for R; R0 X, if 2T (R)then ∆(R) R and X \ R = ;;andif 2T(R)and R0 R with ∆( ) R0 , then 2T(R0 ). Note that locality implies monotonicity.... |

2 |
An optimal randomizedlogarithmic time connectivityalgorithm for the erew pram
- Halperin, Zwick
- 1994
(Show Context)
Citation Context ...The time is O(log 2 n) because we construct planar line segment arrangements (Theorem 3.2). The construction also uses a parallel algorithm for finding the connected components of a graph (see, e.g., =-=[13, 25]-=-) (details will be provided in the full paper). Triangulating non-intersecting (d,1)–simplices in IR d . If the simplices in S are interior disjoint, then Pellegrini [35] notes that a slight modificat... |

2 |
Cutting hyperplanearrangements
- Matouˇsek
- 1991
(Show Context)
Citation Context ..., and in order to compute the conflict lists of T (R) efficiently, we make use of linearization for (X ; T(X )) (first described by Yao and Yao [41], and introduced in geometric sampling by Matouˇsek =-=[27, 29]-=-). The pair (X ;F) is linearizable if there are maps ' from X into IR d ,and from F into L(IR d ) such that for x 2Xand 2F, x\ 6=;iff '(x) 2 ( ). The mapping ' is given by bounded degree polynomials i... |

1 | Computingmany faces in arrangements of lines and segments - Schwarzkopf - 1994 |

1 |
Product range spaces, sensitive sampling, and derandomization
- HervéBrönnimann, Matouˇsek
- 1993
(Show Context)
Citation Context ...nd (iv) of Theorem 2.4, we need to verify that f (A; r) =O(r+k(r=n) 2 ). A (1=r)-approximation A for (S;R(S; T(S))) is also a (1=r)-approximation for (S;R(S;S)). Then, by a result of Brönnimann et al =-=[7]-=- on product range spaces, A can also beused to estimate the number of intersections in A(S) inside any convex region. More precisely, for a set of segments X, letv(X; ) denote the number of intersect... |

1 |
Lam.Finding connectedcomponents in o(logn log log n) time on the EREW PRAM
- Chong, W
- 1993
(Show Context)
Citation Context ...The time is O(log 2 n) because we construct planar line segment arrangements (Theorem 3.2). The construction also uses a parallel algorithm for finding the connected components of a graph (see, e.g., =-=[13, 25]-=-) (details will be provided in the full paper). Triangulating non-intersecting (d,1)–simplices in IR d . If the simplices in S are interior disjoint, then Pellegrini [35] notes that a slight modificat... |

1 |
On a set of almost deterministick-independentrandomvariables
- Joffe
(Show Context)
Citation Context ...hen monotonicity but not locality holds, we need to make use of a particular construction of the sample space for the 0-1 random variables I1;:::;In that determine R. We use the construction of Joffe =-=[26]-=-: Let be a prime number with n <2n; the sample space is Ω( ; 2K) =Z2K. For 1 i and (a0;:::;a2K,1) 2 Ω( ; 2K), let Xi = P2K,1 j=0 ajij mod and let Ii = 1if0 Xi < 2r. The 0-1 random variables I1;:::;I d... |

1 |
A generalapproachto D-dimensionalgeometric queries
- Yao, Yao
(Show Context)
Citation Context ...on. To apply Matouˇsek’s construction for approximations, and in order to compute the conflict lists of T (R) efficiently, we make use of linearization for (X ; T(X )) (first described by Yao and Yao =-=[41]-=-, and introduced in geometric sampling by Matouˇsek [27, 29]). The pair (X ;F) is linearizable if there are maps ' from X into IR d ,and from F into L(IR d ) such that for x 2Xand 2F, x\ 6=;iff '(x) 2... |