## Applications of parametric searching in geometric optimization (1994)

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Venue: | J. Algorithms |

Citations: | 91 - 20 self |

### BibTeX

@ARTICLE{Agarwal94applicationsof,

author = {Pankaj K. Agarwal and Micha Sharir},

title = {Applications of parametric searching in geometric optimization},

journal = {J. Algorithms},

year = {1994},

volume = {17},

pages = {292--318}

}

### Years of Citing Articles

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### Abstract

z Sivan Toledo x

### Citations

1843 |
Computational Geometry: An Introduction
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Citation Context ...ng those vertices of Ulson(ffi) (resp. Urson(ffi)) lying outside Urson(ffi) (resp. Ulson(ffi)). To compute the intersection points of R and B, we construct a segment tree T on the edges of R [ B; see =-=[37]-=- for details. Each node v of T is associated with a horizontal interval \Delta v, a subset Rv of edges in R, and a subset Bv of edges in B. The x-projection of any edge in Rv [ Bv covers the interval ... |

396 | Applicationsof randomsampling in computational geometry
- Clarkson, Shor
- 1989
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Citation Context ...log m). By flipping the roles of G0 and H0, we can obtain another + m0 log n0). solution whose running time is O(n 4+ɛ 0 To obtain a further improved solution, we use the technique of random sampling=-= [16]: -=-we choose a random sample of r surfaces Fη, for some constant parameter r, and triangulate their lower envelope into roughly O(r4+ɛ ) cells, as described above. With high probability each cell inter... |

388 |
Quantifier elimination for real closed fields by cylindrical algebraic decomposition
- Collins
- 1975
(Show Context)
Citation Context ...obtain subquadratic solutions. Chazelle and Sharir [18] have given such a subquadratic solution, which runs in time O(n 1:9999 ) and is based on Collins' cylindrical algebraic decomposition technique =-=[22]-=-. The running time of their algorithm can be improved to O(n 1:9 ) using the results of Chazelle et al. [14] on point location among algebraic surfaces. In this section we give a considerably improved... |

293 |
Triangulating a Simple Polygon in Linear Time
- Chazelle
- 1991
(Show Context)
Citation Context ...i is a convex planar subdivision having O(n) faces, edges and vertices, and it can be computed in O(n log n) time (actually, in O(n) time, using the recent polygon triangulation algorithm of Chazelle =-=[13]-=-). P1 5 6 7 8 `e a (a; 4) (a; 2) (a; 1) (a; 5) (a; 7) (a; 8) M1(a) ` \LambdasP2 1 2 3 4 (a; 6) (a; 3) Figure 1: Polygon and its visibility map It is easy to check that a biggest stick B placed inside ... |

282 |
Algorithms for reporting and counting geometric intersections
- Bentley, Ottmann
- 1979
(Show Context)
Citation Context ... constructing the entire arrangement of the arcs defining the edges of these sets. Finally, an intersection between Kc QP and −Kc PQ can be detected in O((mn) 2 log mn) time by a sweep-line algorith=-=m [10]. H-=-ence, we can conclude Theorem 4.3 Given a collection P of m non-intersecting segments and another collection Q of n non-intersecting segments in the plane, one can determine whether D(P, Q) ≤ δ in ... |

263 | Epsilon-nets and simplex range queries
- Haussler, Welzl
- 1987
(Show Context)
Citation Context ..., if every (open) cell of constant complexity, of the form obtained in the stratification algorithm of [14], which does not intersect any surface of R, intersects at most n=r surfaces of \Sigma ; see =-=[27]-=- for a more formal definition. Haussler and Welzl [27] showed that a random subset of \Sigmasof size O(r log r) is a 1 r -net with high probability. Later Matou^sek [32] gave an O(nr O(1) )-time deter... |

239 | Applying Parallel Computation Algorithms in the Design of Serial Algorithms
- Megiddo
- 1983
(Show Context)
Citation Context ...ersity § School of Mathematical Sciences, Tel Aviv University 0 1 Introduction In this paper we present several applications in computational geometry of the parametric searching technique of Megiddo=-= [32]-=-. This technique, which we briefly review below, is a powerful and ingenious tool for solving efficiently a variety of optimization problems. Although it has been applied successfully to several probl... |

203 |
Linear-time algorithms for linear programming in R3 and related problems
- Megiddo
- 1983
(Show Context)
Citation Context ...istance from e to the points of S.” The problem was posed by D.T. Lee a few years ago. It generalizes the well known notions of a point center (which is the center of the smallest enclosing disk of =-=S [33]-=-) and of a line-center. Finding the point center and the line center of a set S is easy; Page 7sthe segment-center problem appears to be more difficult. Using parametric searching, we present in this ... |

198 | Linear programming in linear time when the dimension is fixed
- MEGIDDO
- 1953
(Show Context)
Citation Context ...n matrix, whose elements satisfy certain monotonicity properties. There are various other extensions of the technique. In particular, Megiddo’s subsequent linear-time algorithm for linear programmin=-=g [34]-=- can be regarded as an optimized variant of the parametric searching technique. Since its design, about 8 years ago, the parametric searching technique have been successfully applied to a variety of o... |

140 |
On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
- Kedem, Livne, et al.
- 1986
(Show Context)
Citation Context ...e 3). Since the relative interiors of the segments in P do not intersect each other, the boundaries of eδ, e ′ δ , for e, e ′ ∈ P, intersect in at most two points (assuming general position of=-= e, e ′ [28]; in -=-any case, it is easy to show that the intersection of the boundaries of eδ, e ′ δ consists of at most two connected components). Therefore, by the result of [28], Pδ has only O(m) Page 6sedges (a... |

127 | Ray shooting and parametric search
- Agarwal, Matousek
- 1992
(Show Context)
Citation Context ...m [18], computing the center of a set of points in 2 and 3 dimensions [35], selecting distances in the plane [1], certain 2-center problems for planar point sets [3], range searching and ray shooting =-=[2]-=-, and extremal polygon containment problems [6]. This is still a relatively small crop, given the large body of literature on geometric optimization problems. In this paper we demonstrate the power of... |

108 |
Parallelism in comparison problems
- Valiant
- 1975
(Show Context)
Citation Context ... P , for a total of P Tp + TpTs log P . In most cases the second term dominates the running time. (Since the parallel algorithm is simulated sequentially, we can allow the comparison model of Valiant =-=[38]-=-, which measures parallelism only in terms of comparisons being made, and ignores all other operations. This observation simplifies the technique considerably.) This brief overview of parametric searc... |

107 | Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes
- Hart, Sharir
- 1986
(Show Context)
Citation Context ...ew of the above lemma, Upi , Lpi can be computed by a sequential algorithm in in O(nλ3(n) log n) time [24], or by a parallel algorithm in time O(log n) using O(λ4(n)) processors. It has been shown i=-=n [23, 5] that λ3(-=-n) = Θ(nα(n)) and λ4(n) = Θ(n2α(n) ). After having computed Upi , Lpi , a ‘good’ point can also be determined within the same time bound. Hence, the fixed size problem can be solved in time O... |

106 |
Slowing down sorting networks to obtain faster sorting algorithms
- Cole
- 1987
(Show Context)
Citation Context ... This observation simplifies the technique considerably.) This brief overview of parametric searching does not cover all aspects of the technique. Among the issues we left out are a trick due to Cole =-=[17],-=- which in certain cases improves the running time of the procedure by a logarithmic factor, a variant due to Matouˇsek [31] and others which replaces in certain applications the parallel generic algo... |

98 | An efficient parallel biconnectivity algorithm
- Tarjan, Vishkin
- 1985
(Show Context)
Citation Context ...l graph of the arrangement, and then a spanning tree of the dual graph, which we convert to an Eulerian path. Once we have an Eulerian path, we can compute cf using any parallel prefix algorithm. See =-=[1, 37] for-=- details. The total time spent in computing cf is O(log m) using O(m 2 ) processors. It is easily seen that an edge γ of the arrangement is in Pδ if cf = 0 for one of the faces adjacent to γ. Pδ t... |

91 |
The upper envelope of Voronoi surfaces and its applications. Discrete Comput
- Huttenlocher, Kedem, et al.
- 1993
(Show Context)
Citation Context ...mum Hausdorff distance under translation between two polygonal regions in the plane under the Euclidean metric. This is a hard instance of a general pattern matching problem. It was left untreated in =-=[26]-=-, and solved by a brute-force inefficient method in [8]. We solve it in time O((mn) 2 log 3 (mn)), where m and n are the number of edges of the given polygons. This is about 3 orders of magnitude fast... |

79 |
Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences
- Agarwal, Sharir, et al.
- 1989
(Show Context)
Citation Context ...ew of the above lemma, Upi , Lpi can be computed by a sequential algorithm in in O(nλ3(n) log n) time [24], or by a parallel algorithm in time O(log n) using O(λ4(n)) processors. It has been shown i=-=n [23, 5] that λ3(-=-n) = Θ(nα(n)) and λ4(n) = Θ(n2α(n) ). After having computed Upi , Lpi , a ‘good’ point can also be determined within the same time bound. Hence, the fixed size problem can be solved in time O... |

77 |
Visibility and intersection problems in plane geometry
- Chazelle, Guibas
- 1985
(Show Context)
Citation Context ...tting first (the interior of) some fixed edge of Pi. It is easy to show that each Mi is a convex subdivision having O(n) faces, edges and vertices, and that it can be computed in O(n log n) time; see =-=[14]-=-. Now let us fix a length d > 0, and consider the subproblem of determining whether a line segment of length d can be placed inside P so that it also intersects e. It is easy to show that if such a pl... |

76 | Efficient partition trees - Matoušek - 1992 |

72 |
Approximate matching of polygonal shapes
- ALT, BEHRENDS, et al.
- 1995
(Show Context)
Citation Context ...ygonal regions in the plane under the Euclidean metric. This is a hard instance of a general pattern matching problem. It was left untreated in [26], and solved by a brute-force inefficient method in =-=[8]. -=-We solve it in time O((mn) 2 log 3 (mn)), where m and n are the number of edges of the given polygons. This is about 3 orders of magnitude faster than the algorithm of [8]. • Solving the 1-segment c... |

65 | Generalized selection and ranking: sorted matrices - Frederickson, Johnson - 1984 |

61 |
Cascading Divide-and-Conquer: A Technique for Designinng Parallel Algorthms
- Atallah, Cole, et al.
(Show Context)
Citation Context ...KPQ in O(log mn) time using O((mn) 2 ) processors. Finally, an intersection between Kc QP and −Kc PQ can be detected in O(log mn) time with O((mn) 2 ) processors using the algorithm of Atallah et al=-=. [9]-=-. Applying the parametric search technique to the resulting algorithm, we can thus conclude Theorem 4.4 Given a collection P of m non-intersecting segments and another collection Q of n non-intersecti... |

50 |
Algorithms for bichromatic line segment problems and polyhedral terrains
- Chazelle, Edelsbrunner, et al.
- 1994
(Show Context)
Citation Context ...arcs, corresponding to the edges of M2, determine whether there exists a pair of arcs, γ ∈ G, η ∈ H, such that η passes above γ. We solve this problem in two stages. First, following the techn=-=ique of [13], we-=- construct a ‘hereditary segment tree’ on the x-projections of the edges of M1 and M2. In particular, we construct a segment tree T on the interval decomposition of the x-axis induced by the x-coo... |

49 |
Approximations and optimal geometric divide-andconquer
- Matoušek
- 1991
(Show Context)
Citation Context ...t n=r surfaces of \Sigma ; see [27] for a more formal definition. Haussler and Welzl [27] showed that a random subset of \Sigmasof size O(r log r) is a 1 r -net with high probability. Later Matou^sek =-=[32]-=- gave an O(nr O(1) )-time deterministic algorithm for computing a 1 r -net of size O(r log r). Applications of Parametric Searching June 17, 2002sThe Biggest Stick Problem 10 Note that P o/ io/ = i an... |

47 | Efficient hidden surface removal for objects with small union size
- Katz, Overmars, et al.
- 1992
(Show Context)
Citation Context ...visibility map of the spheres, as seen from p, is O(n). The idea is now to compute the visibility map of the spheres from each center ci, using the outputsensitive hidden surface removal algorithm of =-=[27]-=-. The algorithm runs in time O((n+k) log 2 n), where k is the combinatorial complexity of the visibility map. If the algorithm runs for too long, we stop it and conclude, in view of the preceding lemm... |

39 |
The polygon containment problem
- Chazelle
- 1983
(Show Context)
Citation Context ... of determining whether P ≡ conv(S) can be placed (by translations and rotations) inside ed; see Figure 4. The problem has thus been reduced to a polygon containment problem, such as those studied i=-=n [6, 11, 30]-=-, with the twist that the environment ed in which P has to be placed is not polygonal. Nevertheless, techniques similar to those used in [30, 36], which compute all critical free placements of P , can... |

37 |
width, closest line pair, and parametric searching, Discrete Comput
- Chazelle, Edelsbrunner, et al.
- 1993
(Show Context)
Citation Context ...model. Consider the first algorithm. Since r is chosen to be some constant, the set R can be computed in polylogarithmic -2time using O(n)[2]: Micha! WHat is the exponent?? processors as described in =-=[16]-=-, and the sets \Sigma o/ and So/ can be computed in constant parallel time using a linear number of processors. We then obtain a collection of independent subproblems, which can all be processed in pa... |

32 |
A singly exponential stratification scheme for real semi-algebraic varieties and its applications
- Chazelle, Edelsbrunner, et al.
- 1991
(Show Context)
Citation Context ... Let M ∗ 2,i (v) ⊆ M ∗ 2 (v) be the set of segments h for which M1,i(v) is used to represent the set of segments in M1(v) inter(v). It has sected by h. Similarly define M2,i(v), M ∗ 1,i been s=-=hown in [12, 3] that -=-the intersection points of segments in M1,i(v), M ∗ 2,i (v), and in M2,i(v), M ∗ 1,i (v) contain all intersection points of M1, M2. Consider one of the subsets, say M1,i(v), M ∗ 2,i (v). Since e... |

32 |
Roundness algorithms using the Voronoi diagrams
- Ebara, Fukuyama, et al.
- 1989
(Show Context)
Citation Context ...nnulus that contains a given set of n points in the plane. We give a randomized algorithm with expected running time O(n 8/5+ɛ ), for any ɛ > 0, considerably improving the quadratic-time algorithm o=-=f [19]. -=-• Finding the minimum Hausdorff distance under translation between two polygonal regions in the plane under the Euclidean metric. This is a hard instance of a general pattern matching problem. It wa... |

32 |
Optimal algorithms for tree partitioning
- Frederickson
- 1991
(Show Context)
Citation Context ... [31] and others which replaces in certain applications the parallel generic algorithm by a randomized (sequential) one, leading to simplified solutions, and a variant due to Frederickson and Johnson =-=[20, 21], -=-where the optimal solution d ⋆ is an element of an implicitly given matrix, whose elements satisfy certain monotonicity properties. There are various other extensions of the technique. In particular... |

30 | Randomized optimal algorithm for slope selection - Matousek - 1991 |

27 | Divide-and-conquer in multidimensional space
- Bentley, Shamos
- 1976
(Show Context)
Citation Context ...nters of the) processors are predetermined, and we want to determine how large can the processors be if every pair is to be able to communicate optically. By running any closest-pair algorithm, e.g., =-=[10]-=-, we can determine in O(n log n) time the largest radius r0 so that the interiors of all spheres of radius r0 centered at the points of P are pairwise disjoint. Therefore, we only have to determine th... |

23 | Using approximation algorithms to design parallel algorithms that may ignore processor allocation - Goodrich - 1991 |

21 |
Selecting distances in the plane
- Agarwal, Aronov, et al.
- 1993
(Show Context)
Citation Context ...efly review below, is a powerful and ingenious tool for solving efficiently a variety of optimization problems. Although it has been applied successfully to several problems in computational geometry =-=[1, 3, 6, 18]-=-, its potential for problems in geometric optimization does not seem to be widely recognized as yet. Many problems of this kind, which could be easily attacked by the technique, are either solved by m... |

21 |
On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space, Discrete Comput
- Leven, Sharir
- 1987
(Show Context)
Citation Context ... of determining whether P ≡ conv(S) can be placed (by translations and rotations) inside ed; see Figure 4. The problem has thus been reduced to a polygon containment problem, such as those studied i=-=n [6, 11, 30]-=-, with the twist that the environment ed in which P has to be placed is not polygonal. Nevertheless, techniques similar to those used in [30, 36], which compute all critical free placements of P , can... |

20 |
Finding the upper envelope of n line segments
- Hershberger
- 1989
(Show Context)
Citation Context ...of Upi , Lpi have O(λ4(n)) breakpoints. See the full version for a proof of the above lemma. In view of the above lemma, Upi , Lpi can be computed by a sequential algorithm in in O(nλ3(n) log n) tim=-=e [24], or by a p-=-arallel algorithm in time O(log n) using O(λ4(n)) processors. It has been shown in [23, 5] that λ3(n) = Θ(nα(n)) and λ4(n) = Θ(n2α(n) ). After having computed Upi , Lpi , a ‘good’ point can... |

20 | Extremal polygon containment problems
- Toledo
- 1991
(Show Context)
Citation Context ...sity; Current address: Laboratory for Computer Science, Massachusetts Institute of Technology. 1sIntroduction 2 Although it has been applied successfully to several problems in computational geometry =-=[1, 2, 3, 4, 20, 21, 36, 38]-=-, its potential for problems in geometric optimization does not seem to be widely recognized as yet. Many problems of this kind, which could be easily attacked by the technique, are either solved by m... |

18 | Planar geometric location problems
- Agarwal, Sharir
- 1990
(Show Context)
Citation Context ...efly review below, is a powerful and ingenious tool for solving efficiently a variety of optimization problems. Although it has been applied successfully to several problems in computational geometry =-=[1, 3, 6, 18]-=-, its potential for problems in geometric optimization does not seem to be widely recognized as yet. Many problems of this kind, which could be easily attacked by the technique, are either solved by m... |

18 |
Randomized optimal algorithm for slope selection
- Matouˇsek
- 1991
(Show Context)
Citation Context ...ects of the technique. Among the issues we left out are a trick due to Cole [17], which in certain cases improves the running time of the procedure by a logarithmic factor, a variant due to Matouˇsek=-= [31]-=- and others which replaces in certain applications the parallel generic algorithm by a randomized (sequential) one, leading to simplified solutions, and a variant due to Frederickson and Johnson [20, ... |

15 |
An algorithm for generalized point location and its applications
- Chazelle, Sharir
- 1990
(Show Context)
Citation Context ...iggest line segment that can be placed inside a simple n-gon. We present a randomized algorithm with expected running time O(n 8/5+ɛ ), for any ɛ > 0, considerably improving the previous algorithm o=-=f [15] wh-=-ose running time is O(n 1.999878 ). 1 • Computing the smallest-width annulus that contains a given set of n points in the plane. We give a randomized algorithm with expected running time O(n 8/5+ɛ ... |

15 |
Efficiently computing the Hausdorff distance for point sets under translation
- Huttenlocher, Kedem
- 1990
(Show Context)
Citation Context ...olygons.. The value of D(P, Q) gives a measure of the resemblance between P and Q, so its (efficient) computation has applications in pattern recognition, computer vision, etc. Huttenlocher and Kedem =-=[25] sho-=-wed that if P and Q are sets of points, then D(P, Q) can be computed in O((mn) 2 α(mn)) time, where α(·) is the inverse Ackermann function. This bound has been recently improved to O(mn(m+n)α(mn) ... |

10 | Counting circular arc intersections
- Agarwal, Sharir
- 1991
(Show Context)
Citation Context ...the recursion, one can show that the expected running time of this step is O(m 4/5+ɛ 0 n 4/5+ɛ 0 + (m0 + n0) 1+ɛ ) for any ɛ > 0. We omit the details of the analysis; similar analyses can be found=-= in [4]. We ne-=-xt sum this bound over all subproblems. Since we have � m0 = � n0 = O(n log 2 n), we readily conclude that the overall running time of the algorithm is O(n8/5+ɛ′ ) for another, still arbitraril... |

6 |
Optimal slope selection
- Cole, Salowe, et al.
- 1989
(Show Context)
Citation Context ...efly review below, is a powerful and ingenious tool for solving efficiently a variety of optimization problems. Although it has been applied successfully to several problems in computational geometry =-=[1, 3, 6, 18]-=-, its potential for problems in geometric optimization does not seem to be widely recognized as yet. Many problems of this kind, which could be easily attacked by the technique, are either solved by m... |

6 | Computing a segment-center for a planar point set - Agarwal, Efrat, et al. - 1993 |

5 |
Finding the kth shortest paths and p-centers by generating and searching good data structures
- Frederickson, Johnson
- 1983
(Show Context)
Citation Context ... [31] and others which replaces in certain applications the parallel generic algorithm by a randomized (sequential) one, leading to simplified solutions, and a variant due to Frederickson and Johnson =-=[20, 21], -=-where the optimal solution d ⋆ is an element of an implicitly given matrix, whose elements satisfy certain monotonicity properties. There are various other extensions of the technique. In particular... |

3 |
The biggest diagonal in a simple polygon
- Aggarwal, Suri
- 1990
(Show Context)
Citation Context ...ction we give a considerably improved solution, whose running time is O(n 8=5+ffl ). We note that if the endpoints of the stick are constrained to lie at vertices of P then a faster solution is known =-=[6]-=-. Our solution is based on the following approach, also used by the previous algorithms mentioned above. We find, in linear time, a chord e that partitions P into two subpolygons, P1, P2, such that ea... |

2 |
Computing the center of a point set in three dimensions
- Naor, Sharir
- 1990
(Show Context)
Citation Context ...ssfully applied to a variety of optimization problems. In computational geometry it has been applied to the slope selection problem [18], computing the center of a set of points in 2 and 3 dimensions =-=[35]-=-, selecting distances in the plane [1], certain 2-center problems for planar point sets [3], range searching and ray shooting [2], and extremal polygon containment problems [6]. This is still a relati... |

2 |
A polygon cutting theorem
- Chazelle
- 1982
(Show Context)
Citation Context ...lowing approach, also used by the previous algorithms mentioned above. We find, in linear time, a chord e that partitions P into two subpolygons, P1, P2, such that each contains at most 2n=3 vertices =-=[11]-=-. We recursively find the biggest stick in P1 and in P2. Then we compute the biggest stick within P which crosses e, and the final answer is the largest of these three candidate sticks. To compute a b... |

1 |
1-Segment covering problem
- Imai, Lee, et al.
- 1989
(Show Context)
Citation Context ...zes the largest distance from the segment to the given points. We present an algorithm for this problem whose time complexity is O(n 2 α(n) log 3 n). It improves the previous algorithm of Imai et al.=-= [29] by -=-roughly two orders of magnitude. 1 Throughout this paper, ɛ denotes an arbitrarily small positive constant. The multiplicative constants in the asymptotic bounds may depend on ɛ. Page 2s• Solving ... |

1 |
Extremal polygon containment problems, manuscript
- Sharir, Toledo
- 1991
(Show Context)
Citation Context ...lygon containment problem, such as those studied in [6, 11, 30], with the twist that the environment ed in which P has to be placed is not polygonal. Nevertheless, techniques similar to those used in =-=[30, 36]-=-, which compute all critical free placements of P , can be developed. Without loss of generality we assume that ed is placed such that e lies on the x-axis and its enpoints are at (0, 0) and (1, 0). I... |