## Algorithmic techniques for geometric optimization (1995)

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Venue: | In Computer Science Today: Recent Trends and Developments, Lecture Notes in Computer Science |

Citations: | 5 - 3 self |

### BibTeX

@INPROCEEDINGS{Agarwal95algorithmictechniques,

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

title = {Algorithmic techniques for geometric optimization},

booktitle = {In Computer Science Today: Recent Trends and Developments, Lecture Notes in Computer Science},

year = {1995},

pages = {234--253},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

### Citations

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The art of computer programming
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Citation Context ... left of x = if and only if (i) > (j). In other words, the number of intersection points to the left of x = can be counted, in O(n log n) time, by counting the number of inversions in the permutation =-=[78]-=-: Construct a balanced binary tree T storing the lines of L in its leaves, in the decreasing slope order `1�:::�`n, by adding the lines one after the other, in the order ` (1)�:::�` (n). When a line `... |

1871 | Randomized Algorithms
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Citation Context ...he past decade, randomized algorithms have been developed for a wide variety of problems in computational geometry and in other elds� see, e.g., the books by Mulmuley [96] and by Motwani and Raghavan =-=[95]-=-. In particular, Clarkson [31] and Seidel [103] gave randomized algorithms for linear programming, whose expected time is linear in any xed dimension, which aremuch simpler than their earlier determin... |

688 |
Algorithms in Combinatorial Geometry
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Citation Context ... the lines, so that no three lines are concurrent, and no two intersection points have the same x-coordinate.) We are thus seeking the k-th leftmost vertexofthearrangement A(L) of the lines in L� see =-=[45, 108]-=- for more details concerning arrangements. (The name of the problem comes from its primal setting, where we are given a set of n points and a parameter k as above, and wish to determine a segment conn... |

647 | A new polynomial-time algorithm for linear programming, Combinatorica 4
- Karmarkar
- 1984
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Citation Context ...constitute an important step toward the still open goal of obtaining strongly-polynomial algorithms for linear programming. (Recall that the polynomial-time algorithms by Khachiyan [73] and Karmarkar =-=[69]-=- are not strongly polynomial, as the number of arithmetic operations performed by these algorithms depends on the size of coe cients of the input constraints.) This new technique is presented in Secti... |

416 | DavenportSchinzel Sequence and Their Geometric Applications
- Sharir, Agarwal
- 1995
(Show Context)
Citation Context ... the lines, so that no three lines are concurrent, and no two intersection points have the same x-coordinate.) We are thus seeking the k-th leftmost vertexofthearrangement A(L) of the lines in L� see =-=[45, 108]-=- for more details concerning arrangements. (The name of the problem comes from its primal setting, where we are given a set of n points and a parameter k as above, and wish to determine a segment conn... |

387 | Applications of random sampling in computational geometry
- Clarkson, Shor
- 1989
(Show Context)
Citation Context ...etric searching has certain drawbacks, which we discuss below. Consequently, there have been several recent attempts to replace parametric searching by alternative techniques, including randomization =-=[11, 33,79]-=-, expander graphs [16, 70,71,72], geometric cuttings [12, 22], and ? Pankaj Agarwal has been supported by National Science Foundation Grant CCR-93{01259, an NYI award, and by matching funds from Xerox... |

375 |
A polynomial algorithm in linear programming
- Khachiyan
- 1979
(Show Context)
Citation Context ...dimension, so they constitute an important step toward the still open goal of obtaining strongly-polynomial algorithms for linear programming. (Recall that the polynomial-time algorithms by Khachiyan =-=[73]-=- and Karmarkar [69] are not strongly polynomial, as the number of arithmetic operations performed by these algorithms depends on the size of coe cients of the input constraints.) This new technique is... |

373 | Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Quantifier elimination and cylindrical algebraic decomposition - Collins - 1993 |

286 |
Computational Geometry: An Introduction Through Randomized Algorithms
- Mulmuley
- 1994
(Show Context)
Citation Context ... techniques reviewed so far. In the past decade, randomized algorithms have been developed for a wide variety of problems in computational geometry and in other elds� see, e.g., the books by Mulmuley =-=[96]-=- and by Motwani and Raghavan [95]. In particular, Clarkson [31] and Seidel [103] gave randomized algorithms for linear programming, whose expected time is linear in any xed dimension, which aremuch si... |

232 | Applying parallel computation algorithms in the design of serial algorithms
- Megiddo
- 1983
(Show Context)
Citation Context ... Although restricted forms of parametric searching already existed earlier [49, 57,58], the parametric searching in its full generality was proposed by Megiddo in the late 1970's and the early 1980's =-=[87, 88]-=-. The technique was originally motivated by so-called parametric optimization problems in combinatorial optimization, and did not receive much attention by the computational geometry community until t... |

194 | Linear programming in linear time when the dimension is fixed
- Megiddo
- 1984
(Show Context)
Citation Context ...3. Almost concurrently with the development of the parametric searching technique, Megiddo devised another ingenious technique for solving linear programming and several related optimization problems =-=[89, 90]-=-. This technique, now known as decimation or prune-and-search, was later re ned and extended by Dyer [41], Clarkson [30], and others. The technique can be viewed as an optimized version of parametric ... |

185 |
Linear programming and convex hulls made easy
- Seidel
- 1990
(Show Context)
Citation Context ...n developed for a wide variety of problems in computational geometry and in other elds� see, e.g., the books by Mulmuley [96] and by Motwani and Raghavan [95]. In particular, Clarkson [31] and Seidel =-=[103]-=- gave randomized algorithms for linear programming, whose expected time is linear in any xed dimension, which aremuch simpler than their earlier deterministic counterparts, and the dependence of their... |

173 | Smallest enclosing disks (balls and ellipsoids), in: H. Maurer (Ed.), New Results and New Trends
- Welzl
- 1991
(Show Context)
Citation Context ...algorithm of Megiddo gives Cd =22d,whichwas improved by Clarkson [30] andDyer [41] to3d2. Using randomization techniques, a number of simpler randomized algorithms have been developed for the problem =-=[31, 43,103,117]-=-, with a better dependence on d, of which the best expected running time, O(d2n + dd=2+O(1) log n), is due to Clarkson [31]. By derandomizing the algorithms in [31, 43], one can obtain dO(d) n-time de... |

164 | A subexponential bound for linear programming
- MatouSek, Sharir, et al.
- 1992
(Show Context)
Citation Context ...ill exponential). Additional signi cant progress was made about four years ago, when new randomized algorithms for linear programming were obtained independently by Kalai [68], and by Matousek et al. =-=[86, 110]-=- (these two algorithms are essentially dual versions of the same technique). The expected number of arithmetic operations performed by these algorithms is `subexponential' in the input size, and is st... |

127 | Ray shooting and parametric search
- Agarwal, Matouˇsek
- 1993
(Show Context)
Citation Context ...d = 2, and later extended his approach to d = 3 [106]. Using multi-dimensional parametric searching, his approach can be generalized to higher dimensions. 5.5 Query Type Problems Agarwal and Matousek =-=[5]-=- gave a general technique, based on parametric searching, to answer ray-shooting queries (where we wish to preprocess a given set of objects in R d , so that the rst object hit by a query ray can be c... |

121 | Combinatorial Optimization with Rational Objective Functions - MEGIDDO - 1979 |

117 | On the complexity of some common geometric location problems
- Meggido, Supowit
- 1984
(Show Context)
Citation Context ... of the L1 or the L2 norm of those `deviations'. If p is considered as part of the input, most facility location problems are known to be NP-hard, even when the supply objects are points in the plane =-=[94]-=-. However, for xed values of p, most of these problems can be solved in polynomial time. In this subsection we review e cient algorithms for some special cases of these problems. p-center. Here we wis... |

112 |
Sorting in c log n parallel steps
- Ajtai, Komlos, et al.
- 1983
(Show Context)
Citation Context ...rsions. Since the insertion of a line can be done in O(log n) time, the whole decision procedure takes O(n log n) time. Any parallel sorting algorithm, which runsinO(log n) time using O(n) processors =-=[15]-=-, can count the number of inversions within the same time and processor bounds. (Notice that the construction of T itself does not involve any comparison that depends on the value of , and so need not... |

105 |
Slowing down sorting networks to obtain faster sorting algorithms
- Cole
- 1987
(Show Context)
Citation Context ...he generic execution.) Plugging these algorithms into the parametric searching paradigm, we obtain an O(n log 3 n)-time algorithm for the slope selection problem. 2.3 Improvements and Extensions Cole =-=[35]-=- observed that in certain applications of parametric searching, including the slope selection problem, the running time can be improved to O((P +Ts)Tp), as follows. Consider a parallel step of the abo... |

105 |
Parallelism in comparison problems
- Valiant
- 1975
(Show Context)
Citation Context ...erally quadratic in the original complexity. To speed up the execution, Megiddo proposes to implement the generic algorithm by a parallel algorithm Ap (under Valiant's comparison model of computation =-=[115]-=-). If Ap uses P processors and runs in Tp parallel steps, then each parallel step involves at most P independent comparisons� that is, we do not need to know the output of such a comparison to be able... |

102 | A Las Vegas algorithm for linear and integer programming when the dimension is small
- CLARKSON
- 1995
(Show Context)
Citation Context ...orithms have been developed for a wide variety of problems in computational geometry and in other elds� see, e.g., the books by Mulmuley [96] and by Motwani and Raghavan [95]. In particular, Clarkson =-=[31]-=- and Seidel [103] gave randomized algorithms for linear programming, whose expected time is linear in any xed dimension, which aremuch simpler than their earlier deterministic counterparts, and the de... |

91 | On linear-time deterministic algorithms for optimization problems in fixed dimension
- Chazelle, Matotiek
- 1993
(Show Context)
Citation Context ...pected running time, O(d2n + dd=2+O(1) log n), is due to Clarkson [31]. By derandomizing the algorithms in [31, 43], one can obtain dO(d) n-time deterministic algorithms for linear programming in R d =-=[13, 26]-=-. In Section 6, we will describe further improved randomized algorithms, due to Kalai [68] and to Matousek et al. [86] (see also [110]), with subexponential expected running time.s5 Applications In th... |

91 |
A subexponential randomized simplex algorithm
- Kalai
- 1992
(Show Context)
Citation Context ...imension is better (though still exponential). Additional signi cant progress was made about four years ago, when new randomized algorithms for linear programming were obtained independently by Kalai =-=[68]-=-, and by Matousek et al. [86, 110] (these two algorithms are essentially dual versions of the same technique). The expected number of arithmetic operations performed by these algorithms is `subexponen... |

87 |
E.: A combinatorial bound for linear programming and related problems
- Sharir, Welzl
- 1992
(Show Context)
Citation Context ...ill exponential). Additional signi cant progress was made about four years ago, when new randomized algorithms for linear programming were obtained independently by Kalai [68], and by Matousek et al. =-=[86, 110]-=- (these two algorithms are essentially dual versions of the same technique). The expected number of arithmetic operations performed by these algorithms is `subexponential' in the input size, and is st... |

80 | Range searching with semi-algebraic sets
- Agarwal, Matousek
- 1994
(Show Context)
Citation Context ... answer ray-shooting queries (where we wish to preprocess a given set of objects in R d , so that the rst object hit by a query ray can be computed e - ciently). This technique, further elaborated in =-=[6, 9]-=-, has yielded fast algorithms for several related problems, including hidden surface removal, nearest neighbor searching, computing convex layers, and computing higher-order Voronoi diagrams� see [5, ... |

72 |
Approximate matching of polygonal shapes
- Alt, Behrends, et al.
- 1995
(Show Context)
Citation Context ...ish to compute minv H(P +v� Q). The problem has been solved in [12], using parametric searching, in O((mn) 2 log 3 (mn)) time, which is signi cantly faster than the previously best known algorithm of =-=[17]-=-. See [27, 28] for other parametric-searching based results on this problem. 5.3 Statistical Estimators and Related Problems Plane tting. Given a set S of n points in R 3 ,we wish to t a plane h throu... |

63 |
Generalized selection and ranking: sorted matrices
- Frederickson, Johnson
- 1984
(Show Context)
Citation Context ...nal Science Foundation, the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scienti c Research and Development.smatrix searching =-=[50, 51,52, 53, 55]-=-. We mention these alternative techniques in Section 3. Almost concurrently with the development of the parametric searching technique, Megiddo devised another ingenious technique for solving linear p... |

62 |
On k-hulls and related problems
- Cole, Sharir, et al.
- 1987
(Show Context)
Citation Context ...arallel step of Ap is thus O(P + Ts log P ), for a total running time of O(PTp + TpTs log P ). In most cases, the second term dominates the running time. The technique has been generalized further in =-=[10, 37,112,113]-=-. 2.2 An Example: The Slope Selection Problem As an illustration, consider the slope selection problem, which weformulate in a dual setting, as follows: We are ; given a set L of n nonvertical lines i... |

62 | Computing the width of a set
- Toussaint
- 1988
(Show Context)
Citation Context ...ere we wish to compute the smallest real value w such that D can be covered by the union of p strips of width w . For p = 1, this is the classical width problem, which can be solved in O(n log n) time=-=[62]-=-. For p = 2, Agarwal and Sharir [10] (see also [8]) gave anO(n 2 log 5 n)-time algorithm. The runningstime was improved to O(n 2 log 4 n)by Katz and Sharir [72] and by Glozman et al. [55], using expan... |

60 |
On a multidimensional search technique and its application to the Euclidean one-center problem
- Dyer
- 1986
(Show Context)
Citation Context ...genious technique for solving linear programming and several related optimization problems [89, 90]. This technique, now known as decimation or prune-and-search, was later re ned and extended by Dyer =-=[41]-=-, Clarkson [30], and others. The technique can be viewed as an optimized version of parametric searching, in which certain special properties of the problem allows one to improve further the e ciency ... |

59 | Helly-type theorems and generalized linear programming
- Amenta
- 1994
(Show Context)
Citation Context ... a general abstract framework, which ts not only linear programming but many other problems. Such `LP-type' problems are also reviewed in Section 6, including the connection, recently noted by Amenta =-=[19, 20]-=-, between abstract linear programming and `Helly-type' theorems. 2 Parametric Searching 2.1 Outline of the Technique The parametric searching technique of Megiddo [87, 88] can be described in the foll... |

53 | Approximating center points with iterative radon points
- Clarkson, Eppstein, et al.
- 1996
(Show Context)
Citation Context ...e algorithm for computing a center point, using a direct and elegant technique. For computing a center point in three dimensions, near-quadratic algorithms were developed in [37, 97]. Clarkson et al. =-=[32]-=- gave ane cient algorithm for computing an approximate center point. Ham-sandwich cuts. Let A1� ::: �Ad be d point sets in R d . A ham-sandwich cut is a hyperplane that simultaneously bisects all the ... |

50 |
Algorithms for bichromatic line segment problems and polyhedral terrains
- Chazelle, Edelsbrunner, et al.
- 1994
(Show Context)
Citation Context ...og n) calls to the approximating decision procedure is only O(n log n), so this also bounds the running time of the whole algorithm. This technique was subsequently simpli ed in [22]. Chazelle et al. =-=[24]-=- have shown that the algorithm of [36] can be extended to compute, insO(n log n) time, the k-th leftmost vertex in an arrangement ofn line segments. Multi-dimensional parametric searching. The paramet... |

46 | Product range spaces, sensitive sampling, and derandomization
- Brönnimann, Chazelle, et al.
- 1999
(Show Context)
Citation Context ...ndomized algorithm (which is worst-case optimal) was given by Clarkson and Shor [33], but no optimal deterministic algorithm is known. The best known deterministic solution is due to Bronniman et al. =-=[23]-=-, and runs in O(n log 3 n) time. It is based on parametric searching, and uses some interesting derandomization techniques. See also [25, 85, 104] for earlier close-to-linear time algorithms based on ... |

46 | A subexponential algorithm for abstract optimization problems
- Gärtner
- 1995
(Show Context)
Citation Context ...s an LP-type problem, with combinatorial dimension d+1. It is, however, not basis-regular, and a naive implementation of the basis-changing operation may be quite costly (in d). Nevertheless, Gartner =-=[54]-=- showed that this operation can be performed in this case using expected e O(p d) arithmetic operations. Hence, the expected running time of the algorithm is O(d 2 n)+e O(p d log d). There are several... |

45 |
On geometric optimization with few violated constraints
- Matousek
- 1994
(Show Context)
Citation Context ... in time O( O( ) n), provided an additional axiom holds (together with an additional computational assumption). Still, these extra requirements are satis ed in many natural LP-type problems. Matousek =-=[83]-=-investigates the problem of nding the best solution, for abstract LP-type problems, which satis es all but k of the given constraints.sAmenta [18] considers the following extension of the abstract fra... |

44 | Efficient randomized algorithms for some geometric optimization problems - Agarwal - 1995 |

40 | A near linear algorithm for the planar 2-center problem
- Sharir
- 1997
(Show Context)
Citation Context ...rametric searching by randomization. The best near-quadratic solution is due to Jaromczyk and Kowaluk [66], and runs in O(n 2 log n) time. A major progress on this problem was made recently by Sharir =-=[107]-=-, who gave an O(n log 9 n)-time algorithm, by combining the parametric searching technique with several additional tricks, including a variant of the matrix searching algorithm of Frederickson and Joh... |

38 | An expander-based approach to geometric optimization
- Katz, Sharir
- 1997
(Show Context)
Citation Context ...rawbacks, which we discuss below. Consequently, there have been several recent attempts to replace parametric searching by alternative techniques, including randomization [11, 33,79], expander graphs =-=[16, 70,71,72]-=-, geometric cuttings [12, 22], and ? Pankaj Agarwal has been supported by National Science Foundation Grant CCR-93{01259, an NYI award, and by matching funds from Xerox Corp. Micha Sharir has been sup... |

37 |
width, closest line pair, and parametric searching, Discrete Comput
- Chazelle, Edelsbrunner, et al.
- 1993
(Show Context)
Citation Context ...e best known deterministic solution is due to Bronniman et al. [23], and runs in O(n log 3 n) time. It is based on parametric searching, and uses some interesting derandomization techniques. See also =-=[25, 85, 104]-=- for earlier close-to-linear time algorithms based on parametric searching. Closest line pair. GivenasetL of n lines in the R 3 ,we wish to compute a closest pair of lines in L. Independently, Chazell... |

36 |
Algorithms for ham-sandwich cuts
- Lo, Matoušek, et al.
- 1994
(Show Context)
Citation Context ...n log n)-time algorithm when A1 and A2 are not linearly separable. The running time was then improved to linearsby Lo and Steiger [77]. E cient algorithms for higher dimensions are given by Lo et al. =-=[76]-=-. 5.4 Placement and Intersection Polygon placement. Let P be a polygonal object with m edges, and let Q be a closed planar polygonal environment with n edges. We wish to nd the largest similar copy of... |

33 |
Optimal algorithms for tree partitioning
- Frederickson
- 1991
(Show Context)
Citation Context ...nal Science Foundation, the Israel Science Fund administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scienti c Research and Development.smatrix searching =-=[50, 51,52, 53, 55]-=-. We mention these alternative techniques in Section 3. Almost concurrently with the development of the parametric searching technique, Megiddo devised another ingenious technique for solving linear p... |

32 |
Roundness algorithms using the Voronoi diagrams
- Ebara, Fukuyama, et al.
- 1989
(Show Context)
Citation Context ...cle C through S, so that the maximum distance between C and the points of S is minimized. This is equivalent to nding an annulus of minimum width that contains S. This problem was initially solved in =-=[44]-=-, by a quadratic-time algorithm, which was improved, using parametric searching, to O(n 17=11+" ), for any ">0 [1]. Using randomization and an improved analysis, this can be improved to O(n 3=2+" ) ti... |

31 |
Randomized optimal algorithm for slope selection
- Matousek
- 1991
(Show Context)
Citation Context ...etric searching has certain drawbacks, which we discuss below. Consequently, there have been several recent attempts to replace parametric searching by alternative techniques, including randomization =-=[11, 33,79]-=-, expander graphs [16, 70,71,72], geometric cuttings [12, 22], and ? Pankaj Agarwal has been supported by National Science Foundation Grant CCR-93{01259, an NYI award, and by matching funds from Xerox... |

30 | Computing a centerpoint of a finite planar set of points in linear time. Discrete Comput - Jadhav, Mukhopadhyay - 1994 |

28 |
Geometric partitioning made easier, even in parallel
- Goodrich
- 1993
(Show Context)
Citation Context ...arwal et al. [2] gave anO(n 4=3 log 4=3 n) expected-time randomized algorithm for the decision problem, which yielded an O(n 4=3 log 8=3 n)-time algorithm for the distance selection problem. Goodrich =-=[56]-=- derandomized this algorithm. Katz and Sharir [71] gave an expander-based O(n 4=3 log 3+" n)-time algorithm for this problem, for any ">0. See also [102]. Minimum Hausdor distance between polygons. Le... |

28 |
Linear optimization queries
- Matoušek, Schwarzkopf
- 1992
(Show Context)
Citation Context ... A (k;1) d generically, suchthat varies over ^ h, one can determine whether intersects ^ h, and, if not, determine which of the two halfspaces contains . The details of this algorithm can be found in =-=[13, 34,81,98]-=-. Recently, Toledo [114] showed how to handle nonlinear polynomials, using Collins' cylindrical algebraic decomposition scheme [35]. 3 Alternatives Approaches to Parametric Searching Despite its power... |

27 |
Partitioning with two lines in the plane
- Megiddo
- 1985
(Show Context)
Citation Context ... the existence of such a cut. Several prune-and-search algorithms have been proposed for computing a ham-sandwich cut in the plane. For the special case when A1 and A2 are linearly separable, Megiddo =-=[91]-=- gave a linear time algorithm. Modifying his algorithm, Edelsbrunner and Waupotitsch [46]gave anO(n log n)-time algorithm when A1 and A2 are not linearly separable. The running time was then improved ... |

27 | Mathematical techniques for efficient record segmentation in large shared databases - Eisner, Severance - 1976 |

25 | Geometric pattern matching in d-dimensional space
- Chew, Dor, et al.
- 1999
(Show Context)
Citation Context ...pute minv H(P +v� Q). The problem has been solved in [12], using parametric searching, in O((mn) 2 log 3 (mn)) time, which is signi cantly faster than the previously best known algorithm of [17]. See =-=[27, 28]-=- for other parametric-searching based results on this problem. 5.3 Statistical Estimators and Related Problems Plane tting. Given a set S of n points in R 3 ,we wish to t a plane h through S so that t... |