## Output-Sensitive Results on Convex Hulls, Extreme Points, and Related Problems (1996)

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Citations: | 66 - 13 self |

### BibTeX

@MISC{Chan96output-sensitiveresults,

author = {T. M. Chan},

title = {Output-Sensitive Results on Convex Hulls, Extreme Points, and Related Problems},

year = {1996}

}

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### Abstract

. We use known data structures for ray-shooting and linear-programming queries to derive new output-sensitive results on convex hulls, extreme points, and related problems. We show that the f -face convex hull of an n-point set P in a fixed dimension d # 2 can be constructed in O(n log f + (nf) 1-1/(#d/2#+1) log O(1) n) time; this is optimal if f = O(n 1/#d/2# / log K n) for some sufficiently large constant K . We also show that the h extreme points of P can be computed in O(n log O(1) h + (nh) 1-1/(#d/2#+1) log O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n 2-# ) time for any constant #<2/(#d/2# 2 + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position. 1. Introduction Let P be a set of n points in d-dimen...

### Citations

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Citation Context ...this polytope is at worst \Theta(n bd=2c ) [27]. In the convex hull problem, we want to construct the facial structure of conv(P ). This problem has been intensively studied in computational geometry =-=[16, 31, 33, 36]-=-, and it has applications to other geometric problems such as computing intersections of halfspaces and computing Voronoi diagrams and Delaunay triangulations. Chazelle [10] has solved the convex hull... |

688 |
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(Show Context)
Citation Context ...this polytope is at worst \Theta(n bd=2c ) [27]. In the convex hull problem, we want to construct the facial structure of conv(P ). This problem has been intensively studied in computational geometry =-=[16, 31, 33, 36]-=-, and it has applications to other geometric problems such as computing intersections of halfspaces and computing Voronoi diagrams and Delaunay triangulations. Chazelle [10] has solved the convex hull... |

503 |
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Citation Context ...this polytope is at worst \Theta(n bd=2c ) [27]. In the convex hull problem, we want to construct the facial structure of conv(P ). This problem has been intensively studied in computational geometry =-=[16, 31, 33, 36]-=-, and it has applications to other geometric problems such as computing intersections of halfspaces and computing Voronoi diagrams and Delaunay triangulations. Chazelle [10] has solved the convex hull... |

387 | Applications of random sampling in computational geometry
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(Show Context)
Citation Context ... and Shi [17] obtained an O(n log 2 f)-time method, and Chazelle and Matousek [11] demonstrated that optimal O(n log f) time is possible by derandomizing an earlier algorithm due to Clarkson and Shor =-=[13]. In any fixed dimen-=-sion, the "gift-wrapping" algorithm of Swart [44] and the "beneath/beyond" algorithm of Seidel [41] achieve O(nf) and O(n 2 + f log n) time respectively. The latter is subsequently... |

286 |
Computational Geometry: An Introduction Through Randomized Algorithms
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(Show Context)
Citation Context |

276 | Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms
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(Show Context)
Citation Context ...heavily on the assumption that the input points or hyperplanes are in general position. For some of the problems we have considered (e.g., convex hulls and k-levels), standard perturbation techniques =-=[18, 20]-=- may be used when this assumption does not hold. However, one should keep in mind that these perturbation methods may increase the output size when there is a large number of degeneracies. The remaind... |

240 |
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(Show Context)
Citation Context ... be done by a simple binary search. Observe that computing the intersection T H is equivalent to computing a convex hull in the dual space, and thus takes O(n log n) time by Graham's scan for example =-=[22]-=-; and the binary search takes O(log n) time. Hence, this method requires O(n log n) preprocessing time, O(n) space, and O(log n) query time. The same preprocessing time, space, and query time can be o... |

232 | Applying parallel computation algorithms in the design of serial algorithms
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Citation Context ... structure for halfspace-emptiness queries (satisfying some reasonable conditions) can be used to answer linear programming queries by a multidimensional version of Megiddo's parametric search method =-=[28]. The re-=-sulting query time is given by O(t(n; m)��(n; m) d log d ��(n; m)), which, in our case, is O( n m 1=bd=2c log O(1) m log d n). 2 Corollary 3.4 A sequence of q linear programming queries on a s... |

194 | Linear programming in linear time when the dimension is fixed
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(Show Context)
Citation Context ...problem of computing the extreme points of P , i.e., the vertices of conv(P ) (or equivalently, the set of points p 2 P with conv(P \Gamma fpg) 6= conv(P )). By Megiddo's linear programming algorithm =-=[29]-=-, we can test whether a given point is an extreme point of P in linear time; this immediately yields an algorithm for the extreme point problem that runs in O(n 2 ) time. Matousek reduced the bound to... |

185 |
Linear programming and convex hulls made easy
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- 1990
(Show Context)
Citation Context ... we describe how the log d k factor can be eliminated by using Sharir and Welzl's randomized algorithm for generalized linear programming [43] (which is based on Seidel's linear programming algorithm =-=[42]-=-). This in turn improves the query time in Lemma 3.3. We first observe that the problem of finding an extremum in a nonempty intersection of k convex objects in E d belongs to the class of LP-type pro... |

173 | Smallest enclosing disks (balls and ellipsoids), in: H. Maurer (Ed.), New Results and New Trends
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- 1991
(Show Context)
Citation Context ...g O(1) m) time. Because O(k) violation tests and basis computations are expected to be performed by Sharir and Welzl's algorithm (the expected number of basis computations is actually only O(log d k) =-=[45]-=-), we obtain a randomized O(k m 1\Gamma1=bd=2c log O(1) m)-time solution to our linear programming problem on k polytopes. The above method carries through if the polytopes are stored in linear-space ... |

159 |
Leeuwen, "Maintenance of Configurations in the Plane
- Overmars, van
- 1981
(Show Context)
Citation Context ...lytopes, where halfspaces may be inserted or deleted. Let n denote a (known) upper bound on the number of halfspaces at any given time. In two dimensions, a data structure by Overmars and van Leeuwen =-=[34]-=- has O(n log n) preprocessing time, O(n) space, O(log 2 n) update time, and O(log n) query time. We can extend the grouping technique to get the following analogues of Lemma 2.1 and Corollary 2.2: Lem... |

146 |
The maximum numbers of faces of a convex polytope
- McMullen
- 1970
(Show Context)
Citation Context ... in general position. The smallest convex set containing P is a polytope conv(P ) called the convex hull of P . It is known that the number of faces, f , in this polytope is at worst \Theta(n bd=2c ) =-=[27]-=-. In the convex hull problem, we want to construct the facial structure of conv(P ). This problem has been intensively studied in computational geometry [16, 31, 33, 36], and it has applications to ot... |

127 | Ray shooting and parametric search
- Agarwal, Matouˇsek
- 1993
(Show Context)
Citation Context ...log f + f) lower bound for d ? 4. Here, we show that the gift-wrapping method can be further improved using the data structures for ray shooting queries in polytopes developed by Agarwal and Matousek =-=[1]-=- and refined by Matousek and Schwarzkopf [26]. Our convex hull algorithm runs in O(n log f + (nf) 1\Gamma1=(bd=2c+1) log O(1) n) time and is optimal when f = O(n 1=bd=2c = log K n) for a sufficiently ... |

105 |
Slowing down sorting networks to obtain faster sorting algorithms
- Cole
- 1987
(Show Context)
Citation Context ...s data structure. The O(q log d+2 n) bound is still correct, but in order to remove the unnecessary log log n factors, we need to apply a multidimensional version of Cole's improved parametric search =-=[14]-=-. We will not go over this in detail, as for very large values of q the bound in Lemma 3.5(ii) is better anyway. In the appendix, we show how to eliminate the log n factors in Lemma 3.3 and thus remov... |

104 |
Decomposable searching problems I: Staticto-dynamic transformation
- Bentley, Saxe
- 1980
(Show Context)
Citation Context ...=2c+1) log O(1) n + q log O(1) n) time. Proof: As in the proof of Lemma 3.3, we consider the halfspace-emptiness problem first. Since, this problem is decomposable, the techniques by Bentley and Saxe =-=[4]-=- may be applied to convert a static structure to a semidynamic one (which increases building time and query time by a logarithmic factor). We then apply Matousek's parametric search to use this struct... |

100 |
The ultimate planar convex hull algorithm
- Kirkpatrick, Seidel
- 1986
(Show Context)
Citation Context ...ever, this bound depends only on the input size n and is insensitive to the output size f . An optimal O(n log f)-time output-sensitive algorithm in two dimensions was given by Kirkpatrick and Seidel =-=[23]-=-. For dimension 3, Edelsbrunner and Shi [17] obtained an O(n log 2 f)-time method, and Chazelle and Matousek [11] demonstrated that optimal O(n log f) time is possible by derandomizing an earlier algo... |

97 |
Convex hulls of finite sets of points in two and three dimensions
- Preparata, Hong
- 1977
(Show Context)
Citation Context ... time. The same preprocessing time, space, and query time can be obtained in three dimensions: in the preprocessing, compute the polytope T H by the dual of Preparata and Hong's convex hull algorithm =-=[35]-=- and construct its Dobkin-Kirkpatrick hierarchical representation [15]; then use the query algorithm from [15]. Our first observation is that a preprocessing time/query time tradeoff is possible using... |

87 |
E.: A combinatorial bound for linear programming and related problems
- Sharir, Welzl
- 1992
(Show Context)
Citation Context ...n O(k m 1\Gamma1=bd=2c log O(1) m log d k)-time solution. Here we describe how the log d k factor can be eliminated by using Sharir and Welzl's randomized algorithm for generalized linear programming =-=[43]-=- (which is based on Seidel's linear programming algorithm [42]). This in turn improves the query time in Lemma 3.3. We first observe that the problem of finding an extremum in a nonempty intersection ... |

77 |
Fast detection of polyhedral intersection
- Dobkin, Kirkpatrick
- 1983
(Show Context)
Citation Context ...ned in three dimensions: in the preprocessing, compute the polytope T H by the dual of Preparata and Hong's convex hull algorithm [35] and construct its Dobkin-Kirkpatrick hierarchical representation =-=[15]; then use-=- the query algorithm from [15]. Our first observation is that a preprocessing time/query time tradeoff is possible using a standard "grouping" technique. Using this observation, we can perfo... |

71 |
Constructing higher dimensional convex hulls at logarithmic cost per face
- Seidel
- 1986
(Show Context)
Citation Context ... time is possible by derandomizing an earlier algorithm due to Clarkson and Shor [13]. In any fixed dimension, the "gift-wrapping" algorithm of Swart [44] and the "beneath/beyond" =-=algorithm of Seidel [41]-=- achieve O(nf) and O(n 2 + f log n) time respectively. The latter is subsequently improved Supported by a Killam Predoctoral Fellowship and an NSERC Postgraduate Scholarship. to O(n 2\Gamma2=(bd=2c+1)... |

60 |
An algorithm for convex polytopes
- Chand, Kapur
- 1970
(Show Context)
Citation Context ... f-face convex hull of an n-point set can be constructed by performing O(f) ray shooting queries in a polytope defined by n halfspaces. The algorithm we use is just the wellknown gift-wrapping method =-=[8, 36, 44] dualized,-=- since a "gift-wrapping operation" corresponds to shooting a ray in the dual polytope. If the ray shooting queries are performed directly by scanning the halfspaces, then we get an O(nf)-tim... |

54 |
shooting in convexpolytopes
- SCHWARZKOPF, Ray
- 1992
(Show Context)
Citation Context ...how that the gift-wrapping method can be further improved using the data structures for ray shooting queries in polytopes developed by Agarwal and Matousek [1] and refined by Matousek and Schwarzkopf =-=[26]-=-. Our convex hull algorithm runs in O(n log f + (nf) 1\Gamma1=(bd=2c+1) log O(1) n) time and is optimal when f = O(n 1=bd=2c = log K n) for a sufficiently large K. Furthermore, it is faster than all p... |

47 | Optimal output-sensitive convex hull algorithms in two and three dimensions, Discrete Comput. Geom. 16
- Chan
- 1996
(Show Context)
Citation Context ...in O(n log f) time. In the plane, our algorithm is as simple as Kirkpatrick and Seidel's, and in three dimensions, our algorithm is simpler than Chazelle and Matousek's. These are reported separately =-=[6]-=-. Next, we turn to the problem of computing the extreme points of P , i.e., the vertices of conv(P ) (or equivalently, the set of points p 2 P with conv(P \Gamma fpg) 6= conv(P )). By Megiddo's linear... |

45 |
On geometric optimization with few violated constraints
- Matousek
- 1994
(Show Context)
Citation Context ...og 2 n) time, if d = 2; ii. O(n log log n + n log k + k 3+" ) time, if d = 3; iii. O(n log log n + n log k) time, if ds4 and k dsn 1=bd=2c\Gamma" . Proof: The depth-first search algorithm by=-= Matousek [25]-=- solves this problem using O(k d ) linear programming/membership queries and O(k d ) insertions/deletions on two dynamic sets of at most n halfspaces. (A membership query is just a special case of a r... |

39 |
Dynamic half-space range reporting and its applications
- Agarwal, Matouˇsek
- 1995
(Show Context)
Citation Context ...: ; i. It is known that this problem can be solved optimally in O(n log n) time by an algorithm of Chazelle [9] for d = 2 and quasi-optimally in O(n 1+" ) time by an algorithm of Agarwal and Mato=-=usek [2]-=- for d = 3. For ds4, Edelsbrunner [16, Problem 10.3(c)] asked whether the vertices of all layers can be identified in o(n 3 ) time. This problem is equivalent to finding the depth of p, i.e., the inde... |

38 |
Constructing Belts in twodimensional arrangements with applications
- Edelsbrunner, Welzl
- 1986
(Show Context)
Citation Context ...most k hyperplanes of H above it (0sk ! n). The 0-level is just the dual of a convex hull. In the plane, an output-sensitive algorithm for constructing the k-level was given by Edelsbrunner and Welzl =-=[19]-=-. We improve its running time from O(n log n+f log 2 n) to O(n log f +f log 2 n), where f denotes the size of the k-level. In higher dimensions, Agarwal and Matousek [2] proposed a method based on ray... |

38 | Dynamic three-dimensional linear programming
- Eppstein
- 1992
(Show Context)
Citation Context ...ling [38], as used in our proof of Lemma 3.1. It is also interesting to compare the techniques here with those used in the previous deterministic and randomized methods by Reichling [39] and Eppstein =-=[21]-=- for the threedimensional problem. The (expected) query time in Lemma 3.3 can now be improved to O((n=m bd=2c ) log O(1) m) since it uses k = dn=me. As a consequence, the O(n log log n) term in Coroll... |

37 | On geometric optimization with few violated constraints, SCG ’94 - Matouˇsek - 1994 |

35 | An efficient approach to removing geometric degeneracies
- Emiris, Canny
- 1992
(Show Context)
Citation Context ...heavily on the assumption that the input points or hyperplanes are in general position. For some of the problems we have considered (e.g., convex hulls and k-levels), standard perturbation techniques =-=[18, 20]-=- may be used when this assumption does not hold. However, one should keep in mind that these perturbation methods may increase the output size when there is a large number of degeneracies. The remaind... |

33 |
Output-sensitive construction of polytopes in four dimensions T.M. Chan and clipped Voronoi diagrams in three
- Chan, Snoeyink, et al.
(Show Context)
Citation Context ...y small constant.) It is in fact possible to reduce the O(n 2\Gamma2=(bd=2c+1)+" ) term to O(n 2\Gamma2=(bd=2c+1) log O(1) n) by using the static structures in [24]. Recently, Chan, Snoeyink, and=-= Yap [7]-=- have obtained an O((n + f) log 2 f)-time algorithm in four dimensions; in higher dimensions, their method is less efficient, running in O((n + f d\Gamma3 ) log d\Gamma2 f) time. Thus, there is still ... |

31 |
Finding the convex hull facet by facet
- Swart
- 1985
(Show Context)
Citation Context ...Matousek [11] demonstrated that optimal O(n log f) time is possible by derandomizing an earlier algorithm due to Clarkson and Shor [13]. In any fixed dimension, the "gift-wrapping" algorithm=-= of Swart [44] and the &-=-quot;beneath/beyond" algorithm of Seidel [41] achieve O(nf) and O(n 2 + f log n) time respectively. The latter is subsequently improved Supported by a Killam Predoctoral Fellowship and an NSERC P... |

28 |
Linear optimization queries
- Matoušek, Schwarzkopf
- 1992
(Show Context)
Citation Context ... f log n) time respectively. The latter is subsequently improved Supported by a Killam Predoctoral Fellowship and an NSERC Postgraduate Scholarship. to O(n 2\Gamma2=(bd=2c+1)+" + f log n) by Mato=-=usek [24] usin-=-g a data structuring technique that he has developed for linear programming queries. (Throughout this paper, " ? 0 denotes an arbitrarily small constant.) It is in fact possible to reduce the O(n... |

26 |
Derandomizing an output-sensitive convex hull algorithm in three dimensions
- Chazelle, Matouˇsek
- 1995
(Show Context)
Citation Context ...-time output-sensitive algorithm in two dimensions was given by Kirkpatrick and Seidel [23]. For dimension 3, Edelsbrunner and Shi [17] obtained an O(n log 2 f)-time method, and Chazelle and Matousek =-=[11] demonstra-=-ted that optimal O(n log f) time is possible by derandomizing an earlier algorithm due to Clarkson and Shor [13]. In any fixed dimension, the "gift-wrapping" algorithm of Swart [44] and the ... |

20 |
Sur l’enveloppe convexe des nuages de points aléatoires dans R n
- Raynaud
- 1970
(Show Context)
Citation Context ... = log K n) for a sufficiently large K. Furthermore, it is faster than all previous methods when f = O(n= log K n) and d ? 4. Note that in many cases, f can in fact be sublinear; for example, Raynaud =-=[37]-=- proved that the expected value of f is O(n (d\Gamma1)=(d+1) ) if the points of P are chosen uniformly at random from a d-dimensional ball. The expected number of hull vertices is only polylogarithmic... |

17 | More output-sensitive geometric algorithms (extended abstract
- Clarkson
- 1994
(Show Context)
Citation Context ...lgorithm, we show that the extreme points can be computed in O(n log O(1) h+ (nh) 1\Gamma1=(bd=2c+1) log O(1) n) time. This O(nh)-time algorithm has recently been discovered independently by Clarkson =-=[12]-=- and Ottmann et al. [32]. We then consider the problem of computing the convex layers of P , defined iteratively as follows: layer 1 is the convex hull of P , and if layer i is nonempty, then layer i ... |

15 |
On the average number of maxima in a set of vectors
- Bentley, Kung, et al.
- 1978
(Show Context)
Citation Context ...osen uniformly at random from a d-dimensional ball. The expected number of hull vertices is only polylogarithmic in n if the points are chosen uniformly from a hypercube or from a normal distribution =-=[3, 37]-=-. Surprisingly, our method leads to new optimal output-sensitive algorithms in two and three dimensions, running in O(n log f) time. In the plane, our algorithm is as simple as Kirkpatrick and Seidel'... |

13 |
An O(nlog2h) time algorithm for the threedimensional convex hull problem
- Edelsbrunner, Shi
(Show Context)
Citation Context ...ize n and is insensitive to the output size f . An optimal O(n log f)-time output-sensitive algorithm in two dimensions was given by Kirkpatrick and Seidel [23]. For dimension 3, Edelsbrunner and Shi =-=[17]-=- obtained an O(n log 2 f)-time method, and Chazelle and Matousek [11] demonstrated that optimal O(n log f) time is possible by derandomizing an earlier algorithm due to Clarkson and Shor [13]. In any ... |

13 | An O(n log 2 h) time algorithm for the three-dimensional convex hull problem - Edelsbrunner, Shi - 1990 |

13 | optimization queries - Linear - 1993 |

11 |
On the detection of a common intersection of k convex objects
- Reichling
- 1988
(Show Context)
Citation Context ... compute the convex polygon \Pi i = T H i for each i, and store each of them in an ordered array. The total preprocessing time is then O( n m (m log m)) = O(n log m), while space is linear. Reichling =-=[38] sho-=-wed that in O(k log 2 m) time, one can detect whether the intersection of k convex m-gons is empty, and if not, report the point in the intersection that is extreme in a given direction ��; his me... |

11 |
k-violation linear programming
- Roos, Peter
- 1994
(Show Context)
Citation Context ... values of k in the two-dimensional case, the time bound of Theorem 7.2(i) can be reduced to O(n log 2 n) using parametric search or slope selection techniques, as Matousek [25] and Roos and Widmayer =-=[40]-=- observed. The techniques here may also be applicable to the infeasible case of linear programming with k violated constraints, or to the smallest k-enclosing circle problem; see Matousek's paper [25]... |

10 | Enumerating extreme points in higher dimensions
- Ottmann, Schuierer, et al.
(Show Context)
Citation Context ...he extreme points can be computed in O(n log O(1) h+ (nh) 1\Gamma1=(bd=2c+1) log O(1) n) time. This O(nh)-time algorithm has recently been discovered independently by Clarkson [12] and Ottmann et al. =-=[32]-=-. We then consider the problem of computing the convex layers of P , defined iteratively as follows: layer 1 is the convex hull of P , and if layer i is nonempty, then layer i + 1 is defined as the co... |

10 |
On the detection of a common intersection of k convex polyhedra
- Reichling
- 1988
(Show Context)
Citation Context ...algorithm by Reichling [38], as used in our proof of Lemma 3.1. It is also interesting to compare the techniques here with those used in the previous deterministic and randomized methods by Reichling =-=[39]-=- and Eppstein [21] for the threedimensional problem. The (expected) query time in Lemma 3.3 can now be improved to O((n=m bd=2c ) log O(1) m) since it uses k = dn=me. As a consequence, the O(n log log... |

8 |
An optimal convex hull algorithm for point sets in any fixed dimension
- Chazelle
- 1991
(Show Context)
Citation Context ...ional geometry [16, 31, 33, 36], and it has applications to other geometric problems such as computing intersections of halfspaces and computing Voronoi diagrams and Delaunay triangulations. Chazelle =-=[10]-=- has solved the convex hull problem optimally in the worst case by giving an O(n log n + n bd=2c )-time algorithm. However, this bound depends only on the input size n and is insensitive to the output... |

6 |
Output sensitive construction of levels and Voronoi diagrams in R d of order 1 to k
- Mulmuley
- 1990
(Show Context)
Citation Context ...igher dimensions; if randomization is allowed, this O(n log log n) term can even be removed. As an aside, we point out that the Matousek's results [25] can be used to improve an algorithm by Mulmuley =-=[30]-=- for constructing ( k)-levels of a nonredundant arrangement of n hyperplanes in E d . The algorithm is an extension of Seidel's output-sensitive convex hull algorithm [41] and runs in O(n 2 k d\Gamma1... |

3 |
An optimal algorithm for computing convex layers
- Chazelle
- 1985
(Show Context)
Citation Context ...d as the convex hull of the points of P that are not vertices of the previous layers 1; : : : ; i. It is known that this problem can be solved optimally in O(n log n) time by an algorithm of Chazelle =-=[9] for -=-d = 2 and quasi-optimally in O(n 1+" ) time by an algorithm of Agarwal and Matousek [2] for d = 3. For ds4, Edelsbrunner [16, Problem 10.3(c)] asked whether the vertices of all layers can be iden... |

1 | and in revised form September 14 - April - 1995 |