The power of geometric duality

Chazelle, Bernard; Guibas, Leo J.; Lee, D. T.

doi:10.1109/sfcs.1983.75

Cited by 105 publications

(126 citation statements)

References 12 publications

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“…, p * n }. Line q * can also be inserted in the arrangement in O(n) time [6]. Let r q denote the vertical upward ray from q and l q its supporting line.…”

Section: Angular Sortingmentioning

confidence: 99%

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Space–Query-Time Tradeoff for Computing the Visibility Polygon

Nouri

Ghodsi²

2009

Frontiers in Algorithmics

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Abstract. Computing the visibility polygon, VP, of a point in a polygonal scene, is a classical problem that has been studied extensively. In this paper, we consider the problem of computing VP for any query point efficiently, with some additional preprocessing phase. The scene consists of a set of obstacles, of total complexity O(n). We show for a query point q, V P (q) can be computed in logarithmic time using O(n 4 ) space and O(n 4 log n) preprocessing time. Furthermore to decrease space usage and preprocessing time, we make a tradeoff between space usage and query time; so by spending O(m) space, we can achieve O(n 2 log(query time, where n 2 ≤ m ≤ n 4 . These results are also applied to angular sorting of a set of points around a query point.

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“…, p * n }. Line q * can also be inserted in the arrangement in O(n) time [6]. Let r q denote the vertical upward ray from q and l q its supporting line.…”

Section: Angular Sortingmentioning

confidence: 99%

“…Using O(n 2 ) preprocessing time and O(n 2 ) space for constructing A(P * ) by an algorithm due to Chazelle et al [6], for any query point q, we can find the angular sorted list of…”

Section: Lemma 2 [Asano Et Al]mentioning

confidence: 99%

Space–Query-Time Tradeoff for Computing the Visibility Polygon

Nouri

Ghodsi²

2009

Frontiers in Algorithmics

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“…The exact algorithm they propose needs time O(n 5 log n). In contrast, we use the concept of geometric duality which Chazelle, Guibas and Lee [2] propose for solving geometrical problems. Hence, we obtain an expected running time of O(n 2 log 2 n) for our exact algorithm and a running time of roughly O(n 2 log n) for our approximation algorithm.…”

Section: Solving the Lqd Geometricallymentioning

confidence: 99%

Computing the least quartile difference estimator in the plane

Bernholt

Nunkesser

Schettlinger

2007

Computational Statistics & Data Analysis

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Abstract. A common problem in linear regression is that largely aberrant values can strongly influence the results. The least quartile difference (LQD) regression estimator is highly robust, since it can resist up to almost 50% largely deviant data values without becoming extremely biased. Additionally, it shows good behavior on Gaussian data -in contrast to many other robust regression methods. However, the LQD is not widely used yet due to the high computational effort needed when using common algorithms, e.g. the subset algorithm of Rousseeuw and Leroy. For computing the LQD estimator for n data points in the plane, we propose a randomized algorithm with expected running time O(n 2 log 2 n) and an approximation algorithm with a running time of roughly O(n 2 log n). It can be expected that the practical relevance of the LQD estimator will strongly increase thereby.

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“…In this case, we s a y that nicely bounds C. More formally, uniquely spans a boundary facet of C and cuts the intersection of the surfaces that contain C at some point on that boundary facet. 2 Thus, there is a one-to-one correspondence between reversible simplices and nice bounding surfaces.…”

Section: Lower Bounds For the General Problemmentioning

confidence: 99%

Better lower bounds on detecting affine and spherical degeneracies

Erickson

Seidel

1995

Discrete Comput Geom

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We show that in the worst case, (n d ) sidedness queries are required to determine whether a set of n points in IR d is a nely degenerate, i.e., whether it contains d + 1 p o i n ts on a common hyperplane. This matches known upper bounds. We g i v e a straightforward adversary argument, based on the explicit construction of a point set containing (n d ) \collapsible" simplices, any o n e o f which can be made degenerate without changing the orientation of any o t h e r simplex. As an immediate corollary, w e h a ve a n ( n d ) l o wer bound on the number of sidedness queries required to determine the order type of a set of n points in IR d . Using similar techniques, we a l s o s h o w t h a t ( n d+1 ) in-sphere queries are required to decide the existence of spherical degeneracies in a set of n points in IR d .

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The power of geometric duality

Cited by 105 publications

References 12 publications

Space–Query-Time Tradeoff for Computing the Visibility Polygon

Space–Query-Time Tradeoff for Computing the Visibility Polygon

Computing the least quartile difference estimator in the plane

Better lower bounds on detecting affine and spherical degeneracies

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