Computing optimal islands

Bautista-Santiago, Crevel; Díaz-Báñez, José Miguel; Lara, Dolores; Pérez-Lantero, Pablo; Urrutia, Jorge; Ventura, Inmaculada

doi:10.1016/j.orl.2011.04.008

Cited by 26 publications

(35 citation statements)

References 17 publications

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“…Thus G ab is constructed in O((1/ε 3/2 ) 2 log n) = O((1/ε 3 ) log n) time. By Lemma 8, each iteration of the lines 13- We next show that each iteration of the repeat-loop (lines [5][6][7][8][9][10][11][12][13][14] takes O(n log 2 n + (n/ε 3 ) log n + n/ε 4 ) time. Since r = O(n), the sample R can be computed in O(n log n) time, as discussed in Section 4.1.…”

Section: Discussionmentioning

confidence: 92%

“…In one iteration of the repeat-loop (lines [5][6][7][8][9][10][11][12][13][14] of the algorithm LargePotato the algorithm finds a convex polygon of area at least (1 − ε)A * (P ) with probability at least 1/4.…”

Section: Discussionmentioning

confidence: 99%

“…Pruning vertices, we can assume that all vertices of H are adjacent to s and below s. We can then use the algorithm of Bautista-Santiago et al [8], which is an improvement over the algorithm of Fischer [24], restricted to the edges that are in H. For completeness, we provide a quick overview of the approach.…”

Section: Largest Convex Polygon In a Visibility Graphmentioning

confidence: 99%

“…The polygon K returned by LargePotato(P, ε, δ) is always a convex polygon contained in P . We have area(K) < (1 − ε) · A * (P ) if and only if all iterations of the repeat-loop (lines [5][6][7][8][9][10][11][12][13][14] fail to find such a (1 − ε)-approximation. Since each such iteration fails with probability at most 3/4 due to Lemma 16, and there are 3 log 2 (1/δ) iterations, we have…”

Section: If These Two Events Occur and S Is A Point Of Smentioning

confidence: 99%

“…The horizontal sides of Ψ have length K * / cos α, while the other sides have length at most 2 · d 2 / cos α. It is wellknown, and an easy consequence of Crofton's formula (see equation (8) in Theorem 18), that if a convex body K 1 is contained in a convex body K 2 , then per(…”

Section: Bounds Depending On the Fatnessmentioning

confidence: 99%

See 4 more Smart Citations

Peeling Potatoes Near-Optimally in Near-Linear Time

Cabello

Cibulka

Kynčl

et al. 2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon P with n vertices. We give a randomized near-linear-time (1 − ε)-approximation algorithm for this problem: in O(n(log 2 n + (1/ε 3 ) log n + 1/ε 4 )) time we find a convex polygon contained in P that, with probability at least 2/3, has area at least (1 − ε) times the area of an optimal solution. We also obtain similar results for the variant of computing a convex polygon inside P with maximum perimeter.To achieve these results we provide new results in geometric probability. The first result is a bound relating the probability that two points chosen uniformly at random inside P are mutually visible and the area of the largest convex body inside P . The second result is a bound on the expected value of the difference between the perimeter of any planar convex body K and the perimeter of the convex hull of a uniform random sample inside K.

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Section: Discussionmentioning

confidence: 92%

Section: Discussionmentioning

confidence: 99%

Section: Largest Convex Polygon In a Visibility Graphmentioning

confidence: 99%

Section: If These Two Events Occur and S Is A Point Of Smentioning

confidence: 99%

Section: Bounds Depending On the Fatnessmentioning

confidence: 99%

See 3 more Smart Citations

Peeling Potatoes Near-Optimally in Near-Linear Time

Cabello

Cibulka

Kynčl

et al. 2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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Maximum Rectilinear Convex Subsets

González-Aguilar

Orden

Pérez-Lantero³

et al. 2019

Fundamentals of Computation Theory

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Let P be a set of n points in the plane. We consider a variation of the classical Erdős-Szekeres problem, presenting efficient algorithms with O(n 3 ) running time and O(n 2 ) space complexity that compute: (1) A subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P , (2) a subset S of P such that the boundary of the rectilinear convex hull of S has the maximum number of points from P and its interior contains no element of P , (3) a subset S of P such that the rectilinear convex hull of S has maximum area and its interior contains no element of P , and (4) when each point of P is assigned a weight, positive or negative, a subset S of P that maximizes the total weight of the points in the rectilinear convex hull of S.

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Minimum Convex Partitions and Maximum Empty Polytopes

Dumitrescu

Har-Peled

Tóth

2012

Algorithm Theory – SWAT 2012

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Let S be a set of n points in R d . A Steiner convex partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a Steiner convex partition with at most ⌈(n − 1)/d⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n). Here we give a (1 − ε)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0,1] d .

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Computing optimal islands

Cited by 26 publications

References 17 publications

Peeling Potatoes Near-Optimally in Near-Linear Time

Peeling Potatoes Near-Optimally in Near-Linear Time

Maximum Rectilinear Convex Subsets

Minimum Convex Partitions and Maximum Empty Polytopes

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