Maximum-area triangle in a convex polygon, revisited

Hoog, Ivor van der; Keikha, Vahideh; Löffler, Maarten; Mohades, Ali; Urhausen, Jérôme

doi:10.1016/j.ipl.2020.105943

Cited by 14 publications

(4 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Apart from their theoretical interest, these problems have found applications in various areas of computer science and mathematics (optimization, packing and covering, approximation algorithms, convexity, computational geometry), see [8,14,24]. In the past decade, several explicit algorithms were proposed for the case of triangles [2,15,20,26].…”

Section: Optimal Covers From a Classmentioning

confidence: 99%

Optimal Embedded and Enclosing Isosceles Triangles

Ambrus,

Csikós,

Kiss

et al. 2023

Int. J. Found. Comput. Sci.

View full text Add to dashboard Cite

Given a triangle [Formula: see text], we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of [Formula: see text] with respect to area and perimeter. This problem was initially posed by Nandakumar [17, 22] and was first studied by Kiss, Pach, and Somlai [13], who showed that if [Formula: see text] is the smallest area isosceles triangle containing [Formula: see text], then [Formula: see text] and [Formula: see text] share a side and an angle. In the present paper, we prove that for any triangle [Formula: see text], every maximum area isosceles triangle embedded in [Formula: see text] and every maximum perimeter isosceles triangle embedded in [Formula: see text] shares a side and an angle with [Formula: see text]. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles [Formula: see text] whose minimum perimeter isosceles containers do not share a side and an angle with [Formula: see text].

show abstract

Section: Optimal Covers From a Classmentioning

confidence: 99%

Optimal Embedded and Enclosing Isosceles Triangles

Ambrus,

Csikós,

Kiss

et al. 2023

Int. J. Found. Comput. Sci.

View full text Add to dashboard Cite

show abstract

“…Our problem also relates to polygon approximation, for which prior work often instead considered approximation in relation to area. For example, given a convex polygon P , [15] gave a near linear time algorithm for finding the three vertices of P whose triangle has the maximum area. To illustrate one the many ways that area approximations differ, observe that the area of the triangle of the three given points of P can be determined in constant time, whereas the computing the furthest point from P to the triangle takes linear time.…”

Section: Other Related Problemsmentioning

confidence: 99%

Fast and Exact Convex Hull Simplification

Klimenko,

Raichel

2021

Preprint

View full text Add to dashboard Cite

Given a point set P in the plane, we seek a subset Q ⊆ P , whose convex hull gives a smaller and thus simpler representation of the convex hull of P . Specifically, let cost(Q, P ) denote the Hausdorff distance between the convex hulls CH(Q) and CH(P ). Then given a value ε > 0 we seek the smallest subset Q ⊆ P such that cost(Q, P ) ≤ ε. We also consider the dual version, where given an integer k, we seek the subset Q ⊆ P which minimizes cost(Q, P ), such that |Q| ≤ k. For these problems, when P is in convex position, we respectively give an O(n log 2 n) time algorithm and an O(n log 3 n) time algorithm, where the latter running time holds with high probability. When there is no restriction on P , we show the problem can be reduced to APSP in an unweighted directed graph, yielding an O(n 2.5302 ) time algorithm when minimizing k and an O(min{n 2.5302 , kn 2.376 }) time algorithm when minimizing ε, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case.

show abstract

“…A significant amount of work has also been done on computing largest-area triangle inside a given polygon [3,5,9,16]. The problems of finding largest area triangle [14,15] and largest area quadrilateral [17] inside a convex polygon are also studied recently. In this paper, we propose a deterministic O(n log n)-time algorithm to find a largest-area triangle inside a given terrain, which improves the best known running time of O(n 2 ), presented in [8].…”

Section: Introductionmentioning

confidence: 99%