Related papers: Maximum-Area Rectangles in a Simple Polygon

We consider the following geometric optimization problem: find a maximum-area rectangle and a maximum-perimeter rectangle contained in a given convex polygon with $n$ vertices. We give exact algorithms that solve these problems in time…

Let $k \geq 2$ be a constant. Given any $k$ convex polygons in the plane with a total of $n$ vertices, we present an $O(n\log^{2k-3}n)$ time algorithm that finds a translation of each of the polygons such that the area of intersection of…

We study the problem of finding maximum-area triangles that can be inscribed in a polygon in the plane. We consider eight versions of the problem: we use either convex polygons or simple polygons as the container; we require the triangles…

Polygon inclusion problems have been studied extensively in geometric optimization. In this paper, we consider the variant of computing the maximum area parallelograms (MAPs) and all the locally maximal area parallelograms (LMAPs) in a…

A convex polygon Q is circumscribed about a convex polygon P if every vertex of P lies on at least one side of Q. We present an algorithm for finding a maximum area convex polygon circumscribed about any given convex n-gon in O(n^3) time.…

We study the problem of computing a convex region with bounded area and diameter that contains the maximum number of points from a given point set $P$. We show that this problem can be solved in $O(n^6k)$ time and $O(n^3k)$ space, where $n$…

Let $P$ be a set of $n$ points in the plane. We consider a variation of the classical Erd\H{o}s-Szekeres problem, presenting efficient algorithms with $O(n^3)$ running time and $O(n^2)$ space complexity that compute: (1) A subset $S$ of $P$…

Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given…

Given a set $P$ of $n$ points on $\mathbb R^{2}$, we address the problem of computing an axis-parallel empty rectangular annulus $A$ of maximum-width such that no point of $P$ lies inside $A$ but all points of $P$ must lie inside, outside…

We propose a technique called Rotate-and-Kill for solving the polygon inclusion and circumscribing problems. By applying this technique, we obtain $O(n)$ time algorithms for computing (1) the maximum area triangle in a given $n$-sided…

Let $P$ be a set of $n$ points in an axis-parallel rectangle $B$ in the plane. We present an $O(n\alpha(n)\log^4 n)$-time algorithm to preprocess $P$ into a data structure of size $O(n\alpha(n)\log^3 n)$, such that, given a query point $q$,…

Let $P$ be a convex polyhedron and $Q$ be a convex polygon with $n$ vertices in total in three-dimensional space. We present a deterministic algorithm that finds a translation vector $v \in \mathbb{R}^3$ maximizing the overlap area $|P \cap…

Given the n vertices of a convex polygon in cyclic order, can the triangle of maximum area inscribed in P be determined by an algorithm with O(n) time complexity? A purported linear-time algorithm by Dobkin and Snyder from 1979 has recently…

This paper attacks the following problem. We are given a large number $N$ of rectangles in the plane, each with horizontal and vertical sides, and also a number $r<N$. The given list of $N$ rectangles may contain duplicates. The problem is…

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 $(1-\varepsilon)$-approximation algorithm for…

We revisit the following problem: Given a convex polygon $P$, find the largest-area inscribed triangle. We show by example that the linear-time algorithm presented in 1979 by Dobkin and Snyder for solving this problem fails. We then proceed…

In this paper, we address the minimum-area rectangular and square annulus problem, which asks a rectangular or square annulus of minimum area, either in a fixed orientation or over all orientations, that encloses a set $P$ of $n$ input…

We show that the maximum number of convex polygons in a triangulation of $n$ points in the plane is $O(1.5029^n)$. This improves an earlier bound of $O(1.6181^n)$ established by van Kreveld, L\"offler, and Pach (2012) and almost matches the…

There is a high demand of space-efficient algorithms in built-in or embedded softwares. In this paper, we consider the problem of designing space-efficient algorithms for computing the maximum area empty rectangle (MER) among a set of…

A terrain is an $x$-monotone polygon whose lower boundary is a single line segment. We present an algorithm to find in a terrain a triangle of largest area in $O(n \log n)$ time, where $n$ is the number of vertices defining the terrain. The…