We consider the problem of finding inscribed boxes and axis-aligned inscribed boxes of maximum volume, inside a compact and solid convex set. Our algorithms are capable of solving these two problems in any such set that can be represented with finite number of convex inequalities. For the axis-aligned case, we formulate the problem for higher dimensions and present an exact optimization algorithm which solves the problem in @math time, where @math is the dimension and @math is the number of inequalities defining the convex set. For the general case, after formulating the problem for higher dimensions we investigate the traditional 2-dimensional problem, which is in the literature merely considered for convex polygons, for a broad range of convex sets. We first present a new exact algorithm that finds the largest inscribed axis-aligned rectangle in such convex sets for any given direction of axes in @math time. Using this exact algorithm as a subroutine, we present an @math -approximation algorithm that computes @math -approximation to the largest inscribed rectangle with computational complexity of @math . Finally, we show that how this running time can be improved to @math with a @math pre-processing time when the convex set is a polygon.
Finding largest inscribed objects inside a polygon is not restricted to neither rectangles nor convex polygons. @cite_5 presented @math algorithms for finding inscribed equilateral triangle and square of maximum area inside a convex polygon and an @math algorithm for finding the largest inscribed equilateral triangle inside a general polygon. @cite_3 developed an @math algorithm for the problem of finding the largest empty rectangle among a point set. Jin proposed an @math algorithm @cite_31 and an @math algorithm @cite_33 for finding all locally maximal area parallelogram inside a convex polygon. Recently, @cite_23 showed that a long lasting linear time algorithm for finding the maximal triangle inside a convex polygon proposed in 1979 by Dobkin and Snyder back @cite_25 was in fact incorrect and then provided an @math algorithm for this problem. Later, Jin @cite_7 proposed another linear time algorithm for the problem of finding the largest triangle inside a convex polygon.
@cite_33 @cite_7 @cite_3 @cite_23 @cite_5 @cite_31 @cite_25
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We revisit the following problem: Given a convex polygon @math , 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 to show that with a small adaptation, their approach does lead to a quadratic-time algorithm. We also present a more involved @math time divide-and-conquer algorithm. Also we show by example that the algorithm presented in 1979 by Dobkin and Snyder for finding the largest-area @math -gon that is inscribed in a convex polygon fails to find the optimal solution for @math . Finally, we discuss the implications of our discoveries on the literature. We consider approximation algorithms for the problem of computing an inscribed rectangle having largest area in a convex polygon on n vertices. If the order of the vertices of the polygon is given, we present a randomized algorithm that computes an inscribed rectangle with area at least ([email protected]) times the optimum with probability t in time O([email protected]) for any constant t<1. We further give a deterministic approximation algorithm that computes an inscribed rectangle of area at least ([email protected]) times the optimum in running time O([email protected]^2logn) and show how this running time can be slightly improved. The widely known linear time algorithm for computing the maximum area triangle in a convex polygon was found incorrect recently by Keikha et. al.(arXiv:1705.11035). We present an alternative algorithm in this paper. Comparing to the only previously known correct solution, ours is much simpler and more efficient. More importantly, our new approach is powerful in solving related problems. We study a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick") or a maximum-area triangle or convex body ("potato"). Exact polynomial-time algorithms are known for some of these problems, but their time bounds are high (e.g., O(n7) for the largest convex polygon in a simple n-gon). We devise efficient approximation algorithms for these problems. In particular, we give near-linear time algorithms for a (1 - ∈)-approximation of the biggest stick, an O(1)-approximation of the maximum-area convex body, and a (1 - ∈)-approximation of the maximum-area fat triangle or rectangle. In addition, we give efficient methods for computing large ellipses inside a polygon (whose vertices are a dense sampling of a closed smooth curve). Our algorithms include both deterministic and randomized methods, one of which has been implemented (for computing large area ellipses in a well sampled closed smooth curve).