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A rectilinear polygon is a polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of right turns plus 4. It is known that any such sequence can be realized by a rectilinear polygon. In this paper, we consider the problem of finding realizations that minimize the perimeter or the area of the polygon or the area of the bounding box of the polygon. We show that all three problems are NP-hard in general. This answers an open question of Patrignani [CGTA 2001], who showed that it is NP-hard to minimize the area of the bounding box of an orthogonal drawing of a given planar graph. We also show that realizing polylines with minimum bounding box area is NP-hard. Then we consider the special cases of $x$-monotone and $xy$-monotone rectilinear polygons. For these, we can optimize the three objectives efficiently.
We consider the problem of finding minimum-link rectilinear paths in rectilinear polygonal domains in the plane. A path or a polygon is rectilinear if all its edges are axis-parallel. Given a set $mathcal{P}$ of $h$ pairwise-disjoint rectilinear poly
We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP) asks for
We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a emph{graph of oriented distances} to encode the distance between pairs of poin
This paper presents an optimal $Theta(n log n)$ algorithm for determining time-minimal rectilinear paths among $n$ transient rectilinear obstacles. An obstacle is transient if it exists in the scene only for a specific time interval, i.e., it appears
We present an $O(nlog n)$-time algorithm that determines whether a given planar $n$-gon is weakly simple. This improves upon an $O(n^2log n)$-time algorithm by Chang, Erickson, and Xu (2015). Weakly simple polygons are required as input for several g