Dilation-Optimal Edge Deletion in Polygonal Cycles

Journal of Combinatorial Mathematics and Combinatorial Computing

In this paper we present optimal algorithms to compute monotone stabbers for convex polygons. More precisely, given a set of m convex polygons with n vertices in total we want to stab the polygons with an x-monotone polygonal chain such that each polygon is entered at its leftmost point and departed at its rightmost point. Since such a stabber does not exist in general, we study two related problems. In the rst problem we want to compute a monotone stabber that stabs as many convex polygons as possible. The second problem is to compute the minimal number of x-monotone stabbers that are necessary to stab all given convex polygons.

Cccg, 2010

The straight skeleton of a simple polygon is defined as the trace of the vertices when the initial polygon is shrunken in self-parallel manner . In this paper, we propose a simple algorithm for drawing the straight skeleton of a monotone polygon. The time and space complexities of our algorithm are O(nlogn) and O(n) respectively.

Computational Geometry

Let P be a crossing-free polygon and C a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of P. A shortcut hull of P is another crossing-free polygon that encloses P and whose oriented boundary is composed of elements from C. Shortcut hulls find their application in geo-related problems such as the simplification of contour lines. We aim at a shortcut hull that linearly balances the enclosed area and perimeter. If no holes in the shortcut hull are allowed, the problem admits a straightforward solution via shortest paths. For the more challenging case that the shortcut hull may contain holes, we present a polynomial-time algorithm that is based on computing a constrained, weighted triangulation of the input polygon's exterior. We use this problem as a starting point for investigating further variants, e.g., restricting the number of edges or bends. We demonstrate that shortcut hulls can be used for drawing the rough extent of point sets as well as for the schematization of polygons.

Proceedings of the sixteenth annual symposium on Computational geometry - SCG '00, 2000

We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's (1991) celebrated optimal deterministic algorithm and, hence, positively answers his question of whether a simpler randomized algorithm for the problem exists. The new algorithm can be viewed as a combination of Chazelle's algorithm and of non-optimal randomized algorithms due to Clarkson et al. (1991) and to Seidel (1991), with the essential innovation that sampling is performed on subchains of the initial polygonal chain, rather than on its edges. It is also essential, as in Chazelle's algorithm, to include a bottom-up preprocessing phase previous to the top-down construction phase.

2012

In this paper, we consider the restricted version of the well-known 2D line simplification problem under area measures and for restricted version. We first propose a unified definition for both of sum-area and difference-area measures that can be used on a general path of n vertices. The optimal simplification runs in O (n3) under both of these measures.

Journal of Discrete Algorithms, 2006

Let P be an x-monotone polygonal path in the plane. For a path Q that approximates P let W A (Q) be the area above P and below Q, and let W B (Q) be the area above Q and below P .

Journal of Algorithms, 1986

Discrete & Computational Geometry, 2001

We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle's [3] celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle's algorithm and of simple nonoptimal randomized algorithms due to Clarkson et al. [6], [7], [9] and to Seidel [20]. As in Chazelle's algorithm, it is indispensable to include a bottom-up preprocessing phase, in addition to the actual top-down construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done.

SIAM Journal on Computing, 1988

Given a simple n-vertex polygon, the triangulation problem is to partition the interior of the polygon into n-2 triangles by adding n-3 nonintersecting diagonals. We propose an O(n log logn)-time algorithm for this problem, improving on the previously best bound of O (n log n) and showing that triangulation is not as hard as sorting. Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from our result. Key words, amortized time, balanced divide and conquer, heterogeneous finger search tree, homogeneous finger search tree, horizontal visibility information, Jordan sorting with error-correction, simplicity testing AMS(MOS) subject classifications. 51M15, 68P05, 68Q25 1. Introduction. Let P be an n-vertex simple polygon, defined by a list Vo,V vnof its vertices in clockwise order around the boundary. (The interior of the polygon is to the right as one walks clockwise around the boundary.) We denote the boundary of P by 0P. We assume throughout this paper (without loss of generality) that the vertices of P have distinct y-coordinates. For convenience we define vn V o. The edges of P are the open line segments whose endpoints are vi,vi+l for 0 < < n. The diagonals of P are the open line segments whose endpoints are vertices and that lie entirely in the interior of P. The triangulation problem is to find n-3 nonintersecting diagonals of P, which partition the interior of P into n-2 triangles. If P is convex, any pair of vertices defines a diagonal, and it is easy to triangulate P in O (n) time. If P is not convex, not all pairs of vertices define diagonals, and even finding one diagonal, let alone triangulating P, is not a trivial problem. In 1978, Garey, Johnson, Preparata and Tarjan [10] presented an O (n log n)-time triangulation algorithm. Since then, work on the problem has proceeded in two directions. Some authors have developed linear-time algorithms for triangulating special classes of polygons, such as monotone polygons [10] and star-shaped polygons [31 ]. Others have devised triangulation algorithms whose running time is O (n log k) for a parameter k that somehow quantifies the complexity of the polygon, such as the number of reflex angles [13] or the "sinuosity" [5]. Since these measures all admit classes of polygons with k 1 (n), the worst case running time of these algorithms is only known to be O(n log n). Determining whether triangulation can be done in o (n log n) time, i.e. asymptotically faster than sorting, has been one of the foremost open problems in computational geometry. In this paper we propose an O (n log logn)-time triangulation algorithm, thereby showing that triangulation is indeed easier than sorting. The paper is a revised and corrected version of a conference paper [27] which erroneously claimed an O (n)-time algorithm. The goal of obtaining a linear-time algorithm remains elusive, but our *

In this paper we present optimal algorithms to compute monotone stabbers for convex polygons. More precisely, given a set of m convex polygons with n vertices in total we want to stab the polygons with an x-monotone polygonal chain such that each polygon is entered at its leftmost point and departed at its rightmost point. Since such a stabber does not exist in general, we study two related problems. In the rst problem we want to compute a monotone stabber that stabs as many convex polygons as possible. The second problem is to compute the minimal number of x-monotone stabbers that are necessary to stab all given convex polygons.