Geodesic-Preserving Polygon Simplification

Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon $\mathcal{P}$ by a polygon $\mathcal{P}'$ such that (1) $\mathcal{P}'$ contains $\mathcal{P}$, (2) $\mathcal{P}'$ has its reflex vertices at the same positions as $\mathcal{P}$, and (3) the number of vertices of $\mathcal{P}'$ is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both $\mathcal{P}$ and $\mathcal{P}'$, our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of $\mathcal{P}$

Planar bichromatic minimum spanning trees

AbstractGiven a set S of n red and blue points in the plane, a planar bichromatic minimum spanning tree is the shortest possible spanning tree of S, such that every edge connects a red and a blue point, and no two edges intersect. We show that computing this tree is NP-hard in general. For points in convex position, a cubic-time algorithm can be easily designed using dynamic programming. We adapt such an algorithm for the special case where the number of red points (m) is much smaller than the number of blue points (n), resulting in an O(nm2) time algorithm. For the general case, we present a factor O(n) approximation algorithm that runs in O(nlognloglogn) time. Finally, we show that if the number of points in one color is bounded by a constant, the optimal tree can be computed in polynomial time

Planar Bichromatic Bottleneck Spanning Trees

Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8?2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck ? can be converted to a planar bichromatic spanning tree of bottleneck at most 8?2 ?

Separating point sets in polygonal environments

SCOPUS: cp.jinfo:eu-repo/semantics/publishe

Separating point sets in polygonal environments

info:eu-repo/semantics/publishe