Local Routing in Convex Subdivisions

In various wireless networking settings, node locations determine a network’s topology, allowing the network to be modelled by a geometric graph drawn in the plane. Without any additional information, local geometric routing algorithms can guarantee delivery to the target node only in restricted classes of geometric graphs, such as triangulations. In order to guarantee delivery on more general classes of geometric graphs (e.g., convex subdivisions or planar subdivisions), previous local geometric routing algorithms required [Formula: see text] state bits to be stored and passed with the message. We present the first local geometric routing algorithm using only one state bit to guarantee delivery on convex subdivisions and, when the algorithm has knowledge of the incoming port (the preceding node on the route), the first stateless local geometric routing algorithm that guarantees delivery on edge-augmented monotone subdivisions (including all convex subdivisions). We also show that [Formula: see text] state bits are necessary in planar subdivisions in which faces may have three or more reflex vertices.

Overlay Algorithm for Simple Subdivision of Arbitrary Polygon in Plane

The overlay algorithm is an important branch in computational geometry field, it is an important process for computing exact Minkowski sum of two convex polyhedrons. By improving the existing plane sweep algorithm, the overlay algorithm for simple subdivision of arbitrary polygon in plane is given. The algorithm can be used to overlay arbitrary polygon after subdivision into simple polygon in the plane. It has lower time complexity than the existing overlay algorithm. The whole algorithm consists of three steps: line segment intersection, reconstructing topology and constructing the DCEL for overlay graph. The results show that the algorithm can compute the overlay of two planar subdivisions in linear time.

FRÉCHET DISTANCE PROBLEMS IN WEIGHTED REGIONS

We discuss two versions of the Fréchet distance problem in weighted planar subdivisions. In the first one, the distance between two points is the weighted length of the line segment joining the points. In the second one, the distance between two points is the length of the shortest path between the points. In both cases, we give algorithms for finding a (1 + ∊)-factor approximation of the Fréchet distance between two polygonal curves. We also consider the Fréchet distance between two polygonal curves among polyhedral obstacles in [Formula: see text] (1/∞ weighted region problem) and present a (1 + ∊)-factor approximation algorithm.

TRAVERSAL OF A QUASI-PLANAR SUBDIVISION WITHOUT USING MARK BITS

The problem of traversal of planar subdivisions or other graph-like structures without using mark bits is central to many real-world applications [7, 8, 11, 12, 13, 17, 18]. The first such algorithms developed were able to traverse triangulated subdivisions [10]. Later these algorithms were extended to traverse vertices of an arrangement or a convex polytope [3]. The research progress culminated to an algorithm that can traverse any planar subdivision [6, 9]. In this paper, we extend the notion of planar subdivision to quasi-planar subdivision in which we allow many edges to cross each other. We generalize the algorithm from [9] to traverse any quasi-planar subdivision that satisfies a simple geometric requirement. If we use techniques from [6] the worst case running time of our algorithm is O(|E| log |E|); matching the running time of the traversal algorithm for planar subdivisions [6].