We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guarantee that a packet is delivered to its destination. These algorithms can be extended to yield algorithms for broadcasting and geocasting that do not require packet duplication. A byproduct of our results is a simple distributed protocol for extracting a planar subgraph of a unit graph. We also present simulation results on the performance of our algorithms.

We present a new approach for computing generalized 2D and 3D Voronoi diagrams using interpolation-based polygon rasterization hardware. We compute a discrete Voronoi diagram by rendering a three dimensional distance mesh for each Voronoi site. The polygonal mesh is a bounded-error approximation of a (possibly) non-linear function of the distance between a site and a 2D planar grid of sample points. For each sample point, we compute the closest site and the distance to that site using polygon scan-conversion and the Z-buffer depth comparison. We construct distance meshes for points, line segments, polygons, polyhedra, curves, and curved surfaces in 2D and 3D. We generalize to weighted and farthest-site Voronoi diagrams, and present efficient techniques for computing the Voronoi boundaries, Voronoi neighbors, and the Delaunay triangulation of points. We also show how to adaptively refine the solution through a simple windowing operation. The algorithm has been implemented on SGI workstations and PCs using OpenGL, and applied to complex datasets. We demonstrate the application of our algorithm to fast motion planning in static and dynamic environments, selection in complex user-interfaces, and creation of dynamic mosaic effects.

Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3-space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.

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We consider online routing strategies for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing strategies, one that works for all Delaunay triangulations and the other that works for all regular triangulations, (2) a randomized memoryless strategy that works for all triangulations, (3) an O(1) memory strategy that works for all convex subdivisions, (4) an O(1) memory strategy that approximates the shortest path in Delaunay triangulations, and (5) theoretical and experimental results on the competitiveness of these strategies.

We present a method for computer generated pen-and-ink illustrations by the simulation of stippling. In a stipple drawing, dots are used to represent tone and also material of surfaces. We create such drawings by generating an initial dot set which is then processed by a relaxation method based on Voronoi diagrams. The point patterns generated are approximations of Poisson disc distributions and can also be used for integrating functions or the positioning of objects. We provide an editor similar to paint systems for interactively creating stipple drawings. This makes it possible to create such drawings within a matter of hours, instead of days or even weeks when the drawing is done manually.

We give a linear-time algorithm for computing the medial axis of a simple polygon P , This answers a long-standing open question -- previously, the best deterministic algorithm ran in O(n log n) time. We decompose P into pseudo-normal histograms, then influence histograms and xy monotone histograms. We can compute the medial axes for xy monotone histograms and merge to obtain the medial axis for P .

Several papers showed how to perform routing in ad hoc wireless networks based on the positions of the mobile hosts. However, all these protocols are likely to fail if the transmission ranges of the mobile hosts vary due to natural or man-made obstacles or weather conditions. These protocols may fail because in routing either some connections are not considered which eectively results in disconnecting the network, or the use of some connections causes livelocks. In this paper, we describe a robust routing protocol that tolerates up to roughly 40% of variation in the transmission ranges of the mobile hosts. More precisely, our protocol guarantees message delivery in a connected adhoc network whenever the ratio of the maximum transmission range to the minimum transmission range is at most 2.

The traditional artistic technique of stippling places small dots of ink onto paper such that their density give the impression of tone. The artist tightly controls the relative placement of the stipples on the paper to produce even tones and avoid artifacts, leading to long creation times for the drawings.

A broad variety of location dependent services will become feasible in the near future due to the use of the Global Position System (GPS), which provides location information (latitude, longitude and possibly height) and global timing to mobile users. Routing is a problem of sending a message from a source to a destination. Geocasting is the problem of sending a message to all nodes located within a region (e.g. circle or square). Recently, several localized GPS based routing and geocasting protocols for a mobile ad hoc network were reported in literature. In directional (DIR) routing and geocasting methods, node A (the source or intermediate node) transmits a message m to all neighbors located between the two tangents from A to the region that could contain the destination. It was shown that memoryless directional methods may create loops in routing process. In two other proposed methods (proven to be loop-free), geographic distance (GEDIR) or most forward progress within radius (MFR)...