Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

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.

. Consider a robot that has to travel from a start location s to a target t in an environment with opaque obstacles that lie in its way. The robot always knows its current absolute position and that of the target. It does not, however, know the positions and extents of the obstacles in advance; rather, it finds out about obstacles as it encounters them. We compare the distance walked by the robot in going from s to t to the length of the shortest (obstacle-free) path between s and t in the scene. We describe and analyze robot strategies that minimize this ratio for different kinds of scenes. In particular, we consider the cases of rectangular obstacles aligned with the axes, rectangular obstacles in more general orientations, and wider classes of convex bodies both in two and three dimensions. For many of these situations, our algorithms are optimal up to constant factors. We study scenes with non-convex obstacles, which are related to the study of maze-traversal. We also show scenes ...

Abstract not found

We consider the problem of localizing a robot in a known environment modeled by a simple polygon P . We assume that the robot has a map of P but is placed at an unknown location inside P . From its initial location, the robot sees a set of points called the visibility polygon V of its location. In general, sensing at a single point will not suffice to uniquely localize the robot, since the set H of points in P with visibility polygon V may have more than one element. Hence, the robot must move around and use range sensing and a compass to determine its position (i.e.

We consider a number of search and exploration problems, from the perspective of robot navigation in a simple polygon. These problems are "on-line" in the sense that the robot does not have access to the map of the polygon; it must make decisions as it proceeds, based only on what it has seen so far. For the problem of exploring a simple rectilinear polygon (under the L 1 norm), Deng, Kameda, and Papadimitriou give a 2-competitive deterministic algorithm; we present a randomized exploration algorithm which is 5=4-competitive. Using similar techniques, we are able to give an algorithm for searching an arbitrary, unknown rectilinear polygon. No constant competitive ratio is attainable in this case, but our algorithm is within a constant factor of optimal in the worst case; in a sense, it is a generalization of some of the strategies of BaezaYates, Culberson, and Rawlins to a much more general class of search spaces. Finally, we examine a type of polygon for which competitive search is po...

We consider online routing algorithms for finding paths between the vertices of plane graphs.

We present a competitive strategy for walking into the kernel of an initially unknown star-shaped polygon. From an arbitrary start point, s, within the polygon, our strategy finds a path to the closest kernel point, k, whose length does not exceed 5.3331 ...times the distance from s to k. This is complemented by a general lower bound of # 2. Our analysis relies on a result about a new and interesting class of curves which are self-approaching in the following sense.

We consider the problem of a searcher exploring an initially unknown weighted planar graph G. When the searcher visits a vertex v, it learns of each edge incident to v. The searcher's goal is to visit each vertex of G, incurring as little cost as possible. We present a constant competitive algorithm for this problem. 1 Introduction In this paper we consider the following situation. A salesperson is assigned to visit all the towns in some rural state that he/she knows nothing about. Of course, the salesperson wishes to accomplish this with as little time spent traveling as possible. The salesperson, however, is not given the benefit of having a map. Hence, when the salesperson visits a town, the only information that he/she may be able to glean about other cities is from the road signs on the roads leaving that town. Each road sign gives the name and the distance to the next city down that road. As the salesperson visits towns, new information may reveal shorter routes and may cause th...

We study how a mobile robot can piecemeal learn an unknown environment. The robot's goal is to learn a complete map of its environment, while satisfying the constraint that it must return every so often to its starting position s #for refueling, say#.