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 algorithms for finding paths between the vertices of plane graphs.

We present a new strategy for searching for a goal in a street. The strategy works in two phases. First it follows an angular bisector, then it uses circular arcs based only on one side of the street. A competitive factor of 1.514 is achieved which is remarkably close to the lower bound of # 2. Secondly, we assume that the location of the goal is known to the robot. We prove a lower bound of # 2 on the competitive ratio of any deterministic strategy for searching in streets with known destination.

A strategy S solving a navigation task T is called competitive with ratio r if the cost of solving any instance t of T does not exceed r times the cost of solving t optimally. The competitive complexity of task T is the smallest possible value r any strategy S can achieve. We discuss this notion, and survey some tasks whose competitive complexities are known.

. We study the problem of on-line searching for a target inside a polygon. In particular we propose a strategy for finding a target of unknown location in a star polygon with a competitive ratio of 14.5, and we further refine it to 12.72. This makes star polygons the first non-trivial class of polygons known to admit constant competitive searches independent of the position of the target. We also provide a lower bound of 9 for the competitive ratio of searching in a star polygon---which is close to the upper bound. A similar task consists of the problem of on-line recognition of star polygons for which we also present a strategy with a constant competitive ratio including negative instances. 1 Introduction In the past years on-line searching has been an active area of research in Computer Science (e.g. [1, 2, 4, 7, 8, 11]). In its full generality, an on-line search problem consists of an agent or robot searching for a target on an unknown terrain. In the worst case a search by a robot...

We consider the problem of a robot which has to find a target in an unknown simple polygon, based only on what it has seen so far. A street is a polygon for which the two boundary chains from start to target are mutually weakly visible. A target inside a street can be found by walking a path that is at most a constant times longer than the shortest path in the street from start to target. We define a strictly larger class of polygons, called generalized streets or G-streets, which are characterized by the property that every point on the boundary of a G-street is visible from a point on a horizontal line segment connecting the two boundary chains. We present an on-line strategy for a robot to find the target in an unknown rectilinear G-street; the length of its path is at most 9 times the length of the shortest path in the L 1 metric, and 9.06 times the length of the L 2 -shortest path. These bounds are optimal. Key words: Simple polygon, street, searching, doubling, competitive...

We present an optimal strategy for searching for a goal in a street which achieves the competitive factor of # 2, thus matching the best lower bound known before. This finally settles an interesting open problem in the area of competitive path planning many authors have been working on.

We present a technique to prove lower bounds for on-line geometric searching problems. It is assumed that a goal which has to be found by a searcher is hidden somewhere in a known environment. The search cost is proportional to the distance traveled by the searcher. We are interested in lower bounds on the competitive ratio, that is, the ratio of the distance traveled by the searcher to the length of the shortest possible path to reach the goal. The technique we present is applicable to a number of problems, such as biased searching on m rays, and searching in `-streets. For each of these problems we prove lower bounds for large classes of search strategies. All our lower bounds match the best known upper bounds.

. We investigate parallel searching on m concurrent rays. We assume that a target t is located somewhere on one of the rays; we are given a group of m point robots each of which has to reach t. Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy S we are interested in the competitive ratio dened as the ratio of the time needed by the robots to reach t using S and the time needed to reach t if the location of t is known in advance. If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9 | independent of m. We show that 9 is a lower bound on the competitive ratio for two large classes of strategies if m 2. If the minimum distance to the target is not known in advance, we show a lower bound on the competitive ratio of 1 + 2(k + 1) k+1 =k k where k = dlog me. We also give a strategy that obtains this ratio. 1 Introduction Searching for a target is an important a...

We consider the on-line navigation problem of a tactile robot searching for a target point g from a starting vertex s in an initially unknown street, which is a simple planar polygon (P, s, g) characterized by the property that the two oriented chains from s to g are mutually weakly visible. We first present a deterministic competitive strategy of searching in unknown streets which achieves an almost optimal competitive ratio of 2+ p 5 2 p 2 (ß 1:498) in the L 2 Euclidean metric and significantly improves the previously best known competitive upper bound [12] of 1:73. Second, we easily modify our strategy into a slightly different on-line algorithm HLS (High Level Strategy) which minimizes the clad (continuous local absolute detour) [8, 11] and has a better competitive factor of 3+ p 5 2 p 2 (ß 1:85) than the best known bound of 2:03 for this class of strategies. These greedy strategies have simple analyses with an optimal lower bound of p 2 (? 1:41) on the competit...