Treasure hunt is the task of finding an inert target by a mobile agent in an unknown environment. We consider treasure hunt in geometric terrains with obstacles. Both the terrain and the obstacles are modeled as polygons and both the agent and the treasure are modeled as points. The agent navigates in the terrain, avoiding obstacles, and finds the treasure when there is a segment of length at most 1 between them, unobstructed by the boundary of the terrain or by the obstacles. The cost of finding the treasure is the length of the trajectory of the agent. We investigate the amount of information that the agent needs a priori in order to find the treasure at cost @math , where @math is the length of the shortest path in the terrain from the initial position of the agent to the treasure, avoiding obstacles. Following the paradigm of algorithms with advice, this information is given to the agent in advance as a binary string, by an oracle cooperating with the agent and knowing the whole environment: the terrain, the position of the treasure and the initial position of the agent. Information complexity of treasure hunt is the minimum length of the advice string (up to multiplicative constants) that enables the agent to find the treasure at cost @math . We first consider treasure hunt in regular terrains which are defined as convex polygons with convex @math -fat obstacles, for some constant @math . A polygon is @math -fat if the ratio of the radius of the smallest disc containing it to the radius of the largest disc contained in it is at most @math . For the class of regular terrains, we establish the exact information complexity of treasure hunt. We then show that information complexity of treasure hunt for the class of arbitrary terrains (even for non-convex polygons without obstacles, and even for those with only horizontal or vertical sides) is exponentially larger than for regular terrains.
Exploration of polygons by a robot with limited vision has been studied, e.g., in @cite_27 @cite_7 @cite_0 . In @cite_27 the authors described an on-line algorithm with competitive ratio @math , where @math is a quantity depending on the perimeter of the polygon, @math is the area seen by the robot, and @math is the area of the polygon. In @cite_0 the author studied exploration of the boundary of a terrain with limited vision. The cost of exploration of arbitrary terrains with obstacles, both for limited and unlimited vision, was studied in @cite_30 .
@cite_0 @cite_27 @cite_30 @cite_7
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We consider the motion planning problem for a point constrained to move along a smooth closed convex path of bounded curvature. The workspace of the moving point is bounded by a convex polygon with m vertices, containing an obstacle in a form of a simple polygon with n vertices. We present an O(m+n) time algorithm finding the path, going around the obstacle, whose curvature is the smallest possible. The paper considers the problem of covering a continuous planar area by a square-shaped tool attached to a mobile robot. Using a tool-based approximation of the work-area, we present an algorithm that covers every point of the approximate area. The algorithm, called spanning tree covering (STC), subdivides the work-area into disjoint cells corresponding to the square-shaped tool, then follows a spanning tree of the graph induced by the cells, while covering every point precisely once. We present and analyze three versions of the STC algorithm. The first version is an off-line algorithm that computes an optimal covering path in linear time O(N), where N is the number of cells comprising the approximate area. The second version is an online or sensor based algorithm, that completes an optimal covering path in time O(N), but requires O(N) memory for its implementation. The third version of STC is "ant"-like, where the robot may leave pheromone-like markers during the coverage process. The ant-like STC algorithm runs in time O(N) and requires only O(1) memory. We present simulation results of the three STC algorithms, demonstrating their effectiveness in cases where the tool size is significantly smaller than the work-area characteristic dimension. A mobile robot represented by a point moving in the plane has to explore an unknown flat terrain with impassable obstacles. Both the terrain and the obstacles are modeled as arbitrary polygons. We consider two scenarios: the unlimited vision, when the robot situated at a point p of the terrain explores (sees) all points q of the terrain for which the segment pq belongs to the terrain, and the limited vision, when we require additionally that the distance between p and q is at most 1. All points of the terrain (except obstacles) have to be explored and the performance of an exploration algorithm, called its complexity, is measured by the length of the trajectory of the robot. For unlimited vision we show an exploration algorithm with complexity O(P+Dk), where P is the total perimeter of the terrain (including perimeters of obstacles), D is the diameter of the convex hull of the terrain, and k is the number of obstacles. We do not assume knowledge of these parameters. We also prove a matching lower bound showing that the above complexity is optimal, even if the terrain is known to the robot. For limited vision we show exploration algorithms with complexity O(P+A+Ak), where A is the area of the terrain (excluding obstacles). Our algorithms work either for arbitrary terrains (if one of the parameters A or k is known) or for c-fat terrains, where c is any constant (unknown to the robot) and no additional knowledge is assumed. (A terrain T with obstacles is c-fat if R r= The context of this work is the exploration of unknown polygonal environments with obstacles. Both the outer boundary and the boundaries of obstacles are piecewise linear. The boundaries can be nonconvex. The exploration problem can be motivated by the following application. Imagine that a robot has to explore the interior of a collapsed building, which has crumbled due to an earthquake, to search for human survivors. It is clearly impossible to have a knowledge of the building's interior geometry prior to the exploration. Thus, the robot must be able to see, with its onboard vision sensors, all points in the building's interior while following its exploration path. In this way, no potential survivors will be missed by the exploring robot. The exploratory path must clearly reflect the topology of the free space, and, therefore, such exploratory paths can be used to guide future robot excursions (such as would arise in our example from a rescue operation).