Searching with Advice: Robot Fence-Jumping

抄録

We study a new problem concerning search in the plane involving a robot and an immobile treasure, initially placed at distance 1 from each other. The length β of an arc (a fence) which is placed within the perimeter of the disk centered at the initial position of the robot, as well as the promise that the treasure is outside the fence, is given as part of the input. The goal is to devise movement trajectories so that the robot locates the treasure in minimum time. Notably, although the presence of the fence limits searching uncertainty, the location of the fence is unknown, and the worst case analysis is determined adversarially. Nevertheless, the robot has the ability to make cross-cuts by moving in the interior of the disk. In particular, the robot can attempt a number of chord-jump moves if it happens to be within the fence or if an endpoint of the fence is discovered. The optimal solution to our question can be obtained as a solution to a complicated optimization problem, which involves trigonometric functions, and trigonometric equations that do not admit closed form solutions. For the 1-Jump Algorithm, we fully describe the optimal trajectory, and provide an analysis of the associated cost as a function of β. Our analysis indicates that the optimal k-Jump Algorithm requires that the robot has enough memory and computation power to compute the optimal chord-jumps. Motivated by this, we give an abstract performance analysis for every k-Jump Algorithm. Subsequently, we present a highly efficient Halving Heuristic k-Jump Algorithm that can effectively approximate the optimal k-Jump Algorithm, with very limited memory and computation requirements. Further, we explore randomized algorithms and analyze their expected worst-case performance.