A competitive online algorithm for exploring a solar map

In this paper, we study the problem of quickly building the 3D model of an outdoor environment from measurements obtained by a robot equipped with a solar panel. The robot knows the angle of the sun and the locations of the objects in the environment. It does not know, however, the height of the objects. For example, it might be possible to use satellite images to obtain locations of trees in a field but not their heights. In order to compute the height of an object, the robot must find the projection of the object´s highest point. This is where the shadow of the object ends. The robot can find it by tracing the shadow (moving parallel to the sun) until the measurement switches from shadow to sun or vice versa. The robot´s goal is to compute the height of every object as quickly as possible using only solar measurements. We formulate this as an online optimization problem. The optimal offline algorithm is given by the Traveling Salesman path of the transition points. The robot does not know these locations a priori. It must search for each of them. We present an algorithm with the property that for n objects, our distance traveled is guaranteed to be within a factor O(log n) of this optimal offline tour. In addition to analytical proofs, we demonstrate the algorithm with simulations using solar data collected from field experiments, and examine its performance for uniformly distributed sites.