We tackle the problem of exploring an unknown environment using a range sensor, where most of the uncertainty is due to the presence of opaque objects of unknown and arbitrary geometry, topology and configuration. Information-gathering control aims to minimize the residual uncertainty, resulting in an optimal control problem that is essentially intractable. We propose a surrogate measure of uncertainty that is efficient to compute and results in guaranteed performance bounds. The resulting algorithm is proven to complete exploration within a prescribed accuracy in a finite number of steps, depending in the complexity of the environment.