In this paper, we present an Extended Kalman Filter (EKF)-based estimator for simultaneous localization and mapping (SLAM) with processing requirements that are linear in the number of features in the map. The proposed algorithm is based on three key ideas. Firstly, by introducing the Global-Map Postponement method, approximations necessary for ensuring linear computational complexity are delayed over many time steps. Then by employing the Power Method, only the most informative of the Kalman vectors, generated during the postponement phase, are retained for updating the covariance matrix. This in effect minimizes the information loss during each approximation epoch. Finally, linear-complexity, rank-2 updates, which minimize the trace of the covariance matrix, are applied to increase the speed of convergence of the estimator. In addition to being consistent, the resulting estimator has processing requirements that can be adjusted to the availability of computational resources. Simulation results are presented that demonstrate the accuracy of the proposed algorithm (Power-SLAM) when compared to the quadratic computational cost standard EKF-based SLAM, and two linear-complexity competing alternatives.