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This paper is concerned with online navigation of a size D mobile robot in an unknown planar environment. A formal means for assessing algorithms for online tasks is competitiveness. For the navigation task, competitiveness measures the algorithm's path length relative to the optimal offline path length. While competitiveness usually means constant relative performance, it is measured in this paper in terms of a quadratic relationship between online performance and optimal offline solution. An online navigation algorithm for a size D robot called CBUG is described. The competitiveness of CBUG is analyzed and shown to be quadratic in the length of the shortest offline path. Moreover, it is shown that, in general, quadratic competitiveness is the best achievable performance over all online navigation algorithms. Thus, up to constants, CBUG achieves optimal competitiveness. The algorithm is improved with some practical speedups, and the usefulness of its competitiveness in terms of path stability is illustrated in office-like environments.