Pereverzev (1995) considered Tikhonov regularization combined with a modified form of a projection method for obtaining stable approximate solutions for ill-posed operator equations. He showed, under a certain a priori choice of the regularization parameter and a specific smoothness assumption on the solution, that the method yields the optimal order with less computational information, in the sense of complexity, than the projection method considered by Plato and Vainikko (1990). In this paper we apply a modified form of the Arcangeli's discrepancy principle for choosing the regularization parameter, and show that the conclusions of Pereverzev still hold. In fact, we do the analysis using a modified form of the generalized Arcangeli's method suggested by Schock (1984) under more flexible smoothness assumption on the solution, as has been done by George and Nair (1998), and derive the optimal result as a special case. Moreover, we compare the computational complexity of the present method with two tradit ional projection methods in the case of a priori parameter choice, and also discuss the computational complexity required to implement the suggested discrepancy principle.