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The problem of optimizing discrete phases in a reconfigurable intelligent surface (RIS) to maximize the received power at a user equipment is addressed. Necessary and sufficient conditions to achieve this maximization are given. These conditions are employed in an algorithm to achieve the maximization. New versions of the algorithm are given that are proven to achieve convergence in N or fewer steps whether the direct link is completely blocked or not, where N is the number of the RIS elements, whereas previously published results achieve this in KN or 2N number of steps where K is the number of discrete phases. Thus, for a discrete-phase RIS, the techniques presented in this paper achieve the optimum received power in the smallest number of steps published in the literature. In addition, in each of those N steps, the techniques presented in this paper determine only one or a small number of phase shifts with a simple elementwise update rule, which result in a substantial reduction of computation time, as compared to the algorithms in the literature. As a secondary result, we define the uniform polar quantization (UPQ) algorithm which is an intuitive quantization algorithm that can approximate the continuous solution with an approximation ratio of sinc2(1/K) and achieve low time-complexity, given perfect knowledge of the channel. 

To maximize the received power at a user equipment, the problem of optimizing a reconfigurable intelligent surface (RIS) with a limited phase range and nonuniform discrete phase shifts with adjustable gains is addressed. Necessary and sufficient conditions to achieve this maximization are given. These conditions are employed in two algorithms to achieve the global optimum in linear time. Depending on the phase range limitation, it is shown that the global optimality is achieved in NK or fewer and N(K + 1) or fewer steps, where N is the number of RIS elements and K is the number of discrete phase shifts which may be placed nonuniformly over the limited phase range. In addition, we define two quantization algorithms that we call nonuniform polar quantization (NPQ) algorithm and extended nonuniform polar quantization (ENPQ) algorithm, where the latter is a novel quantization algorithm for RISs with a significant phase range restriction. With NPQ, we provide a closed-form solution for the approximation ratio with which an arbitrary set of nonuniform discrete phase shifts can approximate the continuous solution. We also show that with a phase range limitation, equal separation among the nonuniform discrete phase shifts maximizes the normalized performance. Furthermore, for a larger RIS phase range limitation, we show that the gain of increasing K is only marginal, whereas, ON/OFF selection for the RIS elements can bring significant performance compared to the case when the RIS elements are strictly ON.