Reconfigurable Intelligent Surfaces (RISs) enable programmable control of the wireless propagation environment through adjustable reflecting elements. A fundamental challenge in RIS-assisted communications is the optimization of discrete phase shifts to maximize the received signal power. While recently developed algorithms can obtain optimal solutions with linear complexity, their computational cost may still be excessive in applications requiring large-scale simulations, real-time adaptation, or repeated optimization over numerous channel realizations. This paper proposes a low-complexity discrete phase-shift optimization algorithm based on multiple reference directions within the feasible optimality sector. By evaluating candidate phase configurations generated from uniformly distributed reference directions, the proposed method systematically improves upon existing approximation approaches that rely on a fixed set of reference directions. The number of reference directions serves as a design parameter that enables a controllable trade-off between computational complexity and optimality. Extensive simulations demonstrate that the proposed algorithm achieves substantial computational speedups while maintaining a small optimality gap relative to the optimal solution. The results further show that increasing the number of reference directions progressively reduces the optimality gap, providing flexible operating points that can be tailored to application-specific complexity and performance requirements.

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