It is observed that the Singular Value Decomposition (SVD) transformation based on min-plus algebra (or called Tropical Semi-Rings) leads to a very good characteristic in zero underestimating the reconstructed matrix. This paper introduces a novel distributed control framework for shared protection in optical networks with reduced routing information based on the Tropical Semi-Rings technique, called Sharing with Reduced Information with Tropical Semi-Rings (SRI-TROP). The design of the proposed framework aims to initiate a compromise between the amount of link-state dissemination and the performance impairment due to the incompleteness of routing information, such that the precision in the link-state matrix reconstruction can efficiently map to the reduction in blocking probability. Based on the framework, a series of novel schemes are proposed, which are verified and compared with the reported counterparts in a simulation. The simulation results show that the performance in terms of the precision in the reconstructed link-state and the resultant blocking probability can be significantly improved.