In this paper, a novel approximate link-state dissemination framework, called TROP, is proposed for shared backup path protection (SBPP) in Multi-Protocol Label Switching (MPLS) networks. While performing dynamic explicit survivable routing in a distributed environment, link-state dissemination may cause non-trivial signaling overhead in the process of exploring spare resource sharing among individual backup Label Switched Paths (LSPs). Several previously reported studies have tackled this problem by initiating a compromise between the amount of dissemination and the achievable extent of resource sharing. The paper first summarizes the previously reported schemes into a compact and general link-state dissemination framework by way of singular value decomposition (SVD). To improve the accuracy of the matrix reconstruction and to eliminate overestimation of the sharable spare capacity along each link, a novel SVD approach based on Min-Plus algebra (or called Tropical Semi-Rings) is intro...