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In this paper, a novel approximate link-state dissemination framework, called TROP, is proposed for shared backup path protection (SBPP) in multiprotocol label switching (MPLS) networks. While performing dynamic explicit survivable routing in a distributed environment, link-state dissemination may cause a nontrivial 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 the overestimation of the sharable spare capacity along each link, a novel SVD approach based on the min-plus algebra (also called tropical semirings) is introduced. Simulation results show that the proposed schemes can achieve a lower blocking probability than that by all the other counterpart schemes while taking the same complexity of link-state dissemination. This great advantage is gained at the expense of a longer computation time for solving a linear program (LP) in each dissemination cycle at the core nodes. We also consider the stale link-state phenomena that may cause imprecision in the routing information at the ingress nodes due to the delay in the periodic/event-driven link-state update message advertisement.