We focus on the Regenerator sharing, adaptive Modulation, Routing, and Spectrum assignment (RMRS) problem in a shared backup path protection based elastic optical network. Three regenerator placement strategies and the mechanisms of how to efficiently use/share the placed regenerators are proposed. An efficient heuristic algorithm is also developed for the RMRS problem to maximize regenerator and spectrum sharing among protection lightpaths with the joint consideration of modulation format and spectrum conversion capabilities of each traversed regenerator along a lightpath. Based on the performance study in terms of bandwidth blocking probability (BBP), we find that the proposed strategies and heuristic algorithm are efficient in signal regenerator utilization to significantly reduce BBP. It is also found that in addition to signal regeneration, the capabilities of modulation format conversion and spectrum conversion provided by each regenerator are helpful in reducing BBP, and as expected, the capability of spectrum conversion plays a more important role than the modulation format conversion for performance improvement. Finally, it is also interesting to find that there exists a saturation trend between the improvement of BBP performance and the increase in the number of regenerators. A limited number of signal regenerators sparsely placed in a network is sufficient to achieve a good BBP performance.