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In low earth orbit (LEO) satellite networks, the issue of inter-satellite link (ISL)-based routing is gathering more and more interest. Most of the existing routing algorithms for LEO systems rely on the periodicity and predictability of satellites' movement, and cannot adapt to unpredictable topological changes, therefore lacking survivability. A destruction-resistant routing algorithm for LEO satellite networks is proposed in this paper, which can automatically avoid invalid ISLs and reroute with the cost as small as possible. Comparing the simulation results of this algorithm with the open shortest path first (OSPF) protocol on signaling overhead, scalability, and routing efficiency, we find that in addition to the enhanced survivability, its performance excels OSPF obviously, therefore it is suitable to be used in LEO systems.