Witnessing　broad　energy-critical　applications　of　barrier　coverage　in　mobile　and　wireless　sensor　networks,　emerging　practical　applications　have　recently　brought　a　new　barrier　coverage　model　which　uses　sink-based　mobile　sensors　for　covering　a　given　barrier　with　the　aim　of　prolonging　the　lifespan　of　the　coverage.　In　the　model,　a　set　of　sink　stations　were　distributed　on　the　plane　in　which　each　sink　can　emit　mobile　sensors　with　an　identical　radius.　The　task　is　to　cover　a　given　line　barrier　with　the　emitted　mobile　sensors,　aiming　to　minimize　the　maximum　movement　of　the　sensors　so　as　to　prolong　the　shortest　lifespan　among　the　sensors　in　coverage.　In　this　paper,　we　first　devise　an　algorithm　for　optimally　solving　the　problem　based　on　the　properties　of　the　structures　called　movement　parity　and　tangent　equilibrium　points　between　the　sinks.　Then　based　on　a　more　sophisticated　geometric　property　of　optimum　solutions,　we　improve　the　runtime　to　a　linear　runtime　O(k)　which　attains　the　possibly　optimum　runtime　of　the　problem　for　k　being　the　number　of　sinks.　At　last,　numerical　experiments　are　carried　out　to　demonstrate　the　practical　performance　gain　of　our　algorithms　against　baselines　in　literature.