On minimizing the sum of sensor movements for barrier coverage of a line segment

A set of sensors establishes barrier coverage of a given line segment if every point of the segment is within the sensing range of a sensor. Given a line segment I, n mobile sensors in arbitrary initial positions on the line (not necessarily inside I) and the sensing ranges of the sensors, we are interested in finding final positions of sensors which establish a barrier coverage of I so that the sum of the distances traveled by all sensors from initial to final positions is minimized. It is shown that the problem is NP complete even to approximate up to constant factor when the sensors may have different sensing ranges. When the sensors have an identical sensing range we give several efficient algorithms to calculate the final destinations so that the sensors either establish a barrier coverage or maximize the coverage of the segment if complete coverage is not feasible while at the same time the sum of the distances traveled by all sensors is minimized. Some open problems are also mentioned.