On the displacement for covering a square with randomly placed sensors

Abstract

Consider n sensors placed randomly and independently with the uniform distribution in a unit square. The sensors have identical sensing range equal to r, for some r > 0. We are interested in moving the sensors from their initial positions to new positions so as to ensure that the unit square is completely covered, i.e., every point in the square is within the range of a sensor. If the i-th sensor is displaced a distance di, what is a displacement of minimum cost? As cost measure for the displacement of the team of sensors we consider the a-total movement defined as the sum Ma:= Σni=1 dai, for some constant a > 0. We assume that r and n are chosen so as to allow full coverage of the square and 0