Optimizing movement in convex and non-convex path-networks to establish connectivity

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We solve a movement problem in which there are n sensors in path network in plane, where any sensor communicates only with its two immediate neighbors and only at a given maximum communication distance λ. Initially, some of the inter-sensor distances may be more than λ. We need to move sensors so that each sensor is in the communication range of its two neighbors, keeping the path topology intact. The problem is to minimize the maximum movement of any sensor. We present an O(n3)-time algorithm to compute the new positions of sensors to establish transmission connectivity, called λ-connectivity in the paper, in a convex path-network which minimizes the maximum movement among the sensors. We also generalize our algorithm for ring, non-convex path, tethered and heterogeneous networks.