On the displacement for covering a d-dimensional cube with randomly placed sensors

Abstract

Consider n sensors placed randomly and independently with the uniform distribution in a d-dimensional unit cube (d ≥ 2). 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 d-dimensional unit cube is completely covered, i.e., every point in the d-dimensional cube is within the range of a sensor. If the ith sensor is displaced a distance di , what is a displacement of minimum cost that ensures coverage? As cost measure for the displacement of the team of sensors we consider the a-total movement defined as the sum Ma:=∑i=1ndia, for some constant a > 0. We assume that r and n are chosen so as to allow full coverage of the d-dimensional unit cube. Motivation for using this cost metric arises from the fact that there might be a terrain affecting the movement of the sensors from their initial to their final destinations (e.g., a terrain surface which is either obstructing or speeding the movement). Therefore the a-total displacement is not more general but can be a more realistic metric than the one previously considered for a=1.The main contribution of this paper is to show the existence of a tradeoff in the d-dimensional cube between sensing radius and a-total movement. Omitting low order terms, the main results can be summarized as follows for the case of the d-dimensional unit cube. 1.If the d-dimensional cube sensing radius is 12n1/d and n=md, for some m∈N, then we present an algorithm that uses Θ(n1-a2d) total expected movement, when a ≥ 1 and O(n1-a2d) total expected movement, when a ∈ (0, 1) (see Algorithm 2, Theorem 5 and Theorem 6).2.If the the d-dimensional cube sensing radius is greater than 33/d(31/d-1)(31/d-1)12n1/d and n is a natural number then the total expected movement is O(n1-a2d(lnnn)a2d) (see Algorithm 3 and Theorem 8). This sharp decline from O(n1-a2d) to O(n1-a2d(lnnn)a2d) in the a-total movement of the sensors to attain complete coverage of the d-dimensional unit cube indicates the presence of an interesting threshold on the sensing radius in a d-dimensional unit cube as it increases from 12n1/d to 33/d(31/d-1)(31/d-1)12n1/d. In addition, we simulate Algorithm 2 and discuss the results of our simulations