On the Exact Asymptotic of Expected Sum of Displacement of Random Sensors for Covering a Unit Interval

In this note we essentially simplify the proof of the main result in one paper from leading computer science conference 25th ACM Symposium on Parallelism in Algorithms and Architectures (see [3].) We also present direct method and give exact asymptotic result. 1 Motivation and algorithm Consider n sensors placed randomly and independently with the uniform distribution in on the unit interval [0, 1]. The sensors have identical sensing range equal to 1 2n ; thus a sensor placed at location x in the unit interval can sense any point at distance at most 1 2n either to the left or right to x. We are interested in moving the sensors from their initial positions to new locations so as to ensure coverage of the unit interval i.e., every point in the unit interval is within the range of a sensor. What is a displacement of minimum cost that ensures coverage? Observe that the only way to attain the coverage is for the sensors to occupy the anchor location ti = i n − 1 2n , for i = 1, 2, . . . , n (see Algorithm 1). Algorithm 1 MV (n) . Require: n mobile sensors with identical sensing radius r = 1 2n placed uniformly and independently at random on the interval [0, 1]. Ensure: The final positions of the sensors are at the locations ( i n − 1 2n ) , 1 ≤ i ≤ n (so as to attain coverage of the interval [0, 1].) 1: sort the initial locations of sensors; the locations after sorting x1, x2, . . . xn, x1 ≤ x2 ≤ · · · ≤ xn. 2: for i = 1 to n do 3: move the sensor Si at position ( i n − 1 2n )