Expected sum and maximum of displacement of random sensors for coverage of a domain

Abstract

Assume that n sensors with identical range r = f(n)⁄2n, for some f(n) ≥ 1 for all n, are thrown randomly and independently with the uniform distribution in the unit interval [0, 1]. They are required to move to new positions so as to cover the entire unit interval in the sense that every point in the interval is within the range of a sensor. We obtain tradeoffs between the expected sum and maximum of displacements of the sensors and their range required to accomplish this task. In particular, when f(n) -- 1 the expected total displacement is shown to be Θ(√n). For senors with larger ranges we present two algorithms that prove the upper bound for the sum drops sharply as f(n) increases. The first of these holds for f(n) ≥ 6 and shows the total movement of the sensors is O(√ ln n/f(n)) while the second holds for 12 ≤ f(n) ≤ ln n -- 2 ln ln n and gives an upper bound of O(lnn⁄ f(n)ef(n)/2). Note that the second algorithm improves upon the first for f(n) > ln ln n -- ln ln ln n. Further we show a lower bound, for any 1 0.For the case of the expected maximum displacement of a sensor when f(n) = 1 our bounds are Ω(n--1/2) and for any e > 0, O(n--1/2+e). For larger sensor ranges (up to (1 -- e) ln n/n, e > 0) the expected maximum displacement is shown to be Θ(ln n/n).We also obtain similar sum and maximum displacement and range tradeoffs for area coverage for sensors thrown at random in a unit square. In this case, for the expected maximum displacement our bounds are tight and for the expected sum they are within a factor of √ln n. Finally, we investigate the related problem of the expected total and maximum displacement for perimeter coverage (whereby only the perimeter of the region need be covered) of a unit square. For example, when n sensors of radius > 2/n are thrown randomly and independently with the uniform distribution in the interior of a unit square, we can show the total expected displacement required to cover the perimeter is n/12 + o(n).