American Mathematical Monthly
Randomized algorithms
Barrier coverage with wireless sensors
Proceedings of the 11th annual international conference on Mobile computing and networking
A First Course in Order Statistics (Classics in Applied Mathematics)
A First Course in Order Statistics (Classics in Applied Mathematics)
Movement-Assisted Connectivity Restoration in Wireless Sensor and Actor Networks
IEEE Transactions on Parallel and Distributed Systems
On Minimizing the Maximum Sensor Movement for Barrier Coverage of a Line Segment
ADHOC-NOW '09 Proceedings of the 8th International Conference on Ad-Hoc, Mobile and Wireless Networks
Optimal movement of mobile sensors for barrier coverage of a planar region
Theoretical Computer Science
On minimizing the sum ofensor movements for barrier coverage of a line segment
ADHOC-NOW'10 Proceedings of the 9th international conference on Ad-hoc, mobile and wireless networks
Algorithms on minimizing the maximum sensor movement for barrier coverage of a linear domain
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
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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 f(n) n of Ω(εf(n)ε--(1+ε)f(n)), ε 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 ε 0, O(n--1/2+ε). For larger sensor ranges (up to (1 -- ε) ln n/n, ε 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).