“Green” Barrier Coverage with Mobile Sensors

Abstract

Mobile sensors are located on a barrier represented by a line segment. Each sensor has a single energy source that can be used for both moving and sensing. A sensor consumes energy in movement in proportion to distance traveled, and it expends energy per time unit for sensing in direct proportion to its radius raised to a constant exponent. We address the problem of energy efficient coverage. The input consists of the initial locations of the sensors and a coverage time requirement t. A feasible solution consists of an assignment of destinations and coverage radii to all sensors such that the barrier is covered. We consider two variants of the problem that are distinguished by whether the radii are given as part of the input. In the fixed radii case, we are also given a radii vector ρ, and the radii assignment r must satisfy r i ∈ {0, ρ i }, for every i, while in the variable radii case the radii assignment is unrestricted. The goal is to cover the barrier for t time in an energy efficient manner. More specifically, we consider two objective functions. In the first the goal is to minimize the sum of the energy spent by all sensors and in the second the goal is to minimize the maximum energy used by any sensor.We present fully polynomial time approximation schemes for the problem of minimizing the energy sum with variable radii and for the problem of minimizing the maximum energy with variable radii. We also show that the latter can be approximated within any additive constant ε > 0. We present a 2-approximation algorithm for the problem of minimizing the maximum energy with fixed radii which also is shown to be strongly NP-hard. We show that the problem of minimizing the energy sum with fixed radii cannot be approximated within a factor of O(n c ), for any constant c, unless P=NP. Additional results are given for three special cases: (i) sensors are stationary, (ii) free movement, and (iii) uniform fixed radii.

Approximation Algorithms for Minimizing Maximum Sensor Movement for Line Barrier Coverage in the Plane

“…The authors presented parametric search algorithms for the cases when the sensors have a predetermined order in the barrier or when sensors are initially located at barrier endpoints. On the other hand, the same authors present two FPTAS respectively for minimizing sumed and maximum energy consumption when the radii of the sensors can be adjusted [2]. When the sensing radii is fixed, i.e.…”

“…Proof: After the procession of S i , any S i ′ ∈ S \ S i must have all its sub-edges appear between the two edges e jh ∈ S i and e jh+1 ∈ S i for some h. Then, after S i is processed, Case (2) can not hold for any edge pair within S i in any other latter iterations. Therefore, the algorithm guarantees that one set contains no sub-edges of Case (2) at one iteration, and hence after n iterations, sub-edges of Case (2) are eliminated.…”

FPTAS for barrier covering problem with equal circles in 2D

“…Monitoring can be carried out using both stationary [7,8,13,16,17,18] and mobile sensors [2,3,4,5,6,9,10,11,13,14,15]. The problem of energy efficient covering a segment with stationary sensors with adjustable circular sensing areas initially located on a segment is considered in [13,17].…”

“…In the literature, two main classes of barrier monitoring problems with mobile devices are considered. The first class is a one-dimensional (1D) problem, when the sensors are initially on the line containing the segment to be covered [2,3,4,6,10,13,15]. In the two-dimensional (2D) case, the problem is considered when the sensors are arbitrarily located on the plane [5,9,11].…”

“…In [3], the barrier is covered with circles, initially located on the line containing the barrier. This takes into account that the sensors consume energy on both movement and monitoring, and two problems are considered: (a) when the radii of the sensors are fixed, (b) when the radii of the sensors can be adjusted.…”